{"id":3129,"date":"2018-12-11T13:51:08","date_gmt":"2018-12-11T18:51:08","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/polynomial-equations\/"},"modified":"2018-12-11T13:51:08","modified_gmt":"2018-12-11T18:51:08","slug":"polynomial-equations","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/polynomial-equations\/","title":{"raw":"Polynomial Equations","rendered":"Polynomial Equations"},"content":{"raw":"\n[latexpage]<div class=\"textbox textbox--learning-objectives\"><h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>By the end of this section, you will be able to: <ul><li>Use the Zero Product Property<\/li><li>Solve quadratic equations by factoring<\/li><li>Solve equations with polynomial functions<\/li><li>Solve applications modeled by polynomial equations<\/li><\/ul><\/div><div data-type=\"note\" id=\"fs-id1167836300333\" class=\"be-prepared\"><p id=\"fs-id1167833272508\">Before you get started, take this readiness quiz.<\/p><ol id=\"fs-id1167836532183\" type=\"1\"><li>Solve: \\(5y-3=0.\\)<div data-type=\"newline\"><br><\/div> If you missed this problem, review <a href=\"\/contents\/9f100e8f-6d15-4cae-bc22-c306e9d7d55c#fs-id1167836432956\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><li>Factor completely: \\({n}^{3}-9{n}^{2}-22n.\\)<div data-type=\"newline\"><br><\/div> If you missed this problem, review <a href=\"\/contents\/5e548626-8f0f-496d-ab87-4f0358ca2fd3#fs-id1167836521479\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><li>If \\(f\\left(x\\right)=8x-16,\\) find \\(f\\left(3\\right)\\) and solve \\(f\\left(x\\right)=0.\\)<div data-type=\"newline\"><br><\/div> If you missed this problem, review <a href=\"\/contents\/da9d6ce0-a078-4ca2-97af-8cb374f040f5#fs-id1167836731462\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><\/ol><\/div><p id=\"fs-id1167833386923\">We have spent considerable time learning how to factor polynomials. We will now look at polynomial equations and solve them using factoring, if possible.<\/p><p id=\"fs-id1167836533739\">A <span data-type=\"term\">polynomial equation<\/span> is an equation that contains a polynomial expression. The <span data-type=\"term\">degree of the polynomial equation<\/span> is the degree of the polynomial.<\/p><div data-type=\"note\" id=\"fs-id1167836493148\"><div data-type=\"title\">Polynomial Equation<\/div><p id=\"fs-id1167836602701\">A <strong data-effect=\"bold\">polynomial equation<\/strong> is an equation that contains a polynomial expression.<\/p><p id=\"fs-id1167836363077\">The <strong data-effect=\"bold\">degree of the polynomial equation<\/strong> is the degree of the polynomial.<\/p><\/div><p id=\"fs-id1167836434319\">We have already solved polynomial equations of <span data-type=\"term\" class=\"no-emphasis\">degree one<\/span>. Polynomial equations of degree one are linear equations are of the form \\(ax+b=c.\\)<\/p><p id=\"fs-id1167836367371\">We are now going to solve polynomial equations of <span data-type=\"term\" class=\"no-emphasis\">degree two<\/span>. A polynomial equation of degree two is called a <span data-type=\"term\">quadratic equation<\/span>. Listed below are some examples of quadratic equations:<\/p><div data-type=\"equation\" class=\"unnumbered\" data-label=\"\">\\({x}^{2}+5x+6=0\\phantom{\\rule{3em}{0ex}}3{y}^{2}+4y=10\\phantom{\\rule{3em}{0ex}}64{u}^{2}-81=0\\phantom{\\rule{3em}{0ex}}n\\left(n+1\\right)=42\\)<\/div><p id=\"fs-id1167836287188\">The last equation doesn\u2019t appear to have the variable squared, but when we simplify the expression on the left we will get \\({n}^{2}+n.\\)<\/p><p id=\"fs-id1167833019237\">The general form of a quadratic equation is \\(a{x}^{2}+bx+c=0,\\) with \\(a\\ne 0.\\) (If \\(a=0,\\) then \\(0\u00b7{x}^{2}=0\\) and we are left with no quadratic term.)<\/p><div data-type=\"note\"><div data-type=\"title\">Quadratic Equation<\/div><p id=\"fs-id1167829904790\">An equation of the form \\(a{x}^{2}+bx+c=0\\) is called a quadratic equation.<\/p><div data-type=\"equation\" id=\"fs-id1167836310424\" class=\"unnumbered\" data-label=\"\">\\(a,b,\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}c\\phantom{\\rule{0.2em}{0ex}}\\text{are real numbers and}\\phantom{\\rule{0.2em}{0ex}}a\\ne 0\\)<\/div><\/div><p id=\"fs-id1167836611159\">To solve quadratic equations we need methods different from the ones we used in solving linear equations. We will look at one method here and then several others in a later chapter.<\/p><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167829828416\"><h3 data-type=\"title\">Use the Zero Product Property<\/h3><p id=\"fs-id1167836516676\">We will first solve some quadratic equations by using the <span data-type=\"term\">Zero Product Property<\/span>. The Zero Product Property says that if the product of two quantities is zero, then at least one of the quantities is zero. The only way to get a product equal to zero is to multiply by zero itself.<\/p><div data-type=\"note\" id=\"fs-id1167833057729\"><div data-type=\"title\">Zero Product Property<\/div><p id=\"fs-id1167833020317\">If \\(a\u00b7b=0,\\) then either \\(a=0\\) or \\(b=0\\) or both.<\/p><\/div><p id=\"fs-id1167836553403\">We will now use the Zero Product Property, to solve a <span data-type=\"term\" class=\"no-emphasis\">quadratic equation<\/span>.<\/p><div data-type=\"example\" id=\"fs-id1167836619617\" class=\"textbox textbox--examples\"><div data-type=\"title\">How to Solve a Quadratic Equation Using the Zero Product Property<\/div><div data-type=\"exercise\" id=\"fs-id1167833056584\"><div data-type=\"problem\" id=\"fs-id1167836727844\"><p id=\"fs-id1167836333497\">Solve: \\(\\left(5n-2\\right)\\left(6n-1\\right)=0.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829936994\"><span data-type=\"media\" id=\"fs-id1167836707067\" data-alt=\"The equation is open parentheses 5n minus 2 close parentheses open parentheses 6n minus 1 close parentheses equals 0. The product equals zero, so at least one factor must equal zero. Step 1 is set each factor equal to zero. So, 5n minus 2 equals 0 and 6n minus 1 equals 0.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_05_001a_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The equation is open parentheses 5n minus 2 close parentheses open parentheses 6n minus 1 close parentheses equals 0. The product equals zero, so at least one factor must equal zero. Step 1 is set each factor equal to zero. So, 5n minus 2 equals 0 and 6n minus 1 equals 0.\"><\/span><span data-type=\"media\" id=\"fs-id1167836393361\" data-alt=\"Step 2 is to solve the linear equations. So, we get n equal to 2 by 5 and n equal to 1 by 6.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_05_001b_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2 is to solve the linear equations. So, we get n equal to 2 by 5 and n equal to 1 by 6.\"><\/span><span data-type=\"media\" id=\"fs-id1167836602556\" data-alt=\"Step 3 is to check by substituting each solution separately into the original equation.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_05_001c_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3 is to check by substituting each solution separately into the original equation.\"><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167824734415\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836625622\"><div data-type=\"problem\" id=\"fs-id1167836521982\"><p id=\"fs-id1167829747240\">Solve: \\(\\left(3m-2\\right)\\left(2m+1\\right)=0.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836340005\"><p id=\"fs-id1167829712286\">\\(m=\\frac{2}{3},m=-\\frac{1}{2}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836516418\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836312584\"><div data-type=\"problem\" id=\"fs-id1167829621115\"><p>Solve: \\(\\left(4p+3\\right)\\left(4p-3\\right)=0.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836285974\"><p id=\"fs-id1167836379067\">\\(p=-\\frac{3}{4},p=\\frac{3}{4}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836408616\" class=\"howto\"><div data-type=\"title\">Use the Zero Product Property.<\/div><ol id=\"fs-id1167824734367\" type=\"1\" class=\"stepwise\"><li>Set each factor equal to zero.<\/li><li>Solve the linear equations.<\/li><li>Check.<\/li><\/ol><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167836294278\"><h3 data-type=\"title\">Solve Quadratic Equations by Factoring<\/h3><p id=\"fs-id1167836554592\">The Zero Product Property works very nicely to solve quadratic equations. The quadratic equation must be factored, with zero isolated on one side. So we be sure to start with the quadratic equation in <span data-type=\"term\" class=\"no-emphasis\">standard form<\/span>, \\(a{x}^{2}+bx+c=0.\\) Then we factor the expression on the left.<\/p><div data-type=\"example\" class=\"textbox textbox--examples\"><div data-type=\"title\">How to Solve a Quadratic Equation by Factoring<\/div><div data-type=\"exercise\" id=\"fs-id1167829719011\"><div data-type=\"problem\" id=\"fs-id1167833047277\"><p id=\"fs-id1167836313413\">Solve: \\(2{y}^{2}=13y+45.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167825835938\"><span data-type=\"media\" id=\"fs-id1167829833272\" data-alt=\"The equation is 2 y squared equals 13y plus 45. Step 1 is to write it in standard form a x squared plus bx plus c. So we have 2 y squared minus 13y minus 45 equals 0.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_05_002a_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The equation is 2 y squared equals 13y plus 45. Step 1 is to write it in standard form a x squared plus bx plus c. So we have 2 y squared minus 13y minus 45 equals 0.\"><\/span><span data-type=\"media\" data-alt=\"Step 2 is to factor the quadratic expression. So we have 2y plus 5, y minus 9 equals 0.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_05_002b_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2 is to factor the quadratic expression. So we have 2y plus 5, y minus 9 equals 0.\"><\/span><span data-type=\"media\" id=\"fs-id1167833022106\" data-alt=\"Step 3 is to use the zero product property. Setting each factor equal to zero, we have two linear equations: 2y plus 5 equals 0 and y minus 9 equals 0.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_05_002c_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3 is to use the zero product property. Setting each factor equal to zero, we have two linear equations: 2y plus 5 equals 0 and y minus 9 equals 0.\"><\/span><span data-type=\"media\" id=\"fs-id1167836635419\" data-alt=\"Step 4 is to solve the linear equations. We get, y equals minus 5 by 2 and y equals 9.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_05_002d_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 4 is to solve the linear equations. We get, y equals minus 5 by 2 and y equals 9.\"><\/span><span data-type=\"media\" data-alt=\"Step 5 is to check by substituting each solution separately into the original equation\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_05_002e_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 5 is to check by substituting each solution separately into the original equation\"><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167829936700\" class=\"try\"><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167836504112\"><p>Solve: \\(3{c}^{2}=10c-8.\\)<\/p><\/div><div data-type=\"solution\"><p id=\"fs-id1167836415111\">\\(c=2,c=\\frac{4}{3}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836323046\" class=\"try\"><div data-type=\"exercise\"><div data-type=\"problem\"><p>Solve: \\(2{d}^{2}-5d=3.\\)<\/p><\/div><div data-type=\"solution\"><p>\\(d=3,d=-\\frac{1}{2}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836515772\" class=\"howto\"><div data-type=\"title\">Solve a quadratic equation by factoring.<\/div><ol type=\"1\" class=\"stepwise\"><li>Write the quadratic equation in standard form, \\(a{x}^{2}+bx+c=0.\\)<\/li><li>Factor the quadratic expression.<\/li><li>Use the Zero Product Property.<\/li><li>Solve the linear equations.<\/li><li>Check. Substitute each solution separately into the original equation.<\/li><\/ol><\/div><p id=\"fs-id1167836386785\">Before we factor, we must make sure the <span data-type=\"term\" class=\"no-emphasis\">quadratic equation<\/span> is in <span data-type=\"term\" class=\"no-emphasis\">standard form<\/span>.<\/p><p>Solving quadratic equations by factoring will make use of all the factoring techniques you have learned in this chapter! Do you recognize the special product pattern in the next example?<\/p><div data-type=\"example\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167833245837\"><div data-type=\"problem\" id=\"fs-id1167836614984\"><p id=\"fs-id1167836538194\">Solve: \\(169{q}^{2}=49.\\)<\/p><\/div><div data-type=\"solution\"><p>\\(\\begin{array}{cccccc}&amp; &amp; &amp; &amp; &amp; \\phantom{\\rule{5.15em}{0ex}}169{x}^{2}=49\\hfill \\\\ \\text{Write the quadratic equation in standard form.}\\hfill &amp; &amp; &amp; &amp; &amp; \\phantom{\\rule{3em}{0ex}}169{x}^{2}-49=0\\hfill \\\\ \\text{Factor. It is a difference of squares.}\\hfill &amp; &amp; &amp; &amp; &amp; \\left(13x-7\\right)\\left(13x+7\\right)=0\\hfill \\\\ \\begin{array}{c}\\text{Use the Zero Product Property to set each factor to}\\phantom{\\rule{0.2em}{0ex}}0.\\hfill \\\\ \\text{Solve each equation.}\\hfill \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\end{array}\\hfill &amp; &amp; &amp; &amp; &amp; \\begin{array}{ccccccccc}\\hfill 13x-7&amp; =\\hfill &amp; 0\\hfill &amp; &amp; &amp; &amp; \\hfill 13x+7&amp; =\\hfill &amp; 0\\hfill \\\\ \\hfill 13x&amp; =\\hfill &amp; 7\\hfill &amp; &amp; &amp; &amp; \\hfill 13x&amp; =\\hfill &amp; -7\\hfill \\\\ \\hfill x&amp; =\\hfill &amp; \\frac{7}{13}\\hfill &amp; &amp; &amp; &amp; \\hfill x&amp; =\\hfill &amp; -\\frac{7}{13}\\hfill \\end{array}\\hfill \\end{array}\\)<\/p><p id=\"fs-id1167833239994\">Check:<\/p><p id=\"fs-id1167836515959\">We leave the check up to you.<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167833018774\" class=\"try\"><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167836493372\"><p>Solve: \\(25{p}^{2}=49.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836296549\"><p id=\"fs-id1167833009374\">\\(p=\\frac{7}{5},\\text{p}=-\\frac{7}{5}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836352178\" class=\"try\"><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167836665581\"><p id=\"fs-id1167833338953\">Solve: \\(36{x}^{2}=121.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836300042\"><p>\\(x=\\frac{11}{6},x=-\\frac{11}{6}\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1167833023151\">In the next example, the left side of the equation is factored, but the right side is not zero. In order to use the <span data-type=\"term\" class=\"no-emphasis\">Zero Product Property<\/span>, one side of the equation must be zero. We\u2019ll multiply the factors and then write the equation in standard form.<\/p><div data-type=\"example\" id=\"fs-id1167829580432\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167836704241\"><div data-type=\"problem\"><p id=\"fs-id1167833158807\">Solve: \\(\\left(3x-8\\right)\\left(x-1\\right)=3x.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836544129\"><p id=\"fs-id1167836524167\">\\(\\begin{array}{cccccc}&amp; &amp; &amp; &amp; &amp; \\phantom{\\rule{0.3em}{0ex}}\\left(3x-8\\right)\\left(x-1\\right)=3x\\hfill \\\\ \\text{Multiply the binomials.}\\hfill &amp; &amp; &amp; &amp; &amp; \\phantom{\\rule{0.6em}{0ex}}3{x}^{2}-11x+8=3x\\hfill \\\\ \\text{Write the quadratic equation in standard form.}\\hfill &amp; &amp; &amp; &amp; &amp; \\phantom{\\rule{0.6em}{0ex}}3{x}^{2}-14x+8=0\\hfill \\\\ \\text{Factor the trinomial.}\\hfill &amp; &amp; &amp; &amp; &amp; \\phantom{\\rule{0.25em}{0ex}}\\left(3x-2\\right)\\left(x-4\\right)=0\\hfill \\\\ \\begin{array}{c}\\text{Use the Zero Product Property to set each factor to 0.}\\hfill \\\\ \\text{Solve each equation.}\\hfill \\end{array}\\hfill &amp; &amp; &amp; &amp; &amp; \\begin{array}{ccccccccc}\\hfill 3x-2&amp; =\\hfill &amp; 0\\hfill &amp; &amp; &amp; &amp; \\hfill x-4&amp; =\\hfill &amp; 0\\hfill \\\\ \\hfill 3x&amp; =\\hfill &amp; 2\\hfill &amp; &amp; &amp; &amp; \\hfill x&amp; =\\hfill &amp; 4\\hfill \\end{array}\\hfill \\\\ &amp; &amp; &amp; &amp; &amp; \\phantom{\\rule{2.7em}{0ex}}x=\\frac{2}{3}\\hfill \\\\ \\text{Check your answers.}\\hfill &amp; &amp; &amp; &amp; &amp; \\text{The check is left to you.}\\hfill \\end{array}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836309718\"><div data-type=\"problem\" id=\"fs-id1167836560393\"><p id=\"fs-id1167836520033\">Solve: \\(\\left(2m+1\\right)\\left(m+3\\right)=12m.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829695518\"><p id=\"fs-id1167836556797\">\\(m=1,m=\\frac{3}{2}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836333805\"><div data-type=\"problem\" id=\"fs-id1167836530331\"><p id=\"fs-id1167836704806\">Solve: \\(\\left(k+1\\right)\\left(k-1\\right)=8.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836558294\"><p>\\(k=3,k=-3\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1167829594693\">In the next example, when we factor the quadratic equation we will get three factors. However the first factor is a constant. We know that factor cannot equal 0.<\/p><div data-type=\"example\" id=\"fs-id1167836691703\" class=\"textbox textbox--examples\"><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167829905872\"><p id=\"fs-id1167836487460\">Solve: \\(3{x}^{2}=12x+63.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836289776\"><p id=\"fs-id1167836628837\">\\(\\begin{array}{cccccc}&amp; &amp; &amp; &amp; &amp; \\phantom{\\rule{5.5em}{0ex}}3{x}^{2}=12x+63\\hfill \\\\ \\text{Write the quadratic equation in standard form.}\\hfill &amp; &amp; &amp; &amp; &amp; \\phantom{\\rule{0.7em}{0ex}}3{x}^{2}-12x-63=0\\hfill \\\\ \\text{Factor the greatest common factor first.}\\hfill &amp; &amp; &amp; &amp; &amp; \\phantom{\\rule{0.7em}{0ex}}3\\left({x}^{2}-4x-21\\right)=0\\hfill \\\\ \\text{Factor the trinomial.}\\hfill &amp; &amp; &amp; &amp; &amp; \\phantom{\\rule{0.8em}{0ex}}3\\left(x-7\\right)\\left(x+3\\right)=0\\hfill \\\\ \\begin{array}{c}\\text{Use the Zero Product Property to set each factor to 0.}\\hfill \\\\ \\text{Solve each equation.}\\hfill \\end{array}\\hfill &amp; &amp; &amp; &amp; &amp; \\begin{array}{ccccccccccccc}3\\ne 0\\hfill &amp; &amp; &amp; &amp; \\hfill x-7&amp; =\\hfill &amp; 0\\hfill &amp; &amp; &amp; &amp; \\hfill x+3&amp; =\\hfill &amp; 0\\hfill \\\\ 3\\ne 0\\hfill &amp; &amp; &amp; &amp; \\hfill x&amp; =\\hfill &amp; 7\\hfill &amp; &amp; &amp; &amp; \\hfill x&amp; =\\hfill &amp; -3\\hfill \\end{array}\\hfill \\\\ \\text{Check your answers.}\\hfill &amp; &amp; &amp; &amp; &amp; \\text{The check is left to you.}\\hfill \\end{array}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836329405\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836376734\"><div data-type=\"problem\" id=\"fs-id1167836288543\"><p id=\"fs-id1167836689175\">Solve: \\(18{a}^{2}-30=-33a.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833310107\"><p>\\(a=-\\frac{5}{2},a=\\frac{2}{3}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836553150\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836333546\"><div data-type=\"problem\" id=\"fs-id1167829694665\"><p id=\"fs-id1167836390484\">Solve: \\(123b=-6-60{b}^{2}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836553353\"><p id=\"fs-id1167836630557\">\\(b=-2,b=-\\frac{1}{20}\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1167829694662\">The <span data-type=\"term\" class=\"no-emphasis\">Zero Product Property<\/span> also applies to the product of three or more factors. If the product is zero, at least one of the factors must be zero. We can solve some equations of degree greater than two by using the Zero Product Property, just like we solved quadratic equations.<\/p><div data-type=\"example\" id=\"fs-id1167829644842\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167836456230\"><div data-type=\"problem\" id=\"fs-id1167836514629\"><p id=\"fs-id1167836700375\">Solve: \\(9{m}^{3}+100m=60{m}^{2}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836518098\"><p id=\"fs-id1167836704559\">\\(\\begin{array}{cccccc}&amp; &amp; &amp; &amp; &amp; \\phantom{\\rule{3.7em}{0ex}}9{m}^{3}+100m=60{m}^{2}\\hfill \\\\ \\text{Bring all the terms to one side so that the other side is zero.}\\hfill &amp; &amp; &amp; &amp; &amp; \\phantom{\\rule{0.35em}{0ex}}9{m}^{3}-60{m}^{2}+100m=0\\hfill \\\\ \\text{Factor the greatest common factor first.}\\hfill &amp; &amp; &amp; &amp; &amp; \\phantom{\\rule{0.35em}{0ex}}m\\left(9{m}^{2}-60m+100\\right)=0\\hfill \\\\ \\text{Factor the trinomial.}\\hfill &amp; &amp; &amp; &amp; &amp; m\\left(3m-10\\right)\\left(3m-10\\right)=0\\hfill \\\\ \\begin{array}{c}\\text{Use the Zero Product Property to set each factor to 0.}\\hfill \\\\ \\text{Solve each equation.}\\hfill \\end{array}\\hfill &amp; &amp; &amp; &amp; &amp; \\phantom{\\rule{0.35em}{0ex}}\\begin{array}{ccccccccccccc}m=0\\hfill &amp; &amp; &amp; &amp; \\hfill 3m-10&amp; =\\hfill &amp; 0\\hfill &amp; &amp; &amp; &amp; \\hfill 3m-10&amp; =\\hfill &amp; 0\\hfill \\\\ m=0\\hfill &amp; &amp; &amp; &amp; \\hfill m&amp; =\\hfill &amp; \\frac{10}{3}\\hfill &amp; &amp; &amp; &amp; \\hfill m&amp; =\\hfill &amp; \\frac{10}{3}\\hfill \\end{array}\\hfill \\\\ \\text{Check your answers.}\\hfill &amp; &amp; &amp; &amp; &amp; \\text{The check is left to you.}\\hfill \\end{array}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836449554\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836363174\"><div data-type=\"problem\" id=\"fs-id1167829621409\"><p id=\"fs-id1167836522448\">Solve: \\(8{x}^{3}=24{x}^{2}-18x.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836293394\"><p id=\"fs-id1167833212958\">\\(x=0,x=\\frac{3}{2}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836389504\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836447280\"><div data-type=\"problem\" id=\"fs-id1167836493814\"><p id=\"fs-id1167836287418\">Solve: \\(16{y}^{2}=32{y}^{3}+2y.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829596286\"><p id=\"fs-id1167829695025\">\\(y=0,y=\\frac{1}{4}\\)<\/p><\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167836525536\"><h3 data-type=\"title\">Solve Equations with Polynomial Functions<\/h3><p id=\"fs-id1167833056834\">As our study of polynomial functions continues, it will often be important to know when the function will have a certain value or what points lie on the graph of the function. Our work with the <span data-type=\"term\" class=\"no-emphasis\">Zero Product Property<\/span> will be help us find these answers.<\/p><div data-type=\"example\" id=\"fs-id1167833061230\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167829599152\"><div data-type=\"problem\" id=\"fs-id1167829719035\"><p id=\"fs-id1167833071676\">For the function \\(f\\left(x\\right)={x}^{2}+2x-2,\\)<\/p><p id=\"fs-id1167836389548\"><span class=\"token\">\u24d0<\/span> find <em data-effect=\"italics\">x<\/em> when \\(f\\left(x\\right)=6\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> find two points that lie on the graph of the function.<\/div><div data-type=\"solution\" id=\"fs-id1167836444737\"><p id=\"fs-id1167836492865\"><span class=\"token\">\u24d0<\/span><\/p><div data-type=\"newline\"><br><\/div>\\(\\begin{array}{cccccc}&amp; &amp; &amp; &amp; &amp; f\\left(x\\right)={x}^{2}+2x-2\\hfill \\\\ \\text{Substitute 6 for}\\phantom{\\rule{0.2em}{0ex}}f\\left(x\\right).\\hfill &amp; &amp; &amp; &amp; &amp; \\phantom{\\rule{1.2em}{0ex}}6={x}^{2}+2x-2\\hfill \\\\ \\text{Put the quadratic in standard form.}\\hfill &amp; &amp; &amp; &amp; &amp; \\phantom{\\rule{0.8em}{0ex}}{x}^{2}+2x-8=0\\hfill \\\\ \\text{Factor the trinomial.}\\hfill &amp; &amp; &amp; &amp; &amp; \\left(x+4\\right)\\left(x-2\\right)=0\\hfill \\\\ \\begin{array}{c}\\text{Use the zero product property.}\\hfill \\\\ \\text{Solve.}\\hfill \\end{array}\\hfill &amp; &amp; &amp; &amp; &amp; \\begin{array}{ccccccccccc}\\hfill x+4&amp; =\\hfill &amp; 0\\hfill &amp; &amp; &amp; \\text{or}\\hfill &amp; &amp; &amp; \\hfill x-2&amp; =\\hfill &amp; 0\\hfill \\\\ \\hfill x&amp; =\\hfill &amp; -4\\hfill &amp; &amp; &amp; \\text{or}\\hfill &amp; &amp; &amp; \\hfill x&amp; =\\hfill &amp; 2\\hfill \\end{array}\\hfill \\\\ \\text{Check:}\\hfill &amp; &amp; &amp; &amp; &amp; \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\phantom{\\rule{4em}{0ex}}\\begin{array}{ccccc}\\phantom{\\rule{0.7em}{0ex}}f\\left(x\\right)={x}^{2}+2x-2\\hfill &amp; &amp; &amp; &amp; f\\left(x\\right)={x}^{2}+2x-2\\hfill \\\\ f\\left(\\text{\u2212}4\\right)={\\left(\\text{\u2212}4\\right)}^{2}+2\\left(\\text{\u2212}4\\right)-2\\hfill &amp; &amp; &amp; &amp; f\\left(2\\right)={2}^{2}+2\u00b72-2\\hfill \\\\ f\\left(\\text{\u2212}4\\right)=16-8-2\\hfill &amp; &amp; &amp; &amp; f\\left(2\\right)=4+4-2\\hfill \\\\ f\\left(\\text{\u2212}4\\right)=6\u2713\\hfill &amp; &amp; &amp; &amp; f\\left(2\\right)=6\u2713\\hfill \\end{array}\\hfill &amp; &amp; &amp; &amp; &amp; \\end{array}\\)<div data-type=\"newline\"><br><\/div><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> Since \\(f\\left(-4\\right)=6\\) and \\(f\\left(2\\right)=6,\\) the points \\(\\left(-4,6\\right)\\) and \\(\\left(2,6\\right)\\) lie on the graph of the function.<\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836609926\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836692073\"><div data-type=\"problem\" id=\"fs-id1167836526181\"><p id=\"fs-id1167829692807\">For the function \\(f\\left(x\\right)={x}^{2}-2x-8,\\)<\/p><p id=\"fs-id1167836392650\"><span class=\"token\">\u24d0<\/span> find <em data-effect=\"italics\">x<\/em> when \\(f\\left(x\\right)=7\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> Find two points that lie on the graph of the function.<\/div><div data-type=\"solution\" id=\"fs-id1167836318962\"><p id=\"fs-id1167836556194\"><span class=\"token\">\u24d0<\/span>\\(x=-3\\) or \\(x=5\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> \\(\\left(-3,7\\right)\\) \\(\\left(5,7\\right)\\)<\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836390310\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836667141\"><div data-type=\"problem\" id=\"fs-id1167836545876\"><p id=\"fs-id1167836559968\">For the function \\(f\\left(x\\right)={x}^{2}-8x+3,\\)<\/p><p id=\"fs-id1167836456286\"><span class=\"token\">\u24d0<\/span> find <em data-effect=\"italics\">x<\/em> when \\(f\\left(x\\right)=-4\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> Find two points that lie on the graph of the function.<\/div><div data-type=\"solution\" id=\"fs-id1167836622813\"><p id=\"fs-id1167836379075\"><span class=\"token\">\u24d0<\/span>\\(x=1\\) or \\(x=7\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> \\(\\left(1,-4\\right)\\) \\(\\left(7,-4\\right)\\)<\/div><\/div><\/div><p id=\"fs-id1167829877554\">The <span data-type=\"term\" class=\"no-emphasis\">Zero Product Property<\/span> also helps us determine where the function is zero. A value of <em data-effect=\"italics\">x<\/em> where the function is 0, is called a <span data-type=\"term\">zero of the function<\/span>.<\/p><div data-type=\"note\" id=\"fs-id1167836544182\"><div data-type=\"title\">Zero of a Function<\/div><p id=\"fs-id1167836552958\">For any function <em data-effect=\"italics\">f<\/em>, if \\(f\\left(x\\right)=0,\\) then <em data-effect=\"italics\">x<\/em> is a <strong data-effect=\"bold\">zero of the function<\/strong>.<\/p><\/div><p id=\"fs-id1167829689269\">When \\(f\\left(x\\right)=0,\\) the point \\(\\left(x,0\\right)\\) is a point on the graph. This point is an <span data-type=\"term\" class=\"no-emphasis\"><em data-effect=\"italics\">x<\/em>-intercept<\/span> of the graph. It is often important to know where the graph of a function crosses the axes. We will see some examples later.<\/p><div data-type=\"example\" id=\"fs-id1167836621947\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167829624424\"><div data-type=\"problem\" id=\"fs-id1167836618856\"><p id=\"fs-id1167829597813\">For the function \\(f\\left(x\\right)=3{x}^{2}+10x-8,\\) find<\/p><p id=\"fs-id1167836449758\"><span class=\"token\">\u24d0<\/span> the zeros of the function,<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> any <em data-effect=\"italics\">x<\/em>-intercepts of the graph of the function<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span> any <em data-effect=\"italics\">y<\/em>-intercepts of the graph of the function<\/div><div data-type=\"solution\" id=\"fs-id1167836321778\"><p id=\"fs-id1167836319053\"><span class=\"token\">\u24d0<\/span> To find the zeros of the function, we need to find when the function value is 0.<\/p><div data-type=\"newline\"><br><\/div> \\(\\begin{array}{cccccc}&amp; &amp; &amp; &amp; &amp; f\\left(x\\right)=3{x}^{2}+10x-8\\hfill \\\\ \\text{Substitute 0 for}\\phantom{\\rule{0.2em}{0ex}}f\\left(x\\right).\\hfill &amp; &amp; &amp; &amp; &amp; \\phantom{\\rule{1.2em}{0ex}}0=3{x}^{2}+10x-8\\hfill \\\\ \\text{Factor the trinomial.}\\hfill &amp; &amp; &amp; &amp; &amp; \\left(x+4\\right)\\left(3x-2\\right)=0\\hfill \\\\ \\begin{array}{c}\\text{Use the zero product property.}\\hfill \\\\ \\text{Solve.}\\hfill \\end{array}\\hfill &amp; &amp; &amp; &amp; &amp; \\begin{array}{ccccccccccc}\\hfill x+4&amp; =\\hfill &amp; 0\\hfill &amp; &amp; &amp; \\text{or}\\hfill &amp; &amp; &amp; \\hfill 3x-2&amp; =\\hfill &amp; 0\\hfill \\\\ \\hfill x&amp; =\\hfill &amp; -4\\hfill &amp; &amp; &amp; \\text{or}\\hfill &amp; &amp; &amp; \\hfill x&amp; =\\hfill &amp; \\frac{2}{3}\\hfill \\end{array}\\hfill \\end{array}\\)<div data-type=\"newline\"><br><\/div><div data-type=\"newline\"><br><\/div> <span class=\"token\">\u24d1<\/span> An <em data-effect=\"italics\">x<\/em>-intercept occurs when \\(y=0.\\) Since \\(f\\left(-4\\right)=0\\) and \\(f\\left(\\frac{2}{3}\\right)=0,\\) the points \\(\\left(-4,0\\right)\\) and \\(\\left(\\frac{2}{3},0\\right)\\) lie on the graph. These points are <em data-effect=\"italics\">x<\/em>-intercepts of the function.<div data-type=\"newline\"><br><\/div><div data-type=\"newline\"><br><\/div><div data-type=\"newline\"><br><\/div> <span class=\"token\">\u24d2<\/span> A <em data-effect=\"italics\">y<\/em>-intercept occurs when \\(x=0.\\) To find the <em data-effect=\"italics\">y<\/em>-intercepts we need to find \\(f\\left(0\\right).\\)<div data-type=\"newline\"><br><\/div> \\(\\begin{array}{cccccc}&amp; &amp; &amp; &amp; &amp; f\\left(x\\right)=3{x}^{2}+10x-8\\hfill \\\\ \\text{Find}\\phantom{\\rule{0.2em}{0ex}}f\\left(0\\right)\\phantom{\\rule{0.2em}{0ex}}\\text{by substituting 0 for}\\phantom{\\rule{0.2em}{0ex}}x.\\hfill &amp; &amp; &amp; &amp; &amp; f\\left(0\\right)=3\u00b7{0}^{2}+10\u00b70-8\\hfill \\\\ \\text{Simplify.}\\hfill &amp; &amp; &amp; &amp; &amp; f\\left(0\\right)=-8\\hfill \\end{array}\\)<div data-type=\"newline\"><br><\/div> Since \\(f\\left(0\\right)=-8,\\) the point \\(\\left(0,-8\\right)\\) lies on the graph. This point is the <em data-effect=\"italics\">y<\/em>-intercept of the function.<\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836612535\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167829811740\"><div data-type=\"problem\" id=\"fs-id1167836516254\"><p id=\"fs-id1167836362622\">For the function \\(f\\left(x\\right)=2{x}^{2}-7x+5,\\) find<\/p><p id=\"fs-id1167836601250\"><span class=\"token\">\u24d0<\/span> the zeros of the function<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> any <em data-effect=\"italics\">x<\/em>-intercepts of the graph of the function<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span> any <em data-effect=\"italics\">y<\/em>-intercepts of the graph of the function.<\/div><div data-type=\"solution\"><p id=\"fs-id1167829686321\"><span class=\"token\">\u24d0<\/span>\\(x=1\\) or \\(x=\\frac{5}{2}\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> \\(\\left(1,0\\right),\\) \\(\\left(\\frac{5}{2},0\\right)\\) <span class=\"token\">\u24d2<\/span> \\(\\left(0,5\\right)\\)<\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836391037\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167829693111\"><div data-type=\"problem\" id=\"fs-id1167836625132\"><p id=\"fs-id1167836360997\">For the function \\(f\\left(x\\right)=6{x}^{2}+13x-15,\\) find<\/p><p id=\"fs-id1167836622572\"><span class=\"token\">\u24d0<\/span> the zeros of the function<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> any <em data-effect=\"italics\">x<\/em>-intercepts of the graph of the function<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span> any <em data-effect=\"italics\">y<\/em>-intercepts of the graph of the function.<\/div><div data-type=\"solution\" id=\"fs-id1167829595141\"><p id=\"fs-id1167833060987\"><span class=\"token\">\u24d0<\/span>\\(x=-3\\) or \\(x=\\frac{5}{6}\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> \\(\\left(-3,0\\right),\\) \\(\\left(\\frac{5}{6},0\\right)\\) <span class=\"token\">\u24d2<\/span> \\(\\left(0,-15\\right)\\)<\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167833381187\"><h3 data-type=\"title\">Solve Applications Modeled by Polynomial Equations<\/h3><p id=\"fs-id1167833054651\">The problem-solving strategy we used earlier for applications that translate to linear equations will work just as well for applications that translate to polynomial equations. We will copy the problem-solving strategy here so we can use it for reference.<\/p><div data-type=\"note\" id=\"fs-id1167836511805\" class=\"howto\"><div data-type=\"title\">Use a problem solving strategy to solve word problems.<\/div><ol id=\"fs-id1167829790276\" type=\"1\" class=\"stepwise\"><li><strong data-effect=\"bold\">Read<\/strong> the problem. Make sure all the words and ideas are understood.<\/li><li><strong data-effect=\"bold\">Identify<\/strong> what we are looking for.<\/li><li><strong data-effect=\"bold\">Name<\/strong> what we are looking for. Choose a variable to represent that quantity.<\/li><li><strong data-effect=\"bold\">Translate<\/strong> into an equation. It may be helpful to restate the problem in one sentence with all the important information. Then, translate the English sentence into an algebraic equation.<\/li><li><strong data-effect=\"bold\">Solve<\/strong> the equation using appropriate algebra techniques.<\/li><li><strong data-effect=\"bold\">Check<\/strong> the answer in the problem and make sure it makes sense.<\/li><li><strong data-effect=\"bold\">Answer<\/strong> the question with a complete sentence.<\/li><\/ol><\/div><p id=\"fs-id1167829830361\">We will start with a number problem to get practice translating words into a polynomial equation.<\/p><div data-type=\"example\" id=\"fs-id1167836611600\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167836533796\"><div data-type=\"problem\" id=\"fs-id1167833019354\"><p id=\"fs-id1167836509779\">The product of two consecutive odd integers is 323. Find the integers.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829596980\"><p id=\"fs-id1167836449894\">\\(\\begin{array}{cccccc}\\mathbf{\\text{Step 1. Read}}\\phantom{\\rule{0.2em}{0ex}}\\text{the problem.}\\hfill &amp; &amp; &amp; &amp; &amp; \\\\ \\mathbf{\\text{Step 2. Identify}}\\phantom{\\rule{0.2em}{0ex}}\\text{what we are looking for.}\\hfill &amp; &amp; &amp; &amp; &amp; \\text{We are looking for two consecutive integers.}\\hfill \\\\ \\mathbf{\\text{Step 3. Name}}\\phantom{\\rule{0.2em}{0ex}}\\text{what we are looking for.}\\hfill &amp; &amp; &amp; &amp; &amp; \\text{Let}\\phantom{\\rule{0.2em}{0ex}}n=\\text{the first integer.}\\hfill \\\\ &amp; &amp; &amp; &amp; &amp; n+2=\\phantom{\\rule{0.2em}{0ex}}\\text{next consecutive odd integer}\\hfill \\\\ \\begin{array}{c}\\mathbf{\\text{Step 4. Translate}}\\phantom{\\rule{0.2em}{0ex}}\\text{into an equation. Restate the}\\hfill \\\\ \\text{problem in a sentence.}\\hfill \\end{array}\\hfill &amp; &amp; &amp; &amp; &amp; \\begin{array}{c}\\text{The product of the two consecutive odd}\\hfill \\\\ \\text{integers is 323.}\\hfill \\end{array}\\hfill \\\\ &amp; &amp; &amp; &amp; &amp; \\phantom{\\rule{3.47em}{0ex}}n\\left(n+2\\right)=323\\hfill \\\\ \\mathbf{\\text{Step 5. Solve}}\\phantom{\\rule{0.2em}{0ex}}\\text{the equation.}\\hfill &amp; &amp; &amp; &amp; &amp; \\phantom{\\rule{3.57em}{0ex}}{n}^{2}+2n=323\\hfill \\\\ \\text{Bring all the terms to one side.}\\hfill &amp; &amp; &amp; &amp; &amp; \\phantom{\\rule{0.85em}{0ex}}{n}^{2}+2n-323=0\\hfill \\\\ \\text{Factor the trinomial.}\\hfill &amp; &amp; &amp; &amp; &amp; \\left(n-17\\right)\\left(n+19\\right)=0\\hfill \\\\ \\begin{array}{c}\\text{Use the Zero Product Property.}\\hfill \\\\ \\text{Solve the equations.}\\hfill \\end{array}\\hfill &amp; &amp; &amp; &amp; &amp; \\begin{array}{ccccccccc}\\hfill n-17&amp; =\\hfill &amp; 0\\hfill &amp; &amp; &amp; &amp; \\hfill n+19&amp; =\\hfill &amp; 0\\hfill \\\\ \\hfill n&amp; =\\hfill &amp; 17\\hfill &amp; &amp; &amp; &amp; \\hfill n&amp; =\\hfill &amp; -19\\hfill \\end{array}\\hfill \\end{array}\\)<\/p><div data-type=\"newline\"><br><\/div> There are two values for <em data-effect=\"italics\">n<\/em> that are solutions to this problem. So there are two sets of consecutive odd integers that will work.<div data-type=\"newline\"><br><\/div> \\(\\begin{array}{cccccc}\\text{If the first integer is}\\phantom{\\rule{0.2em}{0ex}}n=17\\hfill &amp; &amp; &amp; &amp; &amp; \\text{If the first integer is}\\phantom{\\rule{0.2em}{0ex}}n=-19\\hfill \\\\ \\text{then the next odd integer is}\\hfill &amp; &amp; &amp; &amp; &amp; \\text{then the next odd integer is}\\hfill \\\\ \\phantom{\\rule{11em}{0ex}}n+2\\hfill &amp; &amp; &amp; &amp; &amp; \\phantom{\\rule{11em}{0ex}}n+2\\hfill \\\\ \\phantom{\\rule{10.5em}{0ex}}17+2\\hfill &amp; &amp; &amp; &amp; &amp; \\phantom{\\rule{10em}{0ex}}\\text{\u2212}19+2\\hfill \\\\ \\phantom{\\rule{11.5em}{0ex}}19\\hfill &amp; &amp; &amp; &amp; &amp; \\phantom{\\rule{11em}{0ex}}\\text{\u2212}17\\hfill \\\\ \\phantom{\\rule{10.5em}{0ex}}17,19\\hfill &amp; &amp; &amp; &amp; &amp; \\phantom{\\rule{10em}{0ex}}\\text{\u2212}17,-19\\hfill \\\\ \\mathbf{\\text{Step 6. Check}}\\phantom{\\rule{0.2em}{0ex}}\\text{the answer.}\\hfill &amp; &amp; &amp; &amp; &amp; \\\\ \\text{The results are consecutive odd integers}\\hfill &amp; &amp; &amp; &amp; &amp; \\\\ 17,19\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}\\text{\u2212}19,-17.\\hfill &amp; &amp; &amp; &amp; &amp; \\\\ 17\u00b719=323\u2713\\phantom{\\rule{4em}{0ex}}\\text{\u2212}19\\left(\\text{\u2212}17\\right)=323\u2713\\hfill &amp; &amp; &amp; &amp; &amp; \\\\ \\text{Both pairs of consecutive integers are solutions.}\\hfill &amp; &amp; &amp; &amp; &amp; \\\\ \\mathbf{\\text{Step 7. Answer}}\\phantom{\\rule{0.2em}{0ex}}\\text{the question}\\hfill &amp; &amp; &amp; &amp; &amp; \\text{The consecutive integers are 17, 19 and}\\phantom{\\rule{0.2em}{0ex}}\\text{\u2212}19,-17.\\hfill \\end{array}\\)<\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167829809872\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836509306\"><div data-type=\"problem\" id=\"fs-id1167833024146\"><p id=\"fs-id1167833224626\">The product of two consecutive odd integers is 255. Find the integers.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836529154\"><p id=\"fs-id1167836520704\">\\(-15,-17\\) and 15, 17<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836620394\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836660223\"><div data-type=\"problem\" id=\"fs-id1167836417134\"><p id=\"fs-id1167829683765\">The product of two consecutive odd integers is 483 Find the integers.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836728273\"><p id=\"fs-id1167836624591\">\\(-23,-21\\) and 21, 23<\/p><\/div><\/div><\/div><p id=\"fs-id1167836447980\">Were you surprised by the pair of negative integers that is one of the solutions to the previous example? The product of the two positive integers and the product of the two negative integers both give positive results.<\/p><p id=\"fs-id1167836393036\">In some applications, negative solutions will result from the algebra, but will not be realistic for the situation.<\/p><div data-type=\"example\" id=\"fs-id1167836492373\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167833086444\"><div data-type=\"problem\" id=\"fs-id1167836507849\"><p id=\"fs-id1167836309170\">A rectangular bedroom has an area 117 square feet. The length of the bedroom is four feet more than the width. Find the length and width of the bedroom.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833059883\"><table id=\"fs-id1167836283397\" class=\"unnumbered unstyled can-break\" summary=\"Step 1: Read the problem. In problems involving geometric figures, a sketch can help you visualize the situation. Make a rectangle with the shorter side equal to w units and the longer one equal to w plus 4 units. Step 2: Identify what you are looking for. We are looking for the length and width. Step 3: Name what you are looking for. The length is four feet more than the width. Let w be the width of the bedroom and w plus 4 be the length of the garden. Step 4: Translate into an equation. Restate the important information in a sentence. The area of the bedroom is 117square feet. Use the formula for the area of a rectangle. A is L times W. Substitute in the variables. 117 equals open parentheses w plus 4 close parentheses w. Step 5: Solve the equation Distribute first. 117 equals w squared plus 4w. Get zero on one side. 0 equals w squared plus 4w minus 117. Factor the trinomial. 0 equals w plus 13, w minus 9. Use the Zero Product Property. 0 equals w plus 13 and 0 equals w plus 9. Solve each equation. Minus 13 equals w or 9 equals w. Since w is the width of the bedroom, it does not make sense for it to be negative. We eliminate that value for w. So w equals 9 and width is 9 feet. Find the value of the length. Length is w plus 4 equals 13. Step 6: Check the answer. Does the answer make sense? Yes, this makes sense. Step 7: Answer the question. The width of the bedroom is 9 feet and the length is 13 feet.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 1. Read<\/strong> the problem. In problems involving<div data-type=\"newline\"><br><\/div>geometric figures, a sketch can help you visualize<div data-type=\"newline\"><br><\/div>the situation.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836691642\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_05_003b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 2. Identify<\/strong> what you are looking for.<\/td><td data-valign=\"top\" data-align=\"left\">We are looking for the length and width.<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 3. Name<\/strong> what you are looking for.<\/td><td data-valign=\"top\" data-align=\"left\">Let \\(w=\\) the width of the bedroom.<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">The length is four feet more than the width.<\/td><td data-valign=\"top\" data-align=\"left\">\\(w+4=\\) the length of the garden<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 4. Translate<\/strong> into an equation.<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Restate the important information in a sentence.<\/td><td data-valign=\"top\" data-align=\"left\">The area of the bedroom is 117 square feet.<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Use the formula for the area of a rectangle.<\/td><td data-valign=\"top\" data-align=\"center\">\\(A=l\u00b7w\\phantom{\\rule{1.2em}{0ex}}\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Substitute in the variables.<\/td><td data-valign=\"top\" data-align=\"center\">\\(\\phantom{\\rule{0.2em}{0ex}}117=\\left(w+4\\right)w\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 5. Solve<\/strong> the equation Distribute first.<\/td><td data-valign=\"top\" data-align=\"center\">\\(117={w}^{2}+4w\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Get zero on one side.<\/td><td data-valign=\"top\" data-align=\"center\">\\(117={w}^{2}+4w\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Factor the trinomial.<\/td><td data-valign=\"top\" data-align=\"center\">\\(\\phantom{\\rule{3.5em}{0ex}}0={w}^{2}+4w-117\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Use the Zero Product Property.<\/td><td data-valign=\"top\" data-align=\"center\">\\(\\phantom{\\rule{4.4em}{0ex}}0=\\left({w}^{2}+13\\right)\\left(w-9\\right)\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Solve each equation.<\/td><td data-valign=\"top\" data-align=\"center\">\\(\\phantom{\\rule{5.6em}{0ex}}0=w+13\\phantom{\\rule{1.5em}{0ex}}0=w-9\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Since <em data-effect=\"italics\">w<\/em> is the width of the bedroom, it does not<div data-type=\"newline\"><br><\/div>make sense for it to be negative. We eliminate that value for <em data-effect=\"italics\">w<\/em>.<\/td><td data-valign=\"top\" data-align=\"center\">\\(\\phantom{\\rule{3em}{0ex}}\\overline{)-13=w}\\phantom{\\rule{3.5em}{0ex}}9=w\\)<\/td><\/tr><tr valign=\"top\"><td><\/td><td data-valign=\"top\" data-align=\"right\">\\(w=9\\)\u2003Width is 9 feet.&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Find the value of the length.<\/td><td data-valign=\"top\" data-align=\"center\">\\(w+4\\phantom{\\rule{1.5em}{0ex}}\\)<div data-type=\"newline\"><br><\/div>\\(9+4\\phantom{\\rule{1.5em}{0ex}}\\)<div data-type=\"newline\"><br><\/div>\\(\\phantom{\\rule{7em}{0ex}}\\)13\u2003\u2003Length is 13 feet.<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 6. Check<\/strong> the answer.<div data-type=\"newline\"><br><\/div>Does the answer make sense?<div data-type=\"newline\"><br><\/div><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167836326705\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_05_003a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><div data-type=\"newline\"><br><\/div>Yes, this makes sense.<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 7. Answer<\/strong> the question.<\/td><td data-valign=\"top\" data-align=\"left\">The width of the bedroom is 9 feet and<div data-type=\"newline\"><br><\/div>the length is 13 feet.<\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167829716003\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836693448\"><div data-type=\"problem\" id=\"fs-id1167836528456\"><p id=\"fs-id1167829748017\">A rectangular sign has area 30 square feet. The length of the sign is one foot more than the width. Find the length and width of the sign.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833047687\"><p id=\"fs-id1167829879335\">The width is 5 feet and length is 6 feet.<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836619400\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836416135\"><div data-type=\"problem\" id=\"fs-id1167829689459\"><p id=\"fs-id1167836697766\">A rectangular patio has area 180 square feet. The width of the patio is three feet less than the length. Find the length and width of the patio.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829691910\"><p id=\"fs-id1167836550460\">The length of the patio is 12 feet and the width 15 feet.<\/p><\/div><\/div><\/div><p id=\"fs-id1167836283441\">In the next example, we will use the Pythagorean Theorem \\(\\left({a}^{2}+{b}^{2}={c}^{2}\\right).\\) This formula gives the relation between the legs and the hypotenuse of a right triangle.<\/p><span data-type=\"media\" id=\"fs-id1167833227285\" data-alt=\"Figure shows a right triangle with the shortest side being a, the second side being b and the hypotenuse being c.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_05_004_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure shows a right triangle with the shortest side being a, the second side being b and the hypotenuse being c.\"><\/span><p id=\"fs-id1167836289692\">We will use this formula to in the next example.<\/p><div data-type=\"example\" id=\"fs-id1167836558176\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167836547336\"><div data-type=\"problem\" id=\"fs-id1167836529160\"><p id=\"fs-id1167836392939\">A boat\u2019s sail is in the shape of a right triangle as shown. The hypotenuse will be 17 feet long. The length of one side will be 7 feet less than the length of the other side. Find the lengths of the sides of the sail.<\/p><span data-type=\"media\" id=\"fs-id1167836688021\" data-alt=\"Figure shows a right triangle with the shortest side being x, the second side being x minus 7 and the hypotenuse being 17.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_05_005_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure shows a right triangle with the shortest side being x, the second side being x minus 7 and the hypotenuse being 17.\"><\/span><\/div><div data-type=\"solution\" id=\"fs-id1167833025641\"><table id=\"fs-id1167829651452\" class=\"unnumbered unstyled can-break\" summary=\"Step 1: Read the problem. Step 2: Identify what you are looking for. We are looking for the lengths of the sides of the sail. Step 3: Name what you are looking for. One side is 7 less than the other. Let x be length of a side of the sail and x minus 7 be length of other side. Step 4: Translate into an equation. Since this is a right triangle we can use the Pythagorean Theorem a squared plus b squared equals c squared. Substitute in the variables: x squared plus open parentheses x minus 7 close parentheses squared equals 17 squared. Step 5: Solve the equation. Simplify: x squared plus x squared minus 14x plus 49 is 289. 2 x squared minus 14x plus 49 equals 289. It is a quadratic equation, so get zero on one side: 2 x squared minus 14x minus 240 equals 0. Factor the greatest common factor. Factor the trinomial. 2 open parentheses x minus 15 close parentheses open parentheses x plus 8 close parentheses. Use the Zero Product Property: 2 is not equal to 0, x minus 15 equals 0 or x plus 8 equal 0. Solve: x is 15 or x is minus 8. Since x is a side of the triangle, x equals minus 8 does not make sense. Find the length of the other side. If the length of one side is 15, then the length of the other side is 15 minus 7 equals 8. Step 6: Check the answer in the problem. Do these numbers make sense? Step 7: Answer the question: The sides of the sail are 8, 15 and 17 feet.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 1. Read<\/strong> the problem<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 2. Identify<\/strong> what you are looking for.<\/td><td data-valign=\"top\" data-align=\"left\">We are looking for the lengths of the<div data-type=\"newline\"><br><\/div>sides of the sail.<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 3. Name<\/strong> what you are looking for.<div data-type=\"newline\"><br><\/div>One side is 7 less than the other.<\/td><td data-valign=\"top\" data-align=\"left\">Let \\(x=\\) length of a side of the sail.<div data-type=\"newline\"><br><\/div>\\(x-7=\\) length of other side<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 4. Translate<\/strong> into an equation. Since this is a<div data-type=\"newline\"><br><\/div>right triangle we can use the Pythagorean Theorem.<\/td><td data-valign=\"bottom\" data-align=\"left\">\\(\\phantom{\\rule{4.8em}{0ex}}{a}^{2}+{b}^{2}={c}^{2}\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Substitute in the variables.<\/td><td data-valign=\"top\" data-align=\"left\">\\(\\phantom{\\rule{2.5em}{0ex}}{x}^{2}+{\\left(x-7\\right)}^{2}={17}^{2}\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 5. Solve<\/strong> the equation<div data-type=\"newline\"><br><\/div>Simplify.<\/td><td data-valign=\"bottom\" data-align=\"left\">\\({x}^{2}+{x}^{2}-14x+49=289\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\">\\(\\phantom{\\rule{1.5em}{0ex}}2{x}^{2}-14x+49=289\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">It is a quadratic equation, so get zero on one side.<\/td><td data-valign=\"top\" data-align=\"left\">\\(\\phantom{\\rule{1em}{0ex}}2{x}^{2}-14x-240=0\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Factor the greatest common factor.<\/td><td data-valign=\"top\" data-align=\"left\">\\(\\phantom{\\rule{0.8em}{0ex}}2\\left({x}^{2}-7x-120\\right)=0\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Factor the trinomial.<\/td><td data-valign=\"top\" data-align=\"left\">\\(\\phantom{\\rule{1.1em}{0ex}}2\\left(x-15\\right)\\left(x+8\\right)=0\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Use the Zero Product Property.<\/td><td data-valign=\"top\" data-align=\"left\">\\(2\\ne 0\\phantom{\\rule{1.5em}{0ex}}x-15=0\\phantom{\\rule{0.6em}{0ex}}x+8=0\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Solve.<\/td><td data-valign=\"top\" data-align=\"left\">\\(2\\ne 0\\phantom{\\rule{1.5em}{0ex}}x=15\\phantom{\\rule{2em}{0ex}}x=-8\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Since <em data-effect=\"italics\">x<\/em> is a side of the triangle, \\(x=-8\\) does not<div data-type=\"newline\"><br><\/div>make sense.<\/td><td data-valign=\"top\" data-align=\"left\">\\(2\\ne 0\\phantom{\\rule{1.5em}{0ex}}x=15\\phantom{\\rule{2em}{0ex}}\\overline{)x=-8}\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Find the length of the other side.<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">\u2003\u2003\u2003If the length of one side is<div data-type=\"newline\"><br><\/div>\u2003\u2003\u2003then the length of the other side is<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832981684\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_05_006l_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167836485100\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_05_006m_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167836545662\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_05_006n_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><div data-type=\"newline\"><br><\/div>8 is the length of the other side.<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 6. Check<\/strong> the answer in the problem<div data-type=\"newline\"><br><\/div>Do these numbers make sense?<div data-type=\"newline\"><br><\/div><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167836551721\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_05_006a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 7. Answer<\/strong> the question<\/td><td data-valign=\"top\" data-align=\"left\">The sides of the sail are 8, 15 and 17 feet.<\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836623889\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836450411\"><div data-type=\"problem\" id=\"fs-id1167836294961\"><p id=\"fs-id1167836560114\">Justine wants to put a deck in the corner of her backyard in the shape of a right triangle. The length of one side of the deck is 7 feet more than the other side. The hypotenuse is 13. Find the lengths of the two sides of the deck.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833381531\"><p>5 feet and 12 feet<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836377845\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167829747291\"><div data-type=\"problem\" id=\"fs-id1167836312153\"><p id=\"fs-id1167836534023\">A meditation garden is in the shape of a right triangle, with one leg 7 feet. The length of the hypotenuse is one more than the length of the other leg. Find the lengths of the hypotenuse and the other leg.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836628723\"><p id=\"fs-id1167836447576\">24 feet and 25 feet<\/p><\/div><\/div><\/div><p id=\"fs-id1171792850594\">The next example uses the function that gives the height of an object as a function of time when it is thrown from 80 feet above the ground.<\/p><div data-type=\"example\" id=\"fs-id1167833022018\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167836503972\"><div data-type=\"problem\" id=\"fs-id1167829748344\"><p id=\"fs-id1167836330850\">Dennis is going to throw his rubber band ball upward from the top of a campus building. When he throws the rubber band ball from 80 feet above the ground, the function \\(h\\left(t\\right)=-16{t}^{2}+64t+80\\) models the height, <em data-effect=\"italics\">h<\/em>, of the ball above the ground as a function of time, <em data-effect=\"italics\">t.<\/em> Find:<\/p><p id=\"fs-id1167836526114\"><span class=\"token\">\u24d0<\/span> the zeros of this function which tell us when the ball hits the ground<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> when the ball will be 80 feet above the ground<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span> the height of the ball at \\(t=2\\) seconds.<\/div><div data-type=\"solution\" id=\"fs-id1167836612483\"><p id=\"fs-id1167836576454\"><span class=\"token\">\u24d0<\/span> The zeros of this function are found by solving \\(h\\left(t\\right)=0.\\) This will tell us when the ball will hit the ground.<\/p><div data-type=\"newline\"><br><\/div> \\(\\begin{array}{cccccc}&amp; &amp; &amp; &amp; &amp; \\phantom{\\rule{7.45em}{0ex}}h\\left(t\\right)=0\\hfill \\\\ \\text{Substitute in the polynomial for}\\phantom{\\rule{0.2em}{0ex}}h\\left(t\\right).\\hfill &amp; &amp; &amp; &amp; &amp; \\phantom{\\rule{2.03em}{0ex}}-16{t}^{2}+64t+80=0\\hfill \\\\ \\text{Factor the GCF,}\\phantom{\\rule{0.2em}{0ex}}-16.\\hfill &amp; &amp; &amp; &amp; &amp; \\phantom{\\rule{2.35em}{0ex}}-16\\left({t}^{2}-4t-5\\right)=0\\hfill \\\\ \\text{Factor the trinomial.}\\hfill &amp; &amp; &amp; &amp; &amp; \\phantom{\\rule{2.1em}{0ex}}-16\\left(t-5\\right)\\left(t+1\\right)=0\\hfill \\\\ \\begin{array}{c}\\text{Use the Zero Product Property.}\\hfill \\\\ \\text{Solve.}\\hfill \\end{array}\\hfill &amp; &amp; &amp; &amp; &amp; \\begin{array}{ccccccccccc}\\hfill t-5&amp; =\\hfill &amp; 0\\hfill &amp; &amp; &amp; &amp; &amp; &amp; \\hfill \\phantom{\\rule{2em}{0ex}}t+1&amp; =\\hfill &amp; 0\\hfill \\\\ \\hfill t&amp; =\\hfill &amp; 5\\hfill &amp; &amp; &amp; &amp; &amp; &amp; \\hfill \\phantom{\\rule{2em}{0ex}}t&amp; =\\hfill &amp; -1\\hfill \\end{array}\\hfill \\end{array}\\)<p id=\"fs-id1167836334175\">The result \\(t=5\\) tells us the ball will hit the ground 5 seconds after it is thrown. Since time cannot be negative, the result \\(t=-1\\) is discarded.<\/p><p id=\"fs-id1167836282342\"><span class=\"token\">\u24d1<\/span> The ball will be 80 feet above the ground when \\(h\\left(t\\right)=80.\\)<\/p><div data-type=\"newline\"><br><\/div> \\(\\begin{array}{cccccc}&amp; &amp; &amp; &amp; &amp; \\phantom{\\rule{7.55em}{0ex}}h\\left(t\\right)=80\\hfill \\\\ \\text{Substitute in the polynomial for}\\phantom{\\rule{0.2em}{0ex}}h\\left(t\\right).\\hfill &amp; &amp; &amp; &amp; &amp; \\phantom{\\rule{2.05em}{0ex}}-16{t}^{2}+64t+80=80\\hfill \\\\ \\text{Subtract 80 from both sides.}\\hfill &amp; &amp; &amp; &amp; &amp; \\phantom{\\rule{4.2em}{0ex}}-16{t}^{2}+64t=0\\hfill \\\\ \\text{Factor the GCF,}\\phantom{\\rule{0.2em}{0ex}}-16t.\\hfill &amp; &amp; &amp; &amp; &amp; \\phantom{\\rule{4.5em}{0ex}}-16t\\left(t-4\\right)=0\\hfill \\\\ \\begin{array}{c}\\text{Use the Zero Product Property.}\\hfill \\\\ \\text{Solve.}\\hfill \\end{array}\\hfill &amp; &amp; &amp; &amp; &amp; \\phantom{\\rule{3em}{0ex}}\\begin{array}{ccccccccc}\\hfill -16t&amp; =\\hfill &amp; 0\\hfill &amp; &amp; &amp; &amp; \\hfill t-4&amp; =\\hfill &amp; 0\\hfill \\\\ \\hfill t&amp; =\\hfill &amp; 0\\hfill &amp; &amp; &amp; &amp; \\hfill t&amp; =\\hfill &amp; 4\\hfill \\end{array}\\hfill \\\\ &amp; &amp; &amp; &amp; &amp; \\begin{array}{c}\\text{The ball will be at 80 feet the moment Dennis}\\hfill \\\\ \\text{tosses the ball and then 4 seconds later, when}\\hfill \\\\ \\text{the ball is falling.}\\hfill \\end{array}\\hfill \\end{array}\\)<p><span class=\"token\">\u24d2<\/span> To find the height ball at \\(t=2\\) seconds we find \\(h\\left(2\\right).\\)<\/p><div data-type=\"newline\"><br><\/div> \\(\\begin{array}{cccccc}&amp; &amp; &amp; &amp; &amp; \\phantom{\\rule{3.2em}{0ex}}h\\left(t\\right)=-16{t}^{2}+64t+80\\hfill \\\\ \\text{To find}\\phantom{\\rule{0.2em}{0ex}}h\\left(2\\right)\\phantom{\\rule{0.2em}{0ex}}\\text{substitute 2 for}\\phantom{\\rule{0.2em}{0ex}}t.\\hfill &amp; &amp; &amp; &amp; &amp; \\phantom{\\rule{3em}{0ex}}h\\left(2\\right)=-16{\\left(2\\right)}^{2}+64\u00b72+80\\hfill \\\\ \\text{Simplify.}\\hfill &amp; &amp; &amp; &amp; &amp; \\phantom{\\rule{3em}{0ex}}h\\left(2\\right)=144\\hfill \\\\ &amp; &amp; &amp; &amp; &amp; \\phantom{\\rule{3em}{0ex}}\\text{After 2 seconds, the ball will be at 144 feet.}\\hfill \\end{array}\\)<\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836289305\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836706000\"><div data-type=\"problem\" id=\"fs-id1167836706002\"><p id=\"fs-id1167836530068\">Genevieve is going to throw a rock from the top a trail overlooking the ocean. When she throws the rock upward from 160 feet above the ocean, the function \\(h\\left(t\\right)=-16{t}^{2}+48t+160\\) models the height, <em data-effect=\"italics\">h<\/em>, of the rock above the ocean as a function of time, <em data-effect=\"italics\">t<\/em>. Find:<\/p><p id=\"fs-id1167836525352\"><span class=\"token\">\u24d0<\/span> the zeros of this function which tell us when the rock will hit the ocean<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> when the rock will be 160 feet above the ocean.<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span> the height of the rock at \\(t=1.5\\) seconds.<\/div><div data-type=\"solution\" id=\"fs-id1167836585148\"><p id=\"fs-id1167829596335\"><span class=\"token\">\u24d0<\/span> 5 <span class=\"token\">\u24d1<\/span> 0;3 <span class=\"token\">\u24d2<\/span> 196<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836550916\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167829744740\"><div data-type=\"problem\" id=\"fs-id1167836684102\"><p id=\"fs-id1167836297652\">Calib is going to throw his lucky penny from his balcony on a cruise ship. When he throws the penny upward from 128 feet above the ground, the function \\(h\\left(t\\right)=-16{t}^{2}+32t+128\\) models the height, <em data-effect=\"italics\">h<\/em>, of the penny above the ocean as a function of time, <em data-effect=\"italics\">t<\/em>. Find:<\/p><p id=\"fs-id1167836545880\"><span class=\"token\">\u24d0<\/span> the zeros of this function which is when the penny will hit the ocean<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> when the penny will be 128 feet above the ocean.<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span> the height the penny will be at \\(t=1\\) seconds which is when the penny will be at its highest point.<\/div><div data-type=\"solution\" id=\"fs-id1167836531169\"><p id=\"fs-id1167836531171\"><span class=\"token\">\u24d0<\/span> 4 <span class=\"token\">\u24d1<\/span> 0;2 <span class=\"token\">\u24d2<\/span> 144<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167833060072\" class=\"media-2\"><p id=\"fs-id1167836698870\">Access this online resource for additional instruction and practice with quadratic equations.<\/p><ul id=\"fs-id1167836630386\" data-display=\"block\"><li><a href=\"https:\/\/openstax.org\/l\/37ZeroProperty\">Beginning Algebra &amp; Solving Quadratics with the Zero Property<\/a><\/li><\/ul><\/div><\/div><div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167829711704\"><h3 data-type=\"title\">Key Concepts<\/h3><ul id=\"fs-id1167836356276\" data-bullet-style=\"bullet\"><li><strong data-effect=\"bold\">Polynomial Equation:<\/strong> A polynomial equation is an equation that contains a polynomial expression. The degree of the polynomial equation is the degree of the polynomial.<\/li><li><strong data-effect=\"bold\">Quadratic Equation:<\/strong> An equation of the form \\(a{x}^{2}+bx+c=0\\) is called a quadratic equation.<div data-type=\"newline\"><br><\/div> <div data-type=\"equation\" id=\"fs-id1167836518426\" class=\"unnumbered\" data-label=\"\">\\(a,b,c\\phantom{\\rule{0.2em}{0ex}}\\text{are real numbers and}\\phantom{\\rule{0.2em}{0ex}}a\\ne 0\\)<\/div><\/li><li><strong data-effect=\"bold\">Zero Product Property:<\/strong> If \\(a\u00b7b=0,\\) then either \\(a=0\\) or \\(b=0\\) or both.<\/li><li><strong data-effect=\"bold\">How to use the Zero Product Property<\/strong><ol type=\"1\" class=\"stepwise\"><li>Set each factor equal to zero.<\/li><li>Solve the linear equations.<\/li><li>Check.<\/li><\/ol><\/li><li><strong data-effect=\"bold\">How to solve a quadratic equation by factoring.<\/strong><ol id=\"fs-id1167836543767\" type=\"1\" class=\"stepwise\"><li>Write the quadratic equation in standard form, \\(a{x}^{2}+bx+c=0.\\)<\/li><li>Factor the quadratic expression.<\/li><li>Use the Zero Product Property.<div data-type=\"newline\"><br><\/div><\/li><li>Solve the linear equations.<\/li><li>Check. Substitute each solution separately into the original equation.<\/li><\/ol><\/li><li><strong data-effect=\"bold\">Zero of a Function:<\/strong> For any function <em data-effect=\"italics\">f<\/em>, if \\(f\\left(x\\right)=0,\\) then <em data-effect=\"italics\">x<\/em> is a zero of the function.<\/li><li><strong data-effect=\"bold\">How to use a problem solving strategy to solve word problems.<\/strong><ol id=\"fs-id1167836549434\" type=\"1\" class=\"stepwise\"><li><strong data-effect=\"bold\">Read<\/strong> the problem. Make sure all the words and ideas are understood.<\/li><li><strong data-effect=\"bold\">Identify<\/strong> what we are looking for.<\/li><li><strong data-effect=\"bold\">Name<\/strong> what we are looking for. Choose a variable to represent that quantity.<\/li><li><strong data-effect=\"bold\">Translate<\/strong> into an equation. It may be helpful to restate the problem in one sentence with all the important information. Then, translate the English sentence into an algebraic equation.<\/li><li><strong data-effect=\"bold\">Solve<\/strong> the equation using appropriate algebra techniques.<\/li><li><strong data-effect=\"bold\">Check<\/strong> the answer in the problem and make sure it makes sense.<\/li><li><strong data-effect=\"bold\">Answer<\/strong> the question with a complete sentence.<\/li><\/ol><\/li><\/ul><\/div><div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167829685381\"><h3 data-type=\"title\">Section Exercises<\/h3><div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1167829656544\"><h4 data-type=\"title\">Practice Makes Perfect<\/h4><p id=\"fs-id1167833018910\"><strong data-effect=\"bold\">Use the Zero Product Property<\/strong><\/p><p id=\"fs-id1167836434570\">In the following exercises, solve.<\/p><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167836391400\"><p id=\"fs-id1167829624077\">\\(\\left(3a-10\\right)\\left(2a-7\\right)=0\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829752365\"><p id=\"fs-id1167836545075\">\\(a=10\\text{\/}3,a=7\\text{\/}2\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833024298\"><div data-type=\"problem\" id=\"fs-id1167829589900\"><p id=\"fs-id1167829589902\">\\(\\left(5b+1\\right)\\left(6b+1\\right)=0\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836623729\"><div data-type=\"problem\" id=\"fs-id1167836623731\"><p id=\"fs-id1167836798088\">\\(6m\\left(12m-5\\right)=0\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836625460\"><p id=\"fs-id1167836525375\">\\(m=0,m=5\\text{\/}12\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836504078\"><div data-type=\"problem\" id=\"fs-id1167836504080\"><p id=\"fs-id1167836626599\">\\(2x\\left(6x-3\\right)=0\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833082106\"><div data-type=\"problem\" id=\"fs-id1167833082108\"><p id=\"fs-id1167832984060\">\\({\\left(2x-1\\right)}^{2}=0\\)<\/p><\/div><div data-type=\"solution\"><p>\\(x=1\\text{\/}2\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829741712\"><div data-type=\"problem\" id=\"fs-id1167829741714\"><p id=\"fs-id1167836620973\">\\({\\left(3y+5\\right)}^{2}=0\\)<\/p><\/div><\/div><p id=\"fs-id1167829627758\"><strong data-effect=\"bold\">Solve Quadratic Equations by Factoring<\/strong><\/p><p id=\"fs-id1167836288545\">In the following exercises, solve.<\/p><div data-type=\"exercise\" id=\"fs-id1167836529760\"><div data-type=\"problem\" id=\"fs-id1167836529763\"><p id=\"fs-id1167824585340\">\\(5{a}^{2}-26a=24\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836449436\"><p id=\"fs-id1167829579647\">\\(a=-5\\text{\/}4,a=6\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836635232\"><div data-type=\"problem\" id=\"fs-id1167836511402\"><p id=\"fs-id1167836511404\">\\(4{b}^{2}+7b=-3\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829717788\"><div data-type=\"problem\" id=\"fs-id1167836408890\"><p id=\"fs-id1167836549429\">\\(4{m}^{2}=17m-15\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167824732278\"><p id=\"fs-id1167829715419\">\\(m=5\\text{\/}4,m=3\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836318326\"><div data-type=\"problem\" id=\"fs-id1167836318328\"><p id=\"fs-id1167833381336\">\\({n}^{2}=5-6n\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829715838\"><div data-type=\"problem\" id=\"fs-id1167836667079\"><p id=\"fs-id1167836667081\">\\(7{a}^{2}+14a=7a\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836662585\"><p id=\"fs-id1167836503843\">\\(a=-1,a=0\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829599736\"><div data-type=\"problem\" id=\"fs-id1167829599738\"><p id=\"fs-id1167829930662\">\\(12{b}^{2}-15b=-9b\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833386007\"><div data-type=\"problem\" id=\"fs-id1167833386009\"><p id=\"fs-id1167833022584\">\\(49{m}^{2}=144\\)<\/p><\/div><div data-type=\"solution\"><p id=\"fs-id1167829747887\">\\(m=12\\text{\/}7,m=-12\\text{\/}7\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836296256\"><div data-type=\"problem\" id=\"fs-id1167836296259\"><p id=\"fs-id1167829744476\">\\(625={x}^{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836543476\"><div data-type=\"problem\" id=\"fs-id1167836543478\"><p id=\"fs-id1167836616320\">\\(16{y}^{2}=81\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836546088\"><p id=\"fs-id1167836546090\">\\(y=-9\\text{\/}4,y=9\\text{\/}4\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836526027\"><div data-type=\"problem\" id=\"fs-id1167829693190\"><p id=\"fs-id1167829693192\">\\(64{p}^{2}=225\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836386707\"><div data-type=\"problem\" id=\"fs-id1167836386709\"><p id=\"fs-id1167833017831\">\\(121{n}^{2}=36\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829748023\"><p id=\"fs-id1167829748025\">\\(n=-6\\text{\/}11,\\text{n}=6\\text{\/}11\\)<\/p><\/div><\/div><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167836524654\"><p id=\"fs-id1167836524656\">\\(100{y}^{2}=9\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833385717\"><div data-type=\"problem\"><p id=\"fs-id1167836597807\">\\(\\left(x+6\\right)\\left(x-3\\right)=-8\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836611503\"><p id=\"fs-id1167833022610\">\\(x=2,x=-5\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829832924\"><div data-type=\"problem\"><p id=\"fs-id1167836292046\">\\(\\left(p-5\\right)\\left(p+3\\right)=-7\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829739370\"><div data-type=\"problem\" id=\"fs-id1167833057313\"><p id=\"fs-id1167833057315\">\\(\\left(2x+1\\right)\\left(x-3\\right)=-4x\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836487200\"><p id=\"fs-id1167836487202\">\\(x=3\\text{\/}2,x=-1\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833181233\"><div data-type=\"problem\" id=\"fs-id1167836299581\"><p id=\"fs-id1167836299583\">\\(\\left(y-3\\right)\\left(y+2\\right)=4y\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836706420\"><div data-type=\"problem\" id=\"fs-id1167833085272\"><p id=\"fs-id1167833085274\">\\(\\left(3x-2\\right)\\left(x+4\\right)=12x\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836357235\"><p id=\"fs-id1167836357237\">\\(x=3\\text{\/}2,x=-1\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836362558\"><div data-type=\"problem\" id=\"fs-id1167836704677\"><p id=\"fs-id1167836704679\">\\(\\left(2y-3\\right)\\left(3y-1\\right)=8y\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833046960\"><div data-type=\"problem\" id=\"fs-id1167836626547\"><p id=\"fs-id1167836626549\">\\(20{x}^{2}-60x=-45\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836504055\"><p id=\"fs-id1167836504057\">\\(x=-2\\text{\/}3\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836450605\"><div data-type=\"problem\" id=\"fs-id1167836340606\"><p id=\"fs-id1167836340608\">\\(3{y}^{2}-18y=-27\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836493013\"><div data-type=\"problem\" id=\"fs-id1167829789955\"><p id=\"fs-id1167829789957\">\\(15{x}^{2}-10x=40\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836529136\"><p id=\"fs-id1167836529138\">\\(x=2,x=-4\\text{\/}3\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836704609\"><div data-type=\"problem\" id=\"fs-id1167833349767\"><p id=\"fs-id1167833349769\">\\(14{y}^{2}-77y=-35\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836407331\"><div data-type=\"problem\" id=\"fs-id1167836319739\"><p id=\"fs-id1167836319741\">\\(18{x}^{2}-9=-21x\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836424061\"><p>\\(x=-3\\text{\/}2,x=1\\text{\/}3\\)<\/p><\/div><\/div><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167836312383\"><p id=\"fs-id1167836312385\">\\(16{y}^{2}+12=-32y\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836286867\"><div data-type=\"problem\" id=\"fs-id1167836329708\"><p id=\"fs-id1167836329710\">\\(16{p}^{3}=24{p}^{2}+9p\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836493553\"><p id=\"fs-id1167829715616\">\\(p=0,p=\u00be\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829595273\"><div data-type=\"problem\" id=\"fs-id1167829595275\"><p id=\"fs-id1167836298187\">\\({m}^{3}-2{m}^{2}=\\text{\u2212}m\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829790573\"><div data-type=\"problem\" id=\"fs-id1167836329438\"><p id=\"fs-id1167836329440\">\\(2{x}^{3}+72x=24{x}^{2}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836574901\"><p id=\"fs-id1167836295294\">\\(x=0,x=6\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836300282\"><div data-type=\"problem\" id=\"fs-id1167836300285\"><p id=\"fs-id1167836515605\">\\(3{y}^{3}+48y=24{y}^{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836439638\"><div data-type=\"problem\" id=\"fs-id1167836439640\"><p id=\"fs-id1167829811262\">\\(36{x}^{3}+24{x}^{2}=-4x\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167824732569\"><p id=\"fs-id1167824732571\">\\(x=0,x=1\\text{\/}3\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836529024\"><div data-type=\"problem\" id=\"fs-id1167829695577\"><p id=\"fs-id1167829695579\">\\(2{y}^{3}+2{y}^{2}=12y\\)<\/p><\/div><\/div><p id=\"fs-id1167836728266\"><strong data-effect=\"bold\">Solve Equations with Polynomial Functions<\/strong><\/p><p id=\"fs-id1167836377496\">In the following exercises, solve.<\/p><div data-type=\"exercise\" id=\"fs-id1167836377499\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167833053850\"><p id=\"fs-id1167833053852\">For the function, \\(f\\left(x\\right)={x}^{2}-8x+8,\\) <span class=\"token\">\u24d0<\/span> find when \\(f\\left(x\\right)=-4\\) <span class=\"token\">\u24d1<\/span> Use this information to find two points that lie on the graph of the function.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836433034\"><p id=\"fs-id1167829744445\"><span class=\"token\">\u24d0<\/span>\\(x=2\\) or \\(x=6\\) <span class=\"token\">\u24d1<\/span> \\(\\left(2,-4\\right)\\) \\(\\left(6,-4\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836311497\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836310449\"><p id=\"fs-id1167836310451\">For the function, \\(f\\left(x\\right)={x}^{2}+11x+20,\\) <span class=\"token\">\u24d0<\/span> find when \\(f\\left(x\\right)=-8\\) <span class=\"token\">\u24d1<\/span> Use this information to find two points that lie on the graph of the function.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836535278\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836282991\"><p id=\"fs-id1167836282993\">For the function, \\(f\\left(x\\right)=8{x}^{2}-18x+5,\\) <span class=\"token\">\u24d0<\/span> find when \\(f\\left(x\\right)=-4\\) <span class=\"token\">\u24d1<\/span> Use this information to find two points that lie on the graph of the function.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836444808\"><p id=\"fs-id1167836444810\"><span class=\"token\">\u24d0<\/span>\\(x=\\frac{3}{2}\\) or \\(x=\\frac{3}{4}\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> \\(\\left(\\frac{3}{2},-4\\right)\\) \\(\\left(\\frac{3}{4},-4\\right)\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836455830\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836455832\"><p id=\"fs-id1167836526816\">For the function, \\(f\\left(x\\right)=18{x}^{2}+15x-10,\\) <span class=\"token\">\u24d0<\/span> find when \\(f\\left(x\\right)=15\\) <span class=\"token\">\u24d1<\/span> Use this information to find two points that lie on the graph of the function.<\/p><\/div><\/div><p id=\"fs-id1167836417930\">In the following exercises, for each function, find: <span class=\"token\">\u24d0<\/span> the zeros of the function <span class=\"token\">\u24d1<\/span> the <em data-effect=\"italics\">x<\/em>-intercepts of the graph of the function <span class=\"token\">\u24d2<\/span> the <em data-effect=\"italics\">y<\/em>-intercept of the graph of the function.<\/p><div data-type=\"exercise\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836595837\"><p id=\"fs-id1167836595839\">\\(f\\left(x\\right)=9{x}^{2}-4\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836717499\"><p id=\"fs-id1167836717501\"><span class=\"token\">\u24d0<\/span>\\(x=\\frac{2}{3}\\) or \\(x=-\\frac{2}{3}\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> \\(\\left(\\frac{2}{3},0\\right)\\), \\(\\left(-\\frac{2}{3},0\\right)\\) <span class=\"token\">\u24d2<\/span> \\(\\left(0,-4\\right)\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829739808\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167829739810\"><p id=\"fs-id1167829739812\">\\(f\\left(x\\right)=25{x}^{2}-49\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836407732\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836407734\"><p id=\"fs-id1167829692537\">\\(f\\left(x\\right)=6{x}^{2}-7x-5\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829596938\"><p id=\"fs-id1167833263514\"><span class=\"token\">\u24d0<\/span>\\(x=\\frac{5}{3}\\) or \\(x=-\\frac{1}{2}\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> \\(\\left(\\frac{5}{3},0\\right)\\), \\(\\left(-\\frac{1}{2},0\\right)\\) <span class=\"token\">\u24d2<\/span> \\(\\left(0,-5\\right)\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836524185\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836524188\"><p id=\"fs-id1167836524190\">\\(f\\left(x\\right)=12{x}^{2}-11x+2\\)<\/p><\/div><\/div><p id=\"fs-id1167829984109\"><strong data-effect=\"bold\">Solve Applications Modeled by Quadratic Equations<\/strong><\/p><p id=\"fs-id1167833086901\">In the following exercises, solve.<\/p><div data-type=\"exercise\" id=\"fs-id1167829908175\" class=\"material-set-2\"><div data-type=\"problem\"><p id=\"fs-id1167836536765\">The product of two consecutive odd integers is 143. Find the integers.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836542378\"><p>\\(-13,-11\\) and 11, 13<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833057832\" class=\"material-set-2\"><div data-type=\"problem\"><p id=\"fs-id1167833057836\">The product of two consecutive odd integers is 195. Find the integers.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829712386\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167826171959\"><p id=\"fs-id1167826171961\">The product of two consecutive even integers is 168. Find the integers.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829586760\"><p id=\"fs-id1167829586762\">\\(-14,-12\\) and 12, 14<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832940587\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167832940589\"><p id=\"fs-id1167836625236\">The product of two consecutive even integers is 288. Find the integers.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833311290\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167829579684\"><p id=\"fs-id1167829579686\">The area of a rectangular carpet is 28 square feet. The length is three feet more than the width. Find the length and the width of the carpet.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833086705\"><p id=\"fs-id1167833086707\">\\(-4\\) and 7<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836288744\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836288746\"><p id=\"fs-id1167836288748\">A rectangular retaining wall has area 15 square feet. The height of the wall is two feet less than its length. Find the height and the length of the wall.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836432337\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836311198\"><p id=\"fs-id1167836311201\">The area of a bulletin board is 55 square feet. The length is four feet less than three times the width. Find the length and the width of the a bulletin board.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833381374\"><p id=\"fs-id1167833381376\">5, 11<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836294735\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836294737\"><p id=\"fs-id1167833050836\">A rectangular carport has area 150 square feet. The height of the carport is five feet less than twice its length. Find the height and the length of the carport.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833050541\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167833050543\"><p id=\"fs-id1167836282646\">A pennant is shaped like a right triangle, with hypotenuse 10 feet. The length of one side of the pennant is two feet longer than the length of the other side. Find the length of the two sides of the pennant.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829746946\"><p id=\"fs-id1167829746948\">6, 8<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836683998\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836684001\"><p id=\"fs-id1167836684003\">A stained glass window is shaped like a right triangle. The hypotenuse is 15 feet. One leg is three more than the other. Find the lengths of the legs.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829589871\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167829589873\"><p id=\"fs-id1167829589875\">A reflecting pool is shaped like a right triangle, with one leg along the wall of a building. The hypotenuse is 9 feet longer than the side along the building. The third side is 7 feet longer than the side along the building. Find the lengths of all three sides of the reflecting pool.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836322358\"><p id=\"fs-id1167836706550\">8, 15, 17<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833024496\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167833024499\"><p id=\"fs-id1167833024501\">A goat enclosure is in the shape of a right triangle. One leg of the enclosure is built against the side of the barn. The other leg is 4 feet more than the leg against the barn. The hypotenuse is 8 feet more than the leg along the barn. Find the three sides of the goat enclosure.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836610606\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836610609\"><p id=\"fs-id1167836610611\">Juli is going to launch a model rocket in her back yard. When she launches the rocket, the function \\(h\\left(t\\right)=-16{t}^{2}+32t\\) models the height, <em data-effect=\"italics\">h<\/em>, of the rocket above the ground as a function of time, <em data-effect=\"italics\">t<\/em>. Find:<\/p><p id=\"fs-id1167829716526\"><span class=\"token\">\u24d0<\/span> the zeros of this function which tells us when the rocket will hit the ground. <span class=\"token\">\u24d1<\/span> the time the rocket will be 16 feet above the ground.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836433972\"><p id=\"fs-id1167836543376\"><span class=\"token\">\u24d0<\/span> 0, 2 <span class=\"token\">\u24d1<\/span> 1<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829747436\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167829747439\"><p id=\"fs-id1167829747441\">Gianna is going to throw a ball from the top floor of her middle school. When she throws the ball from 48 feet above the ground, the function \\(h\\left(t\\right)=-16{t}^{2}+32t+48\\) models the height, <em data-effect=\"italics\">h<\/em>, of the ball above the ground as a function of time, <em data-effect=\"italics\">t<\/em>. Find:<\/p><p id=\"fs-id1167836700379\"><span class=\"token\">\u24d0<\/span> the zeros of this function which tells us when the ball will hit the ground. <span class=\"token\">\u24d1<\/span> the time(s) the ball will be 48 feet above the ground. <span class=\"token\">\u24d2<\/span> the height the ball will be at \\(t=1\\) seconds which is when the ball will be at its highest point.<\/p><\/div><\/div><\/div><div class=\"writing\" data-depth=\"2\" id=\"fs-id1167836362679\"><h4 data-type=\"title\">Writing Exercises<\/h4><div data-type=\"exercise\" id=\"fs-id1167836519198\"><div data-type=\"problem\" id=\"fs-id1167836519200\"><p id=\"fs-id1167833054701\">Explain how you solve a quadratic equation. How many answers do you expect to get for a quadratic equation?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833054706\"><p id=\"fs-id1167836706782\">Answers will vary.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836551135\"><div data-type=\"problem\" id=\"fs-id1167836551137\"><p id=\"fs-id1167836551139\">Give an example of a quadratic equation that has a GCF and none of the solutions to the equation is zero.<\/p><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167836282836\"><h4 data-type=\"title\">Self Check<\/h4><p id=\"fs-id1167836321166\"><span class=\"token\">\u24d0<\/span> After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.<\/p><span data-type=\"media\" id=\"fs-id1167833272086\" data-alt=\"This table has 4 columns, 3 rows and a header row. The header row labels each column: I can, confidently, with some help and no, I don\u2019t get it. The first column has the following statements: solve quadratic equations by using the zero product property, solve quadratic equations by factoring and solve applications modeled by quadratic equations.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_05_201_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This table has 4 columns, 3 rows and a header row. The header row labels each column: I can, confidently, with some help and no, I don\u2019t get it. The first column has the following statements: solve quadratic equations by using the zero product property, solve quadratic equations by factoring and solve applications modeled by quadratic equations.\"><\/span><p id=\"fs-id1167836612849\"><span class=\"token\">\u24d1<\/span> Overall, after looking at the checklist, do you think you are well-prepared for the next section? Why or why not?<\/p><\/div><\/div><div class=\"review-exercises\" data-depth=\"1\" id=\"fs-id1167829614395\"><h3 data-type=\"title\">Chapter Review Exercises<\/h3><div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167836662547\"><h4 data-type=\"title\"><a href=\"\/contents\/b380a9f9-f0f7-4fdf-9a79-5646a9f745f9\" class=\"target-chapter\">Greatest Common Factor and Factor by Grouping<\/a><\/h4><p><strong data-effect=\"bold\">Find the Greatest Common Factor of Two or More Expressions<\/strong><\/p><p id=\"fs-id1167833060062\">In the following exercises, find the greatest common factor.<\/p><div data-type=\"exercise\" id=\"fs-id1167833060065\"><div data-type=\"problem\" id=\"fs-id1167833060067\"><p id=\"fs-id1167836521832\">\\(12{a}^{2}{b}^{3},15a{b}^{2}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167824617105\"><p id=\"fs-id1167836399826\">\\(3a{b}^{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836455876\"><div data-type=\"problem\" id=\"fs-id1167836349187\"><p id=\"fs-id1167836349189\">\\(12{m}^{2}{n}^{3},42{m}^{5}{n}^{3}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836732282\"><div data-type=\"problem\" id=\"fs-id1167833339509\"><p id=\"fs-id1167833339511\">\\(15{y}^{3},21{y}^{2},30y\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829752740\"><p id=\"fs-id1167829752742\">\\(3y\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836628598\"><div data-type=\"problem\" id=\"fs-id1167836628600\"><p id=\"fs-id1167836628602\">\\(45{x}^{3}{y}^{2},15{x}^{4}y,10{x}^{5}{y}^{3}\\)<\/p><\/div><\/div><p id=\"fs-id1167829645054\"><strong data-effect=\"bold\">Factor the Greatest Common Factor from a Polynomial<\/strong><\/p><p id=\"fs-id1167836321656\">In the following exercises, factor the greatest common factor from each polynomial.<\/p><div data-type=\"exercise\" id=\"fs-id1167836321659\"><div data-type=\"problem\" id=\"fs-id1167829740287\"><p id=\"fs-id1167829740289\">\\(35y+84\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836529086\"><p id=\"fs-id1167836665130\">\\(7\\left(5y+12\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829715304\"><div data-type=\"problem\" id=\"fs-id1167829585662\"><p id=\"fs-id1167829585664\">\\(6{y}^{2}+12y-6\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836693737\"><div data-type=\"problem\" id=\"fs-id1167836693739\"><p id=\"fs-id1167836693741\">\\(18{x}^{3}-15x\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836613926\"><p id=\"fs-id1167824732294\">\\(3x\\left(6{x}^{2}-5\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836525208\"><div data-type=\"problem\" id=\"fs-id1167832925972\"><p id=\"fs-id1167832925974\">\\(15{m}^{4}+6{m}^{2}n\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836532823\"><div data-type=\"problem\" id=\"fs-id1167829861895\"><p id=\"fs-id1167829861897\">\\(4{x}^{3}-12{x}^{2}+16x\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829620341\"><p id=\"fs-id1167829620344\">\\(4x\\left({x}^{2}-3x+4\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829738393\"><div data-type=\"problem\" id=\"fs-id1167829738395\"><p id=\"fs-id1167829720761\">\\(-3x+24\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829860032\"><div data-type=\"problem\" id=\"fs-id1167829860034\"><p id=\"fs-id1167829860036\">\\(-3{x}^{3}+27{x}^{2}-12x\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829718550\"><p id=\"fs-id1167829718552\">\\(-3x\\left({x}^{2}-9x+4\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836327058\"><div data-type=\"problem\" id=\"fs-id1167836389930\"><p id=\"fs-id1167836389932\">\\(3x\\left(x-1\\right)+\\phantom{\\rule{0.2em}{0ex}}5\\left(x-1\\right)\\)<\/p><\/div><\/div><p id=\"fs-id1167836620104\"><strong data-effect=\"bold\">Factor by Grouping<\/strong><\/p><p id=\"fs-id1167836621172\">In the following exercises, factor by grouping.<\/p><div data-type=\"exercise\" id=\"fs-id1167836621175\"><div data-type=\"problem\" id=\"fs-id1167829877871\"><p>\\(ax-ay+bx-by\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836318397\"><p id=\"fs-id1167836318399\">\\(\\left(a+b\\right)\\left(x-y\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836511579\"><div data-type=\"problem\" id=\"fs-id1167836511581\"><p id=\"fs-id1167836293668\">\\({x}^{2}y-x{y}^{2}+2x-2y\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836544551\"><div data-type=\"problem\" id=\"fs-id1167836544553\"><p id=\"fs-id1167836544555\">\\({x}^{2}+7x-3x-21\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836524290\"><p id=\"fs-id1167836524292\">\\(\\left(x-3\\right)\\left(x+7\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833059863\"><div data-type=\"problem\" id=\"fs-id1167833059865\"><p id=\"fs-id1167836547604\">\\(4{x}^{2}-16x+3x-12\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836554999\"><div data-type=\"problem\" id=\"fs-id1167836555001\"><p id=\"fs-id1167836511374\">\\({m}^{3}+{m}^{2}+m+1\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829717351\"><p id=\"fs-id1167829717353\">\\(\\left({m}^{2}+1\\right)\\left(m+1\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836310151\"><div data-type=\"problem\" id=\"fs-id1167836310153\"><p id=\"fs-id1167836310155\">\\(5x-5y-y+x\\)<\/p><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167836299806\"><h4 data-type=\"title\"><a href=\"\/contents\/edd9c403-9825-4cf2-b237-2e05552ea3ec\" class=\"target-chapter\">Factor Trinomials<\/a><\/h4><p id=\"fs-id1167836520946\"><strong data-effect=\"bold\">Factor Trinomials of the Form \\({x}^{2}+bx+c\\)<\/strong><\/p><p id=\"fs-id1167829732082\">In the following exercises, factor each trinomial of the form \\({x}^{2}+bx+c.\\)<\/p><div data-type=\"exercise\" id=\"fs-id1167836663209\"><div data-type=\"problem\" id=\"fs-id1167826131854\"><p id=\"fs-id1167826131856\">\\({a}^{2}+14a+33\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829694283\"><p id=\"fs-id1167836322552\">\\(\\left(a+3\\right)\\left(a+11\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833135499\"><div data-type=\"problem\" id=\"fs-id1167833135501\"><p id=\"fs-id1167833135503\">\\({k}^{2}-16k+60\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836289089\"><div data-type=\"problem\" id=\"fs-id1167836289091\"><p id=\"fs-id1167836415377\">\\({m}^{2}+3m-54\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836315108\"><p id=\"fs-id1167836315110\">\\(\\left(m+9\\right)\\left(m-6\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829713135\"><div data-type=\"problem\"><p id=\"fs-id1167829595235\">\\({x}^{2}-3x-10\\)<\/p><\/div><\/div><p id=\"fs-id1167836419022\">In the following examples, factor each trinomial of the form \\({x}^{2}+bxy+c{y}^{2}.\\)<\/p><div data-type=\"exercise\" id=\"fs-id1167836363319\"><div data-type=\"problem\"><p>\\({x}^{2}+12xy+35{y}^{2}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829599193\"><p id=\"fs-id1167829599195\">\\(\\left(x+5y\\right)\\left(x+7y\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836481436\"><div data-type=\"problem\" id=\"fs-id1167836481439\"><p>\\({r}^{2}+3rs-28{s}^{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836660098\"><div data-type=\"problem\" id=\"fs-id1167836660100\"><p id=\"fs-id1167836660102\">\\({a}^{2}+4ab-21{b}^{2}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829789646\"><p id=\"fs-id1167829789648\">\\(\\left(a+7b\\right)\\left(a-3b\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836533387\"><div data-type=\"problem\" id=\"fs-id1167836533389\"><p id=\"fs-id1167833017936\">\\({p}^{2}-5pq-36{q}^{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833279905\"><div data-type=\"problem\" id=\"fs-id1167833279907\"><p id=\"fs-id1167833019257\">\\({m}^{2}-5mn+30{n}^{2}\\)<\/p><\/div><div data-type=\"solution\"><p id=\"fs-id1167836387220\">Prime<\/p><\/div><\/div><p id=\"fs-id1167836387226\"><strong data-effect=\"bold\">Factor Trinomials of the Form \\(a{x}^{2}+bx+c\\) Using Trial and Error<\/strong><\/p><p id=\"fs-id1167829720901\">In the following exercises, factor completely using trial and error.<\/p><div data-type=\"exercise\" id=\"fs-id1167829720904\"><div data-type=\"problem\" id=\"fs-id1167829720906\"><p id=\"fs-id1167836513956\">\\({x}^{3}+5{x}^{2}-24x\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829745235\"><div data-type=\"problem\" id=\"fs-id1167829745237\"><p id=\"fs-id1167829745239\">\\(3{y}^{3}-21{y}^{2}+30y\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836640498\"><p id=\"fs-id1167836640500\">\\(3y\\left(y-5\\right)\\left(y-2\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836355740\"><div data-type=\"problem\" id=\"fs-id1167836355743\"><p id=\"fs-id1167836355745\">\\(5{x}^{4}+10{x}^{3}-75{x}^{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167824732243\"><div data-type=\"problem\" id=\"fs-id1167824732246\"><p id=\"fs-id1167836700642\">\\(5{y}^{2}+14y+9\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832936381\"><p id=\"fs-id1167836613060\">\\(\\left(5y+9\\right)\\left(y+1\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829596060\"><div data-type=\"problem\" id=\"fs-id1167829596063\"><p id=\"fs-id1167829595434\">\\(8{x}^{2}+25x+3\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836282761\"><div data-type=\"problem\" id=\"fs-id1167836282763\"><p id=\"fs-id1167836652632\">\\(10{y}^{2}-53y-11\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836698812\"><p id=\"fs-id1167829740760\">\\(\\left(5y+1\\right)\\left(2y-11\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829811185\"><div data-type=\"problem\" id=\"fs-id1167836294851\"><p id=\"fs-id1167836294853\">\\(6{p}^{2}-19pq+10{q}^{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829789212\"><div data-type=\"problem\" id=\"fs-id1167836597114\"><p id=\"fs-id1167836597116\">\\(-81{a}^{2}+153a+18\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836513741\"><p id=\"fs-id1167836513743\">\\(-9\\left(9a-1\\right)\\left(a+2\\right)\\)<\/p><\/div><\/div><p id=\"fs-id1167829695659\"><strong data-effect=\"bold\">Factor Trinomials of the Form \\(a{x}^{2}+bx+c\\) using the \u2018ac\u2019 Method<\/strong><\/p><p id=\"fs-id1167836295302\">In the following exercises, factor.<\/p><div data-type=\"exercise\" id=\"fs-id1167836295305\"><div data-type=\"problem\" id=\"fs-id1167836295606\"><p id=\"fs-id1167836295608\">\\(2{x}^{2}+9x+4\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836628901\"><div data-type=\"problem\" id=\"fs-id1167836314158\"><p id=\"fs-id1167836314160\">\\(18{a}^{2}-9a+1\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836512081\"><p id=\"fs-id1167836512084\">\\(\\left(3a-1\\right)\\left(6a-1\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836620731\"><div data-type=\"problem\" id=\"fs-id1167836620733\"><p id=\"fs-id1167836620735\">\\(15{p}^{2}+2p-8\\)<\/p><\/div><\/div><div data-type=\"exercise\"><div data-type=\"problem\"><p id=\"fs-id1167829748808\">\\(15{x}^{2}+6x-2\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836537191\"><p id=\"fs-id1167832945806\">\\(\\left(3x-1\\right)\\left(5x+2\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836688452\"><div data-type=\"problem\" id=\"fs-id1167836399698\"><p id=\"fs-id1167836399700\">\\(8{a}^{2}+32a+24\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836297149\"><div data-type=\"problem\" id=\"fs-id1167836544134\"><p id=\"fs-id1167836544136\">\\(3{x}^{2}+3x-36\\)<\/p><\/div><div data-type=\"solution\"><p id=\"fs-id1167833018721\">\\(3\\left(x+4\\right)\\left(x-3\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836613589\"><div data-type=\"problem\" id=\"fs-id1167836509003\"><p id=\"fs-id1167836509005\">\\(48{y}^{2}+12y-36\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836510934\"><div data-type=\"problem\" id=\"fs-id1167836510936\"><p id=\"fs-id1167836510938\">\\(18{a}^{2}-57a-21\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833327203\"><p id=\"fs-id1167833327206\">\\(3\\left(2a-7\\right)\\left(3a+1\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167829894397\"><p id=\"fs-id1167829894399\">\\(3{n}^{4}-12{n}^{3}-96{n}^{2}\\)<\/p><\/div><\/div><p id=\"fs-id1167836747825\"><strong data-effect=\"bold\">Factor using substitution<\/strong><\/p><p id=\"fs-id1167829787750\">In the following exercises, factor using substitution.<\/p><div data-type=\"exercise\" id=\"fs-id1167829787754\"><div data-type=\"problem\" id=\"fs-id1167829787756\"><p id=\"fs-id1167836485811\">\\({x}^{4}-13{x}^{2}-30\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836334488\"><p id=\"fs-id1167836334490\">\\(\\left({x}^{2}-15\\right)\\left({x}^{2}+2\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833082476\"><div data-type=\"problem\" id=\"fs-id1167833082478\"><p id=\"fs-id1167833082480\">\\({\\left(x-3\\right)}^{2}-5\\left(x-3\\right)-36\\)<\/p><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167833055815\"><h4 data-type=\"title\"><a href=\"\/contents\/d844a3e4-0163-4936-91ca-a71142f07358\" class=\"target-chapter\">Factor Special Products<\/a><\/h4><p id=\"fs-id1167836331150\"><strong data-effect=\"bold\">Factor Perfect Square Trinomials<\/strong><\/p><p id=\"fs-id1167836701146\">In the following exercises, factor completely using the perfect square trinomials pattern.<\/p><div data-type=\"exercise\" id=\"fs-id1167833059265\"><div data-type=\"problem\" id=\"fs-id1167833059267\"><p id=\"fs-id1167833059269\">\\(25{x}^{2}+30x+9\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836539601\"><p id=\"fs-id1167836539603\">\\({\\left(5x+3\\right)}^{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833057727\"><div data-type=\"problem\" id=\"fs-id1167836697611\"><p id=\"fs-id1167836697613\">\\(36{a}^{2}-84ab+49{b}^{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829743405\"><div data-type=\"problem\" id=\"fs-id1167829743407\"><p id=\"fs-id1167829743409\">\\(40{x}^{2}+360x+810\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836602252\"><p id=\"fs-id1167836602254\">\\(10{\\left(2x+9\\right)}^{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836282423\"><div data-type=\"problem\" id=\"fs-id1167836282425\"><p id=\"fs-id1167836282427\">\\(5{k}^{3}-70{k}^{2}+245k\\)<\/p><\/div><\/div><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167836531526\"><p id=\"fs-id1167836531528\">\\(75{u}^{4}-30{u}^{3}v+3{u}^{2}{v}^{2}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836551978\"><p id=\"fs-id1167836494113\">\\(3{u}^{2}{\\left(5u-v\\right)}^{2}\\)<\/p><\/div><\/div><p id=\"fs-id1167833086731\"><strong data-effect=\"bold\">Factor Differences of Squares<\/strong><\/p><p id=\"fs-id1167833224512\">In the following exercises, factor completely using the difference of squares pattern, if possible.<\/p><div data-type=\"exercise\" id=\"fs-id1167833224516\"><div data-type=\"problem\" id=\"fs-id1167836597153\"><p id=\"fs-id1167836597155\">\\(81{r}^{2}-25\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833224386\"><div data-type=\"problem\" id=\"fs-id1167833224388\"><p id=\"fs-id1167833224390\">\\(169{m}^{2}-{n}^{2}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836357359\"><p id=\"fs-id1167836357362\">\\(\\left(13m+n\\right)\\left(13m-n\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833047487\"><div data-type=\"problem\" id=\"fs-id1167833047489\"><p>\\(25{p}^{2}-1\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836546848\"><div data-type=\"problem\"><p id=\"fs-id1167829719631\">\\(9-121{y}^{2}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829811950\"><p id=\"fs-id1167836556695\">\\(\\left(3+11y\\right)\\left(3-11y\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836492382\"><div data-type=\"problem\" id=\"fs-id1167829829336\"><p id=\"fs-id1167829829339\">\\(20{x}^{2}-125\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836543391\"><div data-type=\"problem\" id=\"fs-id1167836543393\"><p id=\"fs-id1167833009345\">\\(169{n}^{3}-n\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836321020\"><p id=\"fs-id1167836542091\">\\(n\\left(13n+1\\right)\\left(13n-1\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836548731\"><div data-type=\"problem\" id=\"fs-id1167836548733\"><p id=\"fs-id1167836548735\">\\(6{p}^{2}{q}^{2}-54{p}^{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836524842\"><div data-type=\"problem\" id=\"fs-id1167836524845\"><p id=\"fs-id1167836524847\">\\(24{p}^{2}+54\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836625184\"><p id=\"fs-id1167836625186\">\\(6\\left(4{p}^{2}+9\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829721107\"><div data-type=\"problem\" id=\"fs-id1167829721109\"><p id=\"fs-id1167829721111\">\\(49{x}^{2}-81{y}^{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836546672\"><div data-type=\"problem\" id=\"fs-id1167833059205\"><p id=\"fs-id1167833059208\">\\(16{z}^{4}-1\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836533874\"><p id=\"fs-id1167836533876\">\\(\\left(2z-1\\right)\\left(2z+1\\right)\\left(4{z}^{2}+1\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836309880\"><div data-type=\"problem\" id=\"fs-id1167836309882\"><p>\\(48{m}^{4}{n}^{2}-243{n}^{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836548892\"><div data-type=\"problem\" id=\"fs-id1167836548895\"><p id=\"fs-id1167836548897\">\\({a}^{2}+6a+9-9{b}^{2}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833018367\"><p id=\"fs-id1167833018369\">\\(\\left(a+3-3b\\right)\\left(a+3+3b\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836688854\"><div data-type=\"problem\" id=\"fs-id1167836360981\"><p id=\"fs-id1167836360983\">\\({x}^{2}-16x+64-{y}^{2}\\)<\/p><\/div><\/div><p id=\"fs-id1167836545257\"><strong data-effect=\"bold\">Factor Sums and Differences of Cubes<\/strong><\/p><p id=\"fs-id1167836547592\">In the following exercises, factor completely using the sums and differences of cubes pattern, if possible.<\/p><div data-type=\"exercise\" id=\"fs-id1167833191672\"><div data-type=\"problem\" id=\"fs-id1167833191674\"><p id=\"fs-id1167836322156\">\\({a}^{3}-125\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836574949\"><p id=\"fs-id1167836574951\">\\(\\left(a-5\\right)\\left({a}^{2}+5a+25\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836522701\"><div data-type=\"problem\" id=\"fs-id1167836522703\"><p id=\"fs-id1167836522705\">\\({b}^{3}-216\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836602902\"><div data-type=\"problem\" id=\"fs-id1167836602904\"><p id=\"fs-id1167836602906\">\\(2{m}^{3}+54\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836613722\"><p id=\"fs-id1167836613724\">\\(2\\left(m+3\\right)\\left({m}^{2}-3m+9\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836541992\"><div data-type=\"problem\" id=\"fs-id1167836541994\"><p id=\"fs-id1167836289221\">\\(81{m}^{3}+3\\)<\/p><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167836417911\"><h4 data-type=\"title\"><a href=\"\/contents\/93db7208-2f30-47df-91eb-33984d9fb8b7\" class=\"target-chapter\">General Strategy for Factoring Polynomials<\/a><\/h4><p id=\"fs-id1167836578726\"><strong data-effect=\"bold\">Recognize and Use the Appropriate Method to Factor a Polynomial Completely<\/strong><\/p><p id=\"fs-id1167836508117\">In the following exercises, factor completely.<\/p><div data-type=\"exercise\" id=\"fs-id1167829594300\"><div data-type=\"problem\" id=\"fs-id1167829594303\"><p id=\"fs-id1167829594305\">\\(24{x}^{3}+44{x}^{2}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833114725\"><p id=\"fs-id1167833114727\">\\(4{x}^{2}\\left(6x+11\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833349928\"><div data-type=\"problem\" id=\"fs-id1167833349930\"><p id=\"fs-id1167833349932\">\\(24{a}^{4}-9{a}^{3}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829893749\"><div data-type=\"problem\" id=\"fs-id1167829893751\"><p id=\"fs-id1167829893753\">\\(16{n}^{2}-56mn+49{m}^{2}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836732073\"><p id=\"fs-id1167836445000\">\\({\\left(4n-7m\\right)}^{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833356111\"><div data-type=\"problem\" id=\"fs-id1167833356113\"><p id=\"fs-id1167836558483\">\\(6{a}^{2}-25a-9\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829693743\"><div data-type=\"problem\" id=\"fs-id1167829717356\"><p id=\"fs-id1167829717358\">\\(5{u}^{4}-45{u}^{2}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836283643\"><p id=\"fs-id1167833386290\">\\(5{u}^{2}\\left(u+3\\right)\\left(u-3\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829694732\"><div data-type=\"problem\" id=\"fs-id1167829694734\"><p id=\"fs-id1167829586698\">\\({n}^{4}-81\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833030958\"><div data-type=\"problem\" id=\"fs-id1167833030960\"><p id=\"fs-id1167833030963\">\\(64{j}^{2}+225\\)<\/p><\/div><div data-type=\"solution\"><p>prime<\/p><\/div><\/div><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167829690364\"><p id=\"fs-id1167829690366\">\\(5{x}^{2}+5x-60\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829742656\"><div data-type=\"problem\" id=\"fs-id1167829786977\"><p id=\"fs-id1167829786979\">\\({b}^{3}-64\\)<\/p><\/div><div data-type=\"solution\"><p id=\"fs-id1167829732135\">\\(\\left(b-4\\right)\\left({b}^{2}+4b+16\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836340027\"><div data-type=\"problem\" id=\"fs-id1167836340029\"><p id=\"fs-id1167836520449\">\\({m}^{3}+125\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833009195\"><div data-type=\"problem\" id=\"fs-id1167833009197\"><p id=\"fs-id1167833009200\">\\(2{b}^{2}-2bc+5cb-5{c}^{2}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836609387\"><p id=\"fs-id1167836510088\">\\(\\left(2b+5c\\right)\\left(b-c\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836299787\"><div data-type=\"problem\" id=\"fs-id1167836283360\"><p id=\"fs-id1167836283362\">\\(48{x}^{5}{y}^{2}-243x{y}^{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836325869\"><div data-type=\"problem\" id=\"fs-id1167836325871\"><p id=\"fs-id1167836551985\">\\(5{q}^{2}-15q-90\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836559750\"><p id=\"fs-id1167836520643\">\\(5\\left(q+3\\right)\\left(q-6\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836629042\"><div data-type=\"problem\" id=\"fs-id1167836629045\"><p id=\"fs-id1167833018074\">\\(4{u}^{5}v+4{u}^{2}{v}^{3}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832989135\"><div data-type=\"problem\" id=\"fs-id1167832989138\"><p id=\"fs-id1167832989140\">\\(10{m}^{4}-6250\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833061000\"><p id=\"fs-id1167833061002\">\\(10\\left(m-5\\right)\\left(m+5\\right)\\left({m}^{2}+25\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836612462\"><div data-type=\"problem\" id=\"fs-id1167836580067\"><p id=\"fs-id1167836580069\">\\(60{x}^{2}y-75xy+30y\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829924607\"><div data-type=\"problem\" id=\"fs-id1167829924610\"><p id=\"fs-id1167829924612\">\\(16{x}^{2}-24xy+9{y}^{2}-64\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836737880\"><p id=\"fs-id1167836737882\">\\(\\left(4x-3y+8\\right)\\left(4x-3y-8\\right)\\)<\/p><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167836522886\"><h4 data-type=\"title\"><a href=\"\/contents\/da8478b4-93bc-4919-81a1-5e3267050e7e\" class=\"target-chapter\">Polynomial Equations<\/a><\/h4><p id=\"fs-id1167836399885\"><strong data-effect=\"bold\">Use the Zero Product Property<\/strong><\/p><p id=\"fs-id1167832945786\">In the following exercises, solve.<\/p><div data-type=\"exercise\" id=\"fs-id1167832945789\"><div data-type=\"problem\" id=\"fs-id1167833059297\"><p id=\"fs-id1167833059299\">\\(\\left(a-3\\right)\\left(a+7\\right)=0\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829721041\"><div data-type=\"problem\" id=\"fs-id1167836493342\"><p id=\"fs-id1167836493344\">\\(\\left(5b+1\\right)\\left(6b+1\\right)=0\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836635208\"><p id=\"fs-id1167833338482\">\\(b=-1\\text{\/}5,b=-1\\text{\/}6\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832927596\"><div data-type=\"problem\" id=\"fs-id1167833224761\"><p id=\"fs-id1167833224763\">\\(6m\\left(12m-5\\right)=0\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836558785\"><div data-type=\"problem\" id=\"fs-id1167836558787\"><p id=\"fs-id1167836558790\">\\({\\left(2x-1\\right)}^{2}=0\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829893802\"><p id=\"fs-id1167836434155\">\\(x=1\\text{\/}2\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833197270\"><div data-type=\"problem\" id=\"fs-id1167836525640\"><p id=\"fs-id1167836525642\">\\(3m\\left(2m-5\\right)\\left(m+6\\right)=0\\)<\/p><\/div><\/div><p id=\"fs-id1167836665477\"><strong data-effect=\"bold\">Solve Quadratic Equations by Factoring<\/strong><\/p><p id=\"fs-id1167829597317\">In the following exercises, solve.<\/p><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167829627721\"><p id=\"fs-id1167829627723\">\\({x}^{2}+9x+20=0\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829614600\"><p id=\"fs-id1167829755864\">\\(x=-4,x=-5\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836417814\"><div data-type=\"problem\" id=\"fs-id1167836417816\"><p id=\"fs-id1167836439970\">\\({y}^{2}-y-72=0\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836550168\"><div data-type=\"problem\" id=\"fs-id1167836550171\"><p id=\"fs-id1167836620378\">\\(2{p}^{2}-11p=40\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836378798\"><p id=\"fs-id1167829693685\">\\(p=-\\frac{5}{2},p=8\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829624398\"><div data-type=\"problem\" id=\"fs-id1167829624400\"><p id=\"fs-id1167836318300\">\\({q}^{3}+3{q}^{2}+2q=0\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829850992\"><div data-type=\"problem\" id=\"fs-id1167829645169\"><p id=\"fs-id1167829645171\">\\(144{m}^{2}-25=0\\)<\/p><\/div><div data-type=\"solution\"><p id=\"fs-id1167836521550\">\\(m=\\frac{5}{12},m=-\\frac{5}{12}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836703866\"><div data-type=\"problem\" id=\"fs-id1167836703868\"><p id=\"fs-id1167836703870\">\\(4{n}^{2}=36\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836292807\"><div data-type=\"problem\" id=\"fs-id1167836292809\"><p id=\"fs-id1167836292811\">\\(\\left(x+6\\right)\\left(x-3\\right)=-8\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836282386\"><p id=\"fs-id1167836282388\">\\(x=2,x=-5\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836293165\"><div data-type=\"problem\" id=\"fs-id1167836293167\"><p id=\"fs-id1167836293169\">\\(\\left(3x-2\\right)\\left(x+4\\right)=12x\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836519229\"><div data-type=\"problem\" id=\"fs-id1167836519231\"><p id=\"fs-id1167836519233\">\\(16{p}^{3}=24{p}^{2}+9p\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836507317\"><p id=\"fs-id1167836507319\">\\(p=0,p=\u00be\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833059796\"><div data-type=\"problem\" id=\"fs-id1167833059798\"><p id=\"fs-id1167833059801\">\\(2{y}^{3}+2{y}^{2}=12y\\)<\/p><\/div><\/div><p id=\"fs-id1167836532602\"><strong data-effect=\"bold\">Solve Equations with Polynomial Functions<\/strong><\/p><p id=\"fs-id1167829579731\">In the following exercises, solve.<\/p><div data-type=\"exercise\" id=\"fs-id1167829579735\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836544735\"><p id=\"fs-id1167836544737\">For the function, \\(f\\left(x\\right)={x}^{2}+11x+20,\\) <span class=\"token\">\u24d0<\/span> find when \\(f\\left(x\\right)=-8\\) <span class=\"token\">\u24d1<\/span> Use this information to find two points that lie on the graph of the function.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829586777\"><p id=\"fs-id1167829586779\"><span class=\"token\">\u24d0<\/span>\\(x=-7\\) or \\(x=-4\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> \\(\\left(-7,-8\\right)\\) \\(\\left(-4,-8\\right)\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836323177\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836323179\"><p id=\"fs-id1167836323181\">For the function, \\(f\\left(x\\right)=9{x}^{2}-18x+5,\\) <span class=\"token\">\u24d0<\/span> find when \\(f\\left(x\\right)=-3\\) <span class=\"token\">\u24d1<\/span> Use this information to find two points that lie on the graph of the function.<\/p><\/div><\/div><p id=\"fs-id1167836548728\">In each function, find: <span class=\"token\">\u24d0<\/span> the zeros of the function <span class=\"token\">\u24d1<\/span> the <em data-effect=\"italics\">x<\/em>-intercepts of the graph of the function <span class=\"token\">\u24d2<\/span> the <em data-effect=\"italics\">y<\/em>-intercept of the graph of the function.<\/p><div data-type=\"exercise\" id=\"fs-id1167829716854\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167829716856\"><p id=\"fs-id1167829689062\">\\(f\\left(x\\right)=64{x}^{2}-49\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836525231\"><p id=\"fs-id1167836525233\"><span class=\"token\">\u24d0<\/span>\\(x=\\frac{7}{8}\\) or \\(x=-\\frac{7}{8}\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> \\(\\left(\\frac{7}{8},0\\right),\\) \\(\\left(-\\frac{7}{8},0\\right)\\) <span class=\"token\">\u24d2<\/span> \\(\\left(0,-49\\right)\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836690553\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836690555\"><p id=\"fs-id1167836529480\">\\(f\\left(x\\right)=6{x}^{2}-13x-5\\)<\/p><\/div><\/div><p id=\"fs-id1167836700518\"><strong data-effect=\"bold\">Solve Applications Modeled by Quadratic Equations<\/strong><\/p><p id=\"fs-id1167836704341\">In the following exercises, solve.<\/p><div data-type=\"exercise\" id=\"fs-id1167836704344\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836481188\"><p id=\"fs-id1167836481190\">The product of two consecutive numbers is 399. Find the numbers.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836627120\"><p id=\"fs-id1167836627122\">The numbers are \\(-21\\) and \\(-19\\) or 19 and 21.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836551058\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836551060\"><p id=\"fs-id1167836551062\">The area of a rectangular shaped patio 432 square feet. The length of the patio is 6 feet more than its width. Find the length and width.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836392250\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167829783696\"><p id=\"fs-id1167829783698\">A ladder leans against the wall of a building. The length of the ladder is 9 feet longer than the distance of the bottom of the ladder from the building. The distance of the top of the ladder reaches up the side of the building is 7 feet longer than the distance of the bottom of the ladder from the building. Find the lengths of all three sides of the triangle formed by the ladder leaning against the building.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836692413\"><p id=\"fs-id1167829783701\">The lengths are 8, 15, and 17 ft.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829621112\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836340204\"><p id=\"fs-id1167836340206\">Shruti is going to throw a ball from the top of a cliff. When she throws the ball from 80 feet above the ground, the function \\(h\\left(t\\right)=-16{t}^{2}+64t+80\\) models the height, <em data-effect=\"italics\">h<\/em>, of the ball above the ground as a function of time, <em data-effect=\"italics\">t<\/em>. Find: <span class=\"token\">\u24d0<\/span> the zeros of this function which tells us when the ball will hit the ground. <span class=\"token\">\u24d1<\/span> the time(s) the ball will be 80 feet above the ground. <span class=\"token\">\u24d2<\/span> the height the ball will be at \\(t=2\\) seconds which is when the ball will be at its highest point.<\/p><\/div><\/div><\/div><\/div><div class=\"practice-test\" data-depth=\"1\" id=\"fs-id1167833139379\"><h3 data-type=\"title\">Chapter Practice Test<\/h3><p id=\"fs-id1167836486354\">In the following exercises, factor completely.<\/p><div data-type=\"exercise\" id=\"fs-id1167829787150\"><div data-type=\"problem\" id=\"fs-id1167829787152\"><p id=\"fs-id1167836544360\">\\(80{a}^{2}+120{a}^{3}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829720971\"><p id=\"fs-id1167829720973\">\\(40{a}^{2}\\left(2+3a\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836528609\"><div data-type=\"problem\" id=\"fs-id1167836300253\"><p id=\"fs-id1167836300255\">\\(5m\\left(m-1\\right)+3\\left(m-1\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836493210\"><div data-type=\"problem\" id=\"fs-id1167836493212\"><p id=\"fs-id1167836493214\">\\({x}^{2}+13x+36\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836612999\"><p id=\"fs-id1167836613001\">\\(\\left(x+7\\right)\\left(x+6\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836608432\"><div data-type=\"problem\" id=\"fs-id1167829614267\"><p id=\"fs-id1167829614270\">\\({p}^{2}+pq-12{q}^{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836424174\"><div data-type=\"problem\" id=\"fs-id1167836424176\"><p id=\"fs-id1167836424178\">\\(xy-8y+7x-56\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836527655\"><p id=\"fs-id1167829651446\">\\(\\left(x-8\\right)\\left(y+7\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836423912\"><div data-type=\"problem\" id=\"fs-id1167836423914\"><p id=\"fs-id1167836423916\">\\(40{r}^{2}+810\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833386900\"><div data-type=\"problem\" id=\"fs-id1167833386903\"><p>\\(9{s}^{2}-12s+4\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836557699\"><p id=\"fs-id1167836557702\">\\({\\left(3s-2\\right)}^{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836688178\"><div data-type=\"problem\" id=\"fs-id1167836688180\"><p id=\"fs-id1167836688183\">\\(6{x}^{2}-11x-10\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836689546\"><div data-type=\"problem\" id=\"fs-id1167836689548\"><p id=\"fs-id1167836408252\">\\(3{x}^{2}-75{y}^{2}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836282607\"><p id=\"fs-id1167836282609\">\\(3\\left(x+5y\\right)\\left(x-5y\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836423849\"><div data-type=\"problem\" id=\"fs-id1167836408244\"><p id=\"fs-id1167836408246\">\\(6{u}^{2}+3u-18\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836548214\"><div data-type=\"problem\" id=\"fs-id1167836548216\"><p id=\"fs-id1167836548218\">\\({x}^{3}+125\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836663062\"><p id=\"fs-id1167836663064\">\\(\\left(x+5\\right)\\left({x}^{2}-5x+25\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836293549\"><div data-type=\"problem\" id=\"fs-id1167836293551\"><p id=\"fs-id1167836293553\">\\(32{x}^{5}{y}^{2}-162x{y}^{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833381407\"><div data-type=\"problem\" id=\"fs-id1167833381409\"><p id=\"fs-id1167829808159\">\\(6{x}^{4}-19{x}^{2}+15\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836621163\"><p id=\"fs-id1167836456503\">\\(\\left(3{x}^{2}-5\\right)\\left(2{x}^{2}-3\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836701190\"><div data-type=\"problem\" id=\"fs-id1167836731147\"><p id=\"fs-id1167836731149\">\\(3{x}^{3}-36{x}^{2}+108x\\)<\/p><\/div><\/div><p id=\"fs-id1167836553477\">In the following exercises, solve<\/p><div data-type=\"exercise\" id=\"fs-id1167836553480\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836553482\"><p id=\"fs-id1167829853138\">\\(5{a}^{2}+26a=24\\)<\/p><\/div><div data-type=\"solution\"><p id=\"fs-id1167836518301\">\\(a=4\\text{\/}5,a=-6\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836502587\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836502590\"><p id=\"fs-id1167836502592\">The product of two consecutive integers is 156. Find the integers.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829740680\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167829740682\"><p id=\"fs-id1167836407030\">The area of a rectangular place mat is 168 square inches. Its length is two inches longer than the width. Find the length and width of the placemat.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836407036\"><p id=\"fs-id1167836432100\">The width is 12 inches and the length is 14 inches.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836432105\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836540607\"><p id=\"fs-id1167836540609\">Jing is going to throw a ball from the balcony of her condo. When she throws the ball from 80 feet above the ground, the function \\(h\\left(t\\right)=-16{t}^{2}+64t+80\\) models the height, <em data-effect=\"italics\">h<\/em>, of the ball above the ground as a function of time, <em data-effect=\"italics\">t<\/em>. Find: <span class=\"token\">\u24d0<\/span> the zeros of this function which tells us when the ball will hit the ground. <span class=\"token\">\u24d1<\/span> the time(s) the ball will be 128 feet above the ground. <span class=\"token\">\u24d2<\/span> the height the ball will be at \\(t=4\\) seconds.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829861913\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836540808\"><p id=\"fs-id1167836540810\">For the function, \\(f\\left(x\\right)={x}^{2}-7x+5,\\) <span class=\"token\">\u24d0<\/span> find when \\(f\\left(x\\right)=-7\\) <span class=\"token\">\u24d1<\/span> Use this information to find two points that lie on the graph of the function.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836697643\"><p id=\"fs-id1167833139186\"><span class=\"token\">\u24d0<\/span>\\(x=3\\) or \\(x=4\\) <span class=\"token\">\u24d1<\/span> \\(\\left(3,-7\\right)\\) \\(\\left(4,-7\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829785512\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167829785514\"><p id=\"fs-id1167836502746\">For the function \\(f\\left(x\\right)=25{x}^{2}-81,\\) find: <span class=\"token\">\u24d0<\/span> the zeros of the function <span class=\"token\">\u24d1<\/span> the <em data-effect=\"italics\">x<\/em>-intercepts of the graph of the function <span class=\"token\">\u24d2<\/span> the <em data-effect=\"italics\">y<\/em>-intercept of the graph of the function.<\/p><\/div><\/div><\/div><div data-type=\"glossary\" class=\"textbox shaded\"><h3 data-type=\"glossary-title\">Glossary<\/h3><dl id=\"fs-id1167836610089\"><dt>degree of the polynomial equation<\/dt><dd id=\"fs-id1167836610094\">The degree of the polynomial equation is the degree of the polynomial.<\/dd><\/dl><dl id=\"fs-id1167836699095\"><dt>polynomial equation<\/dt><dd id=\"fs-id1167836699100\">A polynomial equation is an equation that contains a polynomial expression.<\/dd><\/dl><dl id=\"fs-id1167833047228\"><dt>quadratic equation<\/dt><dd id=\"fs-id1167833047234\">Polynomial equations of degree two are called quadratic equations.<\/dd><\/dl><dl id=\"fs-id1167833082445\"><dt>zero of the function<\/dt><dd>A value of \\(x\\) where the function is 0, is called a zero of the function.<\/dd><\/dl><dl id=\"fs-id1167836516138\"><dt>Zero Product Property<\/dt><dd id=\"fs-id1167836516143\">The Zero Product Property says that if the product of two quantities is zero, then at least one of the quantities is zero.<\/dd><\/dl><\/div>\n","rendered":"<div class=\"textbox textbox--learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<p>By the end of this section, you will be able to: <\/p>\n<ul>\n<li>Use the Zero Product Property<\/li>\n<li>Solve quadratic equations by factoring<\/li>\n<li>Solve equations with polynomial functions<\/li>\n<li>Solve applications modeled by polynomial equations<\/li>\n<\/ul>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836300333\" class=\"be-prepared\">\n<p id=\"fs-id1167833272508\">Before you get started, take this readiness quiz.<\/p>\n<ol id=\"fs-id1167836532183\" type=\"1\">\n<li>Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0bbe8f18dbfd63334c25eb089e4134d2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#121;&#45;&#51;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"85\" style=\"vertical-align: -4px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p> If you missed this problem, review <a href=\"\/contents\/9f100e8f-6d15-4cae-bc22-c306e9d7d55c#fs-id1167836432956\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<li>Factor completely: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3bc20ee1662eae6804bd1ef00d251a54_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#110;&#125;&#94;&#123;&#51;&#125;&#45;&#57;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#50;&#110;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"122\" style=\"vertical-align: 0px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p> If you missed this problem, review <a href=\"\/contents\/5e548626-8f0f-496d-ab87-4f0358ca2fd3#fs-id1167836521479\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<li>If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2ba3276668fa4d5773eed00800113915_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#56;&#120;&#45;&#49;&#54;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"124\" style=\"vertical-align: -4px;\" \/> find <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b7454c1320ac593411a470ad12380405_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"35\" style=\"vertical-align: -4px;\" \/> and solve <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-df88f8b453f241085be5afeee625ccb3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"74\" style=\"vertical-align: -4px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p> If you missed this problem, review <a href=\"\/contents\/da9d6ce0-a078-4ca2-97af-8cb374f040f5#fs-id1167836731462\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<\/ol>\n<\/div>\n<p id=\"fs-id1167833386923\">We have spent considerable time learning how to factor polynomials. We will now look at polynomial equations and solve them using factoring, if possible.<\/p>\n<p id=\"fs-id1167836533739\">A <span data-type=\"term\">polynomial equation<\/span> is an equation that contains a polynomial expression. The <span data-type=\"term\">degree of the polynomial equation<\/span> is the degree of the polynomial.<\/p>\n<div data-type=\"note\" id=\"fs-id1167836493148\">\n<div data-type=\"title\">Polynomial Equation<\/div>\n<p id=\"fs-id1167836602701\">A <strong data-effect=\"bold\">polynomial equation<\/strong> is an equation that contains a polynomial expression.<\/p>\n<p id=\"fs-id1167836363077\">The <strong data-effect=\"bold\">degree of the polynomial equation<\/strong> is the degree of the polynomial.<\/p>\n<\/div>\n<p id=\"fs-id1167836434319\">We have already solved polynomial equations of <span data-type=\"term\" class=\"no-emphasis\">degree one<\/span>. Polynomial equations of degree one are linear equations are of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-845ac975e724a9f3ed6d9af3d55698f0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#120;&#43;&#98;&#61;&#99;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"84\" style=\"vertical-align: -2px;\" \/><\/p>\n<p id=\"fs-id1167836367371\">We are now going to solve polynomial equations of <span data-type=\"term\" class=\"no-emphasis\">degree two<\/span>. A polynomial equation of degree two is called a <span data-type=\"term\">quadratic equation<\/span>. Listed below are some examples of quadratic equations:<\/p>\n<div data-type=\"equation\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8823e853cf0e8da0120badeb8901e5ae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#53;&#120;&#43;&#54;&#61;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#51;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#121;&#61;&#49;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#54;&#52;&#123;&#117;&#125;&#94;&#123;&#50;&#125;&#45;&#56;&#49;&#61;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#110;&#92;&#108;&#101;&#102;&#116;&#40;&#110;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#52;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"605\" style=\"vertical-align: -4px;\" \/><\/div>\n<p id=\"fs-id1167836287188\">The last equation doesn\u2019t appear to have the variable squared, but when we simplify the expression on the left we will get <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-768965b44fe9ab0d4f1219b12df5095a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#43;&#110;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"55\" style=\"vertical-align: -2px;\" \/><\/p>\n<p id=\"fs-id1167833019237\">The general form of a quadratic equation is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9e90a580df4e2c09eb1027a5013f4ef1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#120;&#43;&#99;&#61;&#48;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"133\" style=\"vertical-align: -4px;\" \/> with <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ce78e2da43dbf8e758d3c5c14d7f44ac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#92;&#110;&#101;&#32;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"46\" style=\"vertical-align: -4px;\" \/> (If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4c5e413b70b2e3a32f46a8942b947900_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&#48;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"46\" style=\"vertical-align: -4px;\" \/> then <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ade34e09b123224c133c5346627fc94e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&middot;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"59\" style=\"vertical-align: 0px;\" \/> and we are left with no quadratic term.)<\/p>\n<div data-type=\"note\">\n<div data-type=\"title\">Quadratic Equation<\/div>\n<p id=\"fs-id1167829904790\">An equation of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-29c8b40df2f348455a0b8aec28190877_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#120;&#43;&#99;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"129\" style=\"vertical-align: -2px;\" \/> is called a quadratic equation.<\/p>\n<div data-type=\"equation\" id=\"fs-id1167836310424\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-16f9d314a139beb9c231922239945f26_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#44;&#98;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#99;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#114;&#101;&#32;&#114;&#101;&#97;&#108;&#32;&#110;&#117;&#109;&#98;&#101;&#114;&#115;&#32;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#97;&#92;&#110;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"289\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<p id=\"fs-id1167836611159\">To solve quadratic equations we need methods different from the ones we used in solving linear equations. We will look at one method here and then several others in a later chapter.<\/p>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167829828416\">\n<h3 data-type=\"title\">Use the Zero Product Property<\/h3>\n<p id=\"fs-id1167836516676\">We will first solve some quadratic equations by using the <span data-type=\"term\">Zero Product Property<\/span>. The Zero Product Property says that if the product of two quantities is zero, then at least one of the quantities is zero. The only way to get a product equal to zero is to multiply by zero itself.<\/p>\n<div data-type=\"note\" id=\"fs-id1167833057729\">\n<div data-type=\"title\">Zero Product Property<\/div>\n<p id=\"fs-id1167833020317\">If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1178b2b1f3897a309978aa3009660bb1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&middot;&#98;&#61;&#48;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"54\" style=\"vertical-align: -4px;\" \/> then either <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-072bed0ebf929f9d7c14da365c8512a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\" \/> or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-597ae6c01abdebf6816c77f3ce1fd9de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"40\" style=\"vertical-align: 0px;\" \/> or both.<\/p>\n<\/div>\n<p id=\"fs-id1167836553403\">We will now use the Zero Product Property, to solve a <span data-type=\"term\" class=\"no-emphasis\">quadratic equation<\/span>.<\/p>\n<div data-type=\"example\" id=\"fs-id1167836619617\" class=\"textbox textbox--examples\">\n<div data-type=\"title\">How to Solve a Quadratic Equation Using the Zero Product Property<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833056584\">\n<div data-type=\"problem\" id=\"fs-id1167836727844\">\n<p id=\"fs-id1167836333497\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dc58d05954da936a8387ee4dddac4052_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#110;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#110;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"166\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829936994\"><span data-type=\"media\" id=\"fs-id1167836707067\" data-alt=\"The equation is open parentheses 5n minus 2 close parentheses open parentheses 6n minus 1 close parentheses equals 0. The product equals zero, so at least one factor must equal zero. Step 1 is set each factor equal to zero. So, 5n minus 2 equals 0 and 6n minus 1 equals 0.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_05_001a_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The equation is open parentheses 5n minus 2 close parentheses open parentheses 6n minus 1 close parentheses equals 0. The product equals zero, so at least one factor must equal zero. Step 1 is set each factor equal to zero. So, 5n minus 2 equals 0 and 6n minus 1 equals 0.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167836393361\" data-alt=\"Step 2 is to solve the linear equations. So, we get n equal to 2 by 5 and n equal to 1 by 6.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_05_001b_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2 is to solve the linear equations. So, we get n equal to 2 by 5 and n equal to 1 by 6.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167836602556\" data-alt=\"Step 3 is to check by substituting each solution separately into the original equation.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_05_001c_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3 is to check by substituting each solution separately into the original equation.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167824734415\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836625622\">\n<div data-type=\"problem\" id=\"fs-id1167836521982\">\n<p id=\"fs-id1167829747240\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5a1c40fd77fdcd2e7ab7a4b72183290e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#109;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#109;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"176\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836340005\">\n<p id=\"fs-id1167829712286\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e9ea1bffe6ce85b1ef341f7594b96b72_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#44;&#109;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"120\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836516418\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836312584\">\n<div data-type=\"problem\" id=\"fs-id1167829621115\">\n<p>Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-121d5785372cf716da5737c22e81110b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#112;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#112;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"163\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836285974\">\n<p id=\"fs-id1167836379067\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5f42edd7ec8f6a479525c7a43dc47ffb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#112;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#44;&#112;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"107\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836408616\" class=\"howto\">\n<div data-type=\"title\">Use the Zero Product Property.<\/div>\n<ol id=\"fs-id1167824734367\" type=\"1\" class=\"stepwise\">\n<li>Set each factor equal to zero.<\/li>\n<li>Solve the linear equations.<\/li>\n<li>Check.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167836294278\">\n<h3 data-type=\"title\">Solve Quadratic Equations by Factoring<\/h3>\n<p id=\"fs-id1167836554592\">The Zero Product Property works very nicely to solve quadratic equations. The quadratic equation must be factored, with zero isolated on one side. So we be sure to start with the quadratic equation in <span data-type=\"term\" class=\"no-emphasis\">standard form<\/span>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ff55a4ef4f0f0f08cb24335fede5832e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#120;&#43;&#99;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"133\" style=\"vertical-align: -2px;\" \/> Then we factor the expression on the left.<\/p>\n<div data-type=\"example\" class=\"textbox textbox--examples\">\n<div data-type=\"title\">How to Solve a Quadratic Equation by Factoring<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829719011\">\n<div data-type=\"problem\" id=\"fs-id1167833047277\">\n<p id=\"fs-id1167836313413\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6791f6189994340c762c4fcdc7fa1ebd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#51;&#121;&#43;&#52;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"120\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167825835938\"><span data-type=\"media\" id=\"fs-id1167829833272\" data-alt=\"The equation is 2 y squared equals 13y plus 45. Step 1 is to write it in standard form a x squared plus bx plus c. So we have 2 y squared minus 13y minus 45 equals 0.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_05_002a_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The equation is 2 y squared equals 13y plus 45. Step 1 is to write it in standard form a x squared plus bx plus c. So we have 2 y squared minus 13y minus 45 equals 0.\" \/><\/span><span data-type=\"media\" data-alt=\"Step 2 is to factor the quadratic expression. So we have 2y plus 5, y minus 9 equals 0.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_05_002b_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2 is to factor the quadratic expression. So we have 2y plus 5, y minus 9 equals 0.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167833022106\" data-alt=\"Step 3 is to use the zero product property. Setting each factor equal to zero, we have two linear equations: 2y plus 5 equals 0 and y minus 9 equals 0.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_05_002c_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3 is to use the zero product property. Setting each factor equal to zero, we have two linear equations: 2y plus 5 equals 0 and y minus 9 equals 0.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167836635419\" data-alt=\"Step 4 is to solve the linear equations. We get, y equals minus 5 by 2 and y equals 9.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_05_002d_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 4 is to solve the linear equations. We get, y equals minus 5 by 2 and y equals 9.\" \/><\/span><span data-type=\"media\" data-alt=\"Step 5 is to check by substituting each solution separately into the original equation\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_05_002e_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 5 is to check by substituting each solution separately into the original equation\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167829936700\" class=\"try\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167836504112\">\n<p>Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5a65cef1ba1fec89be67803a69af5255_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#123;&#99;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#48;&#99;&#45;&#56;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"108\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167836415111\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2925157f9d148a6d311b4cc684b81c98_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;&#61;&#50;&#44;&#99;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"88\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836323046\" class=\"try\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\">\n<p>Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-90faf0cdd3c247e6f5270f507aa9c5eb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#123;&#100;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#100;&#61;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"102\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6cc88cbba7138a1835e64b21cf9028bf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#100;&#61;&#51;&#44;&#100;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"105\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836515772\" class=\"howto\">\n<div data-type=\"title\">Solve a quadratic equation by factoring.<\/div>\n<ol type=\"1\" class=\"stepwise\">\n<li>Write the quadratic equation in standard form, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ff55a4ef4f0f0f08cb24335fede5832e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#120;&#43;&#99;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"133\" style=\"vertical-align: -2px;\" \/><\/li>\n<li>Factor the quadratic expression.<\/li>\n<li>Use the Zero Product Property.<\/li>\n<li>Solve the linear equations.<\/li>\n<li>Check. Substitute each solution separately into the original equation.<\/li>\n<\/ol>\n<\/div>\n<p id=\"fs-id1167836386785\">Before we factor, we must make sure the <span data-type=\"term\" class=\"no-emphasis\">quadratic equation<\/span> is in <span data-type=\"term\" class=\"no-emphasis\">standard form<\/span>.<\/p>\n<p>Solving quadratic equations by factoring will make use of all the factoring techniques you have learned in this chapter! Do you recognize the special product pattern in the next example?<\/p>\n<div data-type=\"example\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167833245837\">\n<div data-type=\"problem\" id=\"fs-id1167836614984\">\n<p id=\"fs-id1167836538194\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ba0446cb0ed3e50256ebd8dc2d6cd1bc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#54;&#57;&#123;&#113;&#125;&#94;&#123;&#50;&#125;&#61;&#52;&#57;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"87\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ded6f803107edf0532f8e16bd0471edb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#53;&#46;&#49;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#54;&#57;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#61;&#52;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#87;&#114;&#105;&#116;&#101;&#32;&#116;&#104;&#101;&#32;&#113;&#117;&#97;&#100;&#114;&#97;&#116;&#105;&#99;&#32;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#32;&#105;&#110;&#32;&#115;&#116;&#97;&#110;&#100;&#97;&#114;&#100;&#32;&#102;&#111;&#114;&#109;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#54;&#57;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#57;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#70;&#97;&#99;&#116;&#111;&#114;&#46;&#32;&#73;&#116;&#32;&#105;&#115;&#32;&#97;&#32;&#100;&#105;&#102;&#102;&#101;&#114;&#101;&#110;&#99;&#101;&#32;&#111;&#102;&#32;&#115;&#113;&#117;&#97;&#114;&#101;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#51;&#120;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#51;&#120;&#43;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#85;&#115;&#101;&#32;&#116;&#104;&#101;&#32;&#90;&#101;&#114;&#111;&#32;&#80;&#114;&#111;&#100;&#117;&#99;&#116;&#32;&#80;&#114;&#111;&#112;&#101;&#114;&#116;&#121;&#32;&#116;&#111;&#32;&#115;&#101;&#116;&#32;&#101;&#97;&#99;&#104;&#32;&#102;&#97;&#99;&#116;&#111;&#114;&#32;&#116;&#111;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#48;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#111;&#108;&#118;&#101;&#32;&#101;&#97;&#99;&#104;&#32;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#51;&#120;&#45;&#55;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#51;&#120;&#43;&#55;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#51;&#120;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#51;&#120;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#120;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#49;&#51;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#120;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#49;&#51;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"177\" width=\"848\" style=\"vertical-align: -62px;\" \/><\/p>\n<p id=\"fs-id1167833239994\">Check:<\/p>\n<p id=\"fs-id1167836515959\">We leave the check up to you.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167833018774\" class=\"try\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167836493372\">\n<p>Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-35c0b6a146d3b3b12105819333a50c21_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#53;&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#61;&#52;&#57;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"80\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836296549\">\n<p id=\"fs-id1167833009374\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-57030ab3dbf3c3d1ca50e4e71c79f7da_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#112;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#53;&#125;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#112;&#125;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"108\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836352178\" class=\"try\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167836665581\">\n<p id=\"fs-id1167833338953\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5f4bc2bfbc66ec5482647c313d7c2639_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#50;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"90\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836300042\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fe0d60638a84c87d7c3d616d10e896f0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#49;&#125;&#123;&#54;&#125;&#44;&#120;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#49;&#125;&#123;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"123\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167833023151\">In the next example, the left side of the equation is factored, but the right side is not zero. In order to use the <span data-type=\"term\" class=\"no-emphasis\">Zero Product Property<\/span>, one side of the equation must be zero. We\u2019ll multiply the factors and then write the equation in standard form.<\/p>\n<div data-type=\"example\" id=\"fs-id1167829580432\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167836704241\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167833158807\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a09280227c2cdc2e45546f9e6feebfd3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#120;&#45;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#51;&#120;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"166\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836544129\">\n<p id=\"fs-id1167836524167\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d0280c3ee9d99382c37848eb509bddea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#120;&#45;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#51;&#120;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#77;&#117;&#108;&#116;&#105;&#112;&#108;&#121;&#32;&#116;&#104;&#101;&#32;&#98;&#105;&#110;&#111;&#109;&#105;&#97;&#108;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#54;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#49;&#120;&#43;&#56;&#61;&#51;&#120;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#87;&#114;&#105;&#116;&#101;&#32;&#116;&#104;&#101;&#32;&#113;&#117;&#97;&#100;&#114;&#97;&#116;&#105;&#99;&#32;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#32;&#105;&#110;&#32;&#115;&#116;&#97;&#110;&#100;&#97;&#114;&#100;&#32;&#102;&#111;&#114;&#109;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#54;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#52;&#120;&#43;&#56;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#70;&#97;&#99;&#116;&#111;&#114;&#32;&#116;&#104;&#101;&#32;&#116;&#114;&#105;&#110;&#111;&#109;&#105;&#97;&#108;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#120;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#85;&#115;&#101;&#32;&#116;&#104;&#101;&#32;&#90;&#101;&#114;&#111;&#32;&#80;&#114;&#111;&#100;&#117;&#99;&#116;&#32;&#80;&#114;&#111;&#112;&#101;&#114;&#116;&#121;&#32;&#116;&#111;&#32;&#115;&#101;&#116;&#32;&#101;&#97;&#99;&#104;&#32;&#102;&#97;&#99;&#116;&#111;&#114;&#32;&#116;&#111;&#32;&#48;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#111;&#108;&#118;&#101;&#32;&#101;&#97;&#99;&#104;&#32;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#120;&#45;&#50;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#120;&#45;&#52;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#120;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#120;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#46;&#55;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#104;&#101;&#99;&#107;&#32;&#121;&#111;&#117;&#114;&#32;&#97;&#110;&#115;&#119;&#101;&#114;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#104;&#101;&#32;&#99;&#104;&#101;&#99;&#107;&#32;&#105;&#115;&#32;&#108;&#101;&#102;&#116;&#32;&#116;&#111;&#32;&#121;&#111;&#117;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"171\" width=\"794\" style=\"vertical-align: -80px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836309718\">\n<div data-type=\"problem\" id=\"fs-id1167836560393\">\n<p id=\"fs-id1167836520033\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-992cbc57313618f1eb9c6bd916a88db9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#109;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#109;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#49;&#50;&#109;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"192\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829695518\">\n<p id=\"fs-id1167836556797\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ad7605136a9887fd07721ef11d621b69_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#49;&#44;&#109;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"104\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836333805\">\n<div data-type=\"problem\" id=\"fs-id1167836530331\">\n<p id=\"fs-id1167836704806\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e1cc20c20f7db3e18ae7a899a922ee6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#107;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#107;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#56;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"147\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836558294\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b2c94325793bdd81c1fd0591def0d90a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#107;&#61;&#51;&#44;&#107;&#61;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"107\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167829594693\">In the next example, when we factor the quadratic equation we will get three factors. However the first factor is a constant. We know that factor cannot equal 0.<\/p>\n<div data-type=\"example\" id=\"fs-id1167836691703\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167829905872\">\n<p id=\"fs-id1167836487460\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1c369017bd2e5602a6bd0a1c925810a7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#50;&#120;&#43;&#54;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"122\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836289776\">\n<p id=\"fs-id1167836628837\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-61db71e68a4021d561d9a4aef938a339_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#53;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#50;&#120;&#43;&#54;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#87;&#114;&#105;&#116;&#101;&#32;&#116;&#104;&#101;&#32;&#113;&#117;&#97;&#100;&#114;&#97;&#116;&#105;&#99;&#32;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#32;&#105;&#110;&#32;&#115;&#116;&#97;&#110;&#100;&#97;&#114;&#100;&#32;&#102;&#111;&#114;&#109;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#55;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#50;&#120;&#45;&#54;&#51;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#70;&#97;&#99;&#116;&#111;&#114;&#32;&#116;&#104;&#101;&#32;&#103;&#114;&#101;&#97;&#116;&#101;&#115;&#116;&#32;&#99;&#111;&#109;&#109;&#111;&#110;&#32;&#102;&#97;&#99;&#116;&#111;&#114;&#32;&#102;&#105;&#114;&#115;&#116;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#55;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#120;&#45;&#50;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#70;&#97;&#99;&#116;&#111;&#114;&#32;&#116;&#104;&#101;&#32;&#116;&#114;&#105;&#110;&#111;&#109;&#105;&#97;&#108;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#56;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#85;&#115;&#101;&#32;&#116;&#104;&#101;&#32;&#90;&#101;&#114;&#111;&#32;&#80;&#114;&#111;&#100;&#117;&#99;&#116;&#32;&#80;&#114;&#111;&#112;&#101;&#114;&#116;&#121;&#32;&#116;&#111;&#32;&#115;&#101;&#116;&#32;&#101;&#97;&#99;&#104;&#32;&#102;&#97;&#99;&#116;&#111;&#114;&#32;&#116;&#111;&#32;&#48;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#111;&#108;&#118;&#101;&#32;&#101;&#97;&#99;&#104;&#32;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#51;&#92;&#110;&#101;&#32;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#120;&#45;&#55;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#120;&#43;&#51;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#51;&#92;&#110;&#101;&#32;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#120;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#120;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#104;&#101;&#99;&#107;&#32;&#121;&#111;&#117;&#114;&#32;&#97;&#110;&#115;&#119;&#101;&#114;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#104;&#101;&#32;&#99;&#104;&#101;&#99;&#107;&#32;&#105;&#115;&#32;&#108;&#101;&#102;&#116;&#32;&#116;&#111;&#32;&#121;&#111;&#117;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"151\" width=\"906\" style=\"vertical-align: -69px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836329405\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836376734\">\n<div data-type=\"problem\" id=\"fs-id1167836288543\">\n<p id=\"fs-id1167836689175\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b88855b077aba839abdecdeff4ecac3a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#56;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#48;&#61;&#45;&#51;&#51;&#97;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"142\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833310107\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ef5d7c812d8db3878c5f052b7a8db710_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#50;&#125;&#44;&#97;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"107\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836553150\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836333546\">\n<div data-type=\"problem\" id=\"fs-id1167829694665\">\n<p id=\"fs-id1167836390484\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8f652ec0fdbfff02ead2806bb09e9a9c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#50;&#51;&#98;&#61;&#45;&#54;&#45;&#54;&#48;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"138\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836553353\">\n<p id=\"fs-id1167836630557\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-683700f3c3ed78072592e353a0c95047_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;&#61;&#45;&#50;&#44;&#98;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"123\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167829694662\">The <span data-type=\"term\" class=\"no-emphasis\">Zero Product Property<\/span> also applies to the product of three or more factors. If the product is zero, at least one of the factors must be zero. We can solve some equations of degree greater than two by using the Zero Product Property, just like we solved quadratic equations.<\/p>\n<div data-type=\"example\" id=\"fs-id1167829644842\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167836456230\">\n<div data-type=\"problem\" id=\"fs-id1167836514629\">\n<p id=\"fs-id1167836700375\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f748bc2808c3480f5efcaa956b0a6482_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#123;&#109;&#125;&#94;&#123;&#51;&#125;&#43;&#49;&#48;&#48;&#109;&#61;&#54;&#48;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"165\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836518098\">\n<p id=\"fs-id1167836704559\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ae835800c0660acf9afe747193017ac8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#46;&#55;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#57;&#123;&#109;&#125;&#94;&#123;&#51;&#125;&#43;&#49;&#48;&#48;&#109;&#61;&#54;&#48;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#66;&#114;&#105;&#110;&#103;&#32;&#97;&#108;&#108;&#32;&#116;&#104;&#101;&#32;&#116;&#101;&#114;&#109;&#115;&#32;&#116;&#111;&#32;&#111;&#110;&#101;&#32;&#115;&#105;&#100;&#101;&#32;&#115;&#111;&#32;&#116;&#104;&#97;&#116;&#32;&#116;&#104;&#101;&#32;&#111;&#116;&#104;&#101;&#114;&#32;&#115;&#105;&#100;&#101;&#32;&#105;&#115;&#32;&#122;&#101;&#114;&#111;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#51;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#57;&#123;&#109;&#125;&#94;&#123;&#51;&#125;&#45;&#54;&#48;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#48;&#48;&#109;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#70;&#97;&#99;&#116;&#111;&#114;&#32;&#116;&#104;&#101;&#32;&#103;&#114;&#101;&#97;&#116;&#101;&#115;&#116;&#32;&#99;&#111;&#109;&#109;&#111;&#110;&#32;&#102;&#97;&#99;&#116;&#111;&#114;&#32;&#102;&#105;&#114;&#115;&#116;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#51;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#109;&#92;&#108;&#101;&#102;&#116;&#40;&#57;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#48;&#109;&#43;&#49;&#48;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#70;&#97;&#99;&#116;&#111;&#114;&#32;&#116;&#104;&#101;&#32;&#116;&#114;&#105;&#110;&#111;&#109;&#105;&#97;&#108;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#109;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#109;&#45;&#49;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#109;&#45;&#49;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#85;&#115;&#101;&#32;&#116;&#104;&#101;&#32;&#90;&#101;&#114;&#111;&#32;&#80;&#114;&#111;&#100;&#117;&#99;&#116;&#32;&#80;&#114;&#111;&#112;&#101;&#114;&#116;&#121;&#32;&#116;&#111;&#32;&#115;&#101;&#116;&#32;&#101;&#97;&#99;&#104;&#32;&#102;&#97;&#99;&#116;&#111;&#114;&#32;&#116;&#111;&#32;&#48;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#111;&#108;&#118;&#101;&#32;&#101;&#97;&#99;&#104;&#32;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#51;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#109;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#109;&#45;&#49;&#48;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#109;&#45;&#49;&#48;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#109;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#109;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#48;&#125;&#123;&#51;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#109;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#48;&#125;&#123;&#51;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#104;&#101;&#99;&#107;&#32;&#121;&#111;&#117;&#114;&#32;&#97;&#110;&#115;&#119;&#101;&#114;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#104;&#101;&#32;&#99;&#104;&#101;&#99;&#107;&#32;&#105;&#115;&#32;&#108;&#101;&#102;&#116;&#32;&#116;&#111;&#32;&#121;&#111;&#117;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"151\" width=\"997\" style=\"vertical-align: -69px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836449554\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836363174\">\n<div data-type=\"problem\" id=\"fs-id1167829621409\">\n<p id=\"fs-id1167836522448\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cd09ba92a755ba590fdf8d1ba46653e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#61;&#50;&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#56;&#120;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"140\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836293394\">\n<p id=\"fs-id1167833212958\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9cbe79bedf19e50c43ffcfd3e5fd33ac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#48;&#44;&#120;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"93\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836389504\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836447280\">\n<div data-type=\"problem\" id=\"fs-id1167836493814\">\n<p id=\"fs-id1167836287418\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-24953b95ffb4fee553047fd9a46be623_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#54;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#51;&#50;&#123;&#121;&#125;&#94;&#123;&#51;&#125;&#43;&#50;&#121;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"136\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829596286\">\n<p id=\"fs-id1167829695025\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-75934fe8f7f8d78f4b2b8fe545497098_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#48;&#44;&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"92\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167836525536\">\n<h3 data-type=\"title\">Solve Equations with Polynomial Functions<\/h3>\n<p id=\"fs-id1167833056834\">As our study of polynomial functions continues, it will often be important to know when the function will have a certain value or what points lie on the graph of the function. Our work with the <span data-type=\"term\" class=\"no-emphasis\">Zero Product Property<\/span> will be help us find these answers.<\/p>\n<div data-type=\"example\" id=\"fs-id1167833061230\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167829599152\">\n<div data-type=\"problem\" id=\"fs-id1167829719035\">\n<p id=\"fs-id1167833071676\">For the function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-954352054b8aa12d4c67ecf1fbcb68a6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#120;&#45;&#50;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"154\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"fs-id1167836389548\"><span class=\"token\">\u24d0<\/span> find <em data-effect=\"italics\">x<\/em> when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9f6892a61e60f235449cf9b79d44bd58_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"70\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> find two points that lie on the graph of the function.<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836444737\">\n<p id=\"fs-id1167836492865\"><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c5ba394a9a567b626b4fe8561832e1c8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#120;&#45;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#117;&#98;&#115;&#116;&#105;&#116;&#117;&#116;&#101;&#32;&#54;&#32;&#102;&#111;&#114;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#54;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#120;&#45;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#80;&#117;&#116;&#32;&#116;&#104;&#101;&#32;&#113;&#117;&#97;&#100;&#114;&#97;&#116;&#105;&#99;&#32;&#105;&#110;&#32;&#115;&#116;&#97;&#110;&#100;&#97;&#114;&#100;&#32;&#102;&#111;&#114;&#109;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#56;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#120;&#45;&#56;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#70;&#97;&#99;&#116;&#111;&#114;&#32;&#116;&#104;&#101;&#32;&#116;&#114;&#105;&#110;&#111;&#109;&#105;&#97;&#108;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#85;&#115;&#101;&#32;&#116;&#104;&#101;&#32;&#122;&#101;&#114;&#111;&#32;&#112;&#114;&#111;&#100;&#117;&#99;&#116;&#32;&#112;&#114;&#111;&#112;&#101;&#114;&#116;&#121;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#111;&#108;&#118;&#101;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#120;&#43;&#52;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#111;&#114;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#120;&#45;&#50;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#120;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#111;&#114;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#120;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#104;&#101;&#99;&#107;&#58;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#55;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#120;&#45;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#120;&#45;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#50;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&middot;&#50;&#45;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#49;&#54;&#45;&#56;&#45;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#52;&#43;&#52;&#45;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#54;&#10003;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#54;&#10003;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"417\" width=\"886\" style=\"vertical-align: -203px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> Since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-675c64e905a7f6ac02bcf9a30f59267b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"83\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8426205cacfa8c65f47a40e974e6d6f2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#54;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"73\" style=\"vertical-align: -4px;\" \/> the points <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-46dcd9d3898bf55c3e7c3af73128aeca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#44;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-80be894898c77ffda9d4d5843e6791f9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> lie on the graph of the function.<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836609926\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836692073\">\n<div data-type=\"problem\" id=\"fs-id1167836526181\">\n<p id=\"fs-id1167829692807\">For the function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0ad613292664fe4fbf91269e0026265d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#120;&#45;&#56;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"154\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"fs-id1167836392650\"><span class=\"token\">\u24d0<\/span> find <em data-effect=\"italics\">x<\/em> when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-60b9ee7ee1a9bef2487a662f34b077e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"70\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> Find two points that lie on the graph of the function.<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836318962\">\n<p id=\"fs-id1167836556194\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7e135cd6350a4c21195c621240f7aee7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"57\" style=\"vertical-align: 0px;\" \/> or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8ddab230605c435eb8b7408a736d3e77_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"42\" style=\"vertical-align: 0px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f03214ae685c15fbb390fe7d48d59247_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#44;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-17d5ca95974a7b3d2df8384be1677b0d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#44;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836390310\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836667141\">\n<div data-type=\"problem\" id=\"fs-id1167836545876\">\n<p id=\"fs-id1167836559968\">For the function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3ac47640e1ce71728def72d35360e5f0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#56;&#120;&#43;&#51;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"154\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"fs-id1167836456286\"><span class=\"token\">\u24d0<\/span> find <em data-effect=\"italics\">x<\/em> when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8b079a0bd8cb258dd6194cee203d2f7c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"84\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> Find two points that lie on the graph of the function.<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836622813\">\n<p id=\"fs-id1167836379075\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3330a01aa4d7d81947b71297d8623d3b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"42\" style=\"vertical-align: -1px;\" \/> or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-49af7f686347e06977fd3a18757933db_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"43\" style=\"vertical-align: 0px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-95e282826e52860edf4d8703db75fb7f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4edac078cdeabae03cf60148f88b1f29_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#44;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167829877554\">The <span data-type=\"term\" class=\"no-emphasis\">Zero Product Property<\/span> also helps us determine where the function is zero. A value of <em data-effect=\"italics\">x<\/em> where the function is 0, is called a <span data-type=\"term\">zero of the function<\/span>.<\/p>\n<div data-type=\"note\" id=\"fs-id1167836544182\">\n<div data-type=\"title\">Zero of a Function<\/div>\n<p id=\"fs-id1167836552958\">For any function <em data-effect=\"italics\">f<\/em>, if <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4b6a350c5088c4d60bfbe05674203c9c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#48;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"74\" style=\"vertical-align: -4px;\" \/> then <em data-effect=\"italics\">x<\/em> is a <strong data-effect=\"bold\">zero of the function<\/strong>.<\/p>\n<\/div>\n<p id=\"fs-id1167829689269\">When <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4b6a350c5088c4d60bfbe05674203c9c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#48;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"74\" style=\"vertical-align: -4px;\" \/> the point <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3f0d502e1404554e5df388133569ae57_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"39\" style=\"vertical-align: -4px;\" \/> is a point on the graph. This point is an <span data-type=\"term\" class=\"no-emphasis\"><em data-effect=\"italics\">x<\/em>-intercept<\/span> of the graph. It is often important to know where the graph of a function crosses the axes. We will see some examples later.<\/p>\n<div data-type=\"example\" id=\"fs-id1167836621947\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167829624424\">\n<div data-type=\"problem\" id=\"fs-id1167836618856\">\n<p id=\"fs-id1167829597813\">For the function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cfacef9810a901861beb310285c304d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#48;&#120;&#45;&#56;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"172\" style=\"vertical-align: -4px;\" \/> find<\/p>\n<p id=\"fs-id1167836449758\"><span class=\"token\">\u24d0<\/span> the zeros of the function,<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> any <em data-effect=\"italics\">x<\/em>-intercepts of the graph of the function<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span> any <em data-effect=\"italics\">y<\/em>-intercepts of the graph of the function<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836321778\">\n<p id=\"fs-id1167836319053\"><span class=\"token\">\u24d0<\/span> To find the zeros of the function, we need to find when the function value is 0.<\/p>\n<div data-type=\"newline\"><\/div>\n<p> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0b67d6247984f4bbebe8b4aefdd884f5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#48;&#120;&#45;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#117;&#98;&#115;&#116;&#105;&#116;&#117;&#116;&#101;&#32;&#48;&#32;&#102;&#111;&#114;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#48;&#61;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#48;&#120;&#45;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#70;&#97;&#99;&#116;&#111;&#114;&#32;&#116;&#104;&#101;&#32;&#116;&#114;&#105;&#110;&#111;&#109;&#105;&#97;&#108;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#120;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#85;&#115;&#101;&#32;&#116;&#104;&#101;&#32;&#122;&#101;&#114;&#111;&#32;&#112;&#114;&#111;&#100;&#117;&#99;&#116;&#32;&#112;&#114;&#111;&#112;&#101;&#114;&#116;&#121;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#111;&#108;&#118;&#101;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#120;&#43;&#52;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#111;&#114;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#120;&#45;&#50;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#120;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#111;&#114;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#120;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"109\" width=\"671\" style=\"vertical-align: -50px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p> <span class=\"token\">\u24d1<\/span> An <em data-effect=\"italics\">x<\/em>-intercept occurs when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-03bd81b19337bd478ac4d869320205bf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"46\" style=\"vertical-align: -4px;\" \/> Since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8fa462170c5d0a2bf982856da150bb8d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"83\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c72607403b02c1810cf8c069b7ea3aaa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#48;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"77\" style=\"vertical-align: -7px;\" \/> the points <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-91357ad9f837f690ca370a0ee126647e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-de702e3f7fd4f26121cfe64713c191f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"40\" style=\"vertical-align: -7px;\" \/> lie on the graph. These points are <em data-effect=\"italics\">x<\/em>-intercepts of the function.<\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p> <span class=\"token\">\u24d2<\/span> A <em data-effect=\"italics\">y<\/em>-intercept occurs when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d6889ee3f02f0af137641306363d2da7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"47\" style=\"vertical-align: 0px;\" \/> To find the <em data-effect=\"italics\">y<\/em>-intercepts we need to find <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-db4f958b4c4deacd90047ad0c954283a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"43\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8a3de5a4067e03dc9260aab31e9ac015_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#48;&#120;&#45;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#70;&#105;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#98;&#121;&#32;&#115;&#117;&#98;&#115;&#116;&#105;&#116;&#117;&#116;&#105;&#110;&#103;&#32;&#48;&#32;&#102;&#111;&#114;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#51;&middot;&#123;&#48;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#48;&middot;&#48;&#45;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#105;&#109;&#112;&#108;&#105;&#102;&#121;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"63\" width=\"509\" style=\"vertical-align: -26px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p> Since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-38b7314e94c77fd84e7cf35553bfe56b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#56;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"87\" style=\"vertical-align: -4px;\" \/> the point <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-de801b109a641a695bdc89761979b816_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#45;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/> lies on the graph. This point is the <em data-effect=\"italics\">y<\/em>-intercept of the function.<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836612535\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167829811740\">\n<div data-type=\"problem\" id=\"fs-id1167836516254\">\n<p id=\"fs-id1167836362622\">For the function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9e5a3da6d61e8a73ff04e4f90a72d403_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#55;&#120;&#43;&#53;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"163\" style=\"vertical-align: -4px;\" \/> find<\/p>\n<p id=\"fs-id1167836601250\"><span class=\"token\">\u24d0<\/span> the zeros of the function<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> any <em data-effect=\"italics\">x<\/em>-intercepts of the graph of the function<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span> any <em data-effect=\"italics\">y<\/em>-intercepts of the graph of the function.<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167829686321\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3330a01aa4d7d81947b71297d8623d3b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"42\" style=\"vertical-align: -1px;\" \/> or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3fa72629f044a280114c8466fb1eee02_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"43\" style=\"vertical-align: -6px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b2aedbffc1c0be6be268afc470cd42b2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"45\" style=\"vertical-align: -4px;\" \/> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d03f220a14669e7447fa24af6cc39b37_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#50;&#125;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"40\" style=\"vertical-align: -7px;\" \/> <span class=\"token\">\u24d2<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6655aef23bcf82d48b1ff5bf888d5b2a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836391037\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167829693111\">\n<div data-type=\"problem\" id=\"fs-id1167836625132\">\n<p id=\"fs-id1167836360997\">For the function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8e4d5bd314a5e2be51144536839b9b5d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#51;&#120;&#45;&#49;&#53;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"181\" style=\"vertical-align: -4px;\" \/> find<\/p>\n<p id=\"fs-id1167836622572\"><span class=\"token\">\u24d0<\/span> the zeros of the function<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> any <em data-effect=\"italics\">x<\/em>-intercepts of the graph of the function<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span> any <em data-effect=\"italics\">y<\/em>-intercepts of the graph of the function.<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829595141\">\n<p id=\"fs-id1167833060987\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7e135cd6350a4c21195c621240f7aee7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"57\" style=\"vertical-align: 0px;\" \/> or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-469a2bfaa32a89173f00a5ba72ca063a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"43\" style=\"vertical-align: -6px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-03af16e4bcfcd37f0f1dc04e4e3193f5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"59\" style=\"vertical-align: -4px;\" \/> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ade5a27d17c0ef292410edf75f982758_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#54;&#125;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"40\" style=\"vertical-align: -7px;\" \/> <span class=\"token\">\u24d2<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-eea79a1336bcc848762018d0a40fffb8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#45;&#49;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"61\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167833381187\">\n<h3 data-type=\"title\">Solve Applications Modeled by Polynomial Equations<\/h3>\n<p id=\"fs-id1167833054651\">The problem-solving strategy we used earlier for applications that translate to linear equations will work just as well for applications that translate to polynomial equations. We will copy the problem-solving strategy here so we can use it for reference.<\/p>\n<div data-type=\"note\" id=\"fs-id1167836511805\" class=\"howto\">\n<div data-type=\"title\">Use a problem solving strategy to solve word problems.<\/div>\n<ol id=\"fs-id1167829790276\" type=\"1\" class=\"stepwise\">\n<li><strong data-effect=\"bold\">Read<\/strong> the problem. Make sure all the words and ideas are understood.<\/li>\n<li><strong data-effect=\"bold\">Identify<\/strong> what we are looking for.<\/li>\n<li><strong data-effect=\"bold\">Name<\/strong> what we are looking for. Choose a variable to represent that quantity.<\/li>\n<li><strong data-effect=\"bold\">Translate<\/strong> into an equation. It may be helpful to restate the problem in one sentence with all the important information. Then, translate the English sentence into an algebraic equation.<\/li>\n<li><strong data-effect=\"bold\">Solve<\/strong> the equation using appropriate algebra techniques.<\/li>\n<li><strong data-effect=\"bold\">Check<\/strong> the answer in the problem and make sure it makes sense.<\/li>\n<li><strong data-effect=\"bold\">Answer<\/strong> the question with a complete sentence.<\/li>\n<\/ol>\n<\/div>\n<p id=\"fs-id1167829830361\">We will start with a number problem to get practice translating words into a polynomial equation.<\/p>\n<div data-type=\"example\" id=\"fs-id1167836611600\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167836533796\">\n<div data-type=\"problem\" id=\"fs-id1167833019354\">\n<p id=\"fs-id1167836509779\">The product of two consecutive odd integers is 323. Find the integers.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829596980\">\n<p id=\"fs-id1167836449894\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-27a05fa802eb9b196b55a78aae97db58_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#116;&#101;&#112;&#32;&#49;&#46;&#32;&#82;&#101;&#97;&#100;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#32;&#112;&#114;&#111;&#98;&#108;&#101;&#109;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#116;&#101;&#112;&#32;&#50;&#46;&#32;&#73;&#100;&#101;&#110;&#116;&#105;&#102;&#121;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#119;&#104;&#97;&#116;&#32;&#119;&#101;&#32;&#97;&#114;&#101;&#32;&#108;&#111;&#111;&#107;&#105;&#110;&#103;&#32;&#102;&#111;&#114;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#87;&#101;&#32;&#97;&#114;&#101;&#32;&#108;&#111;&#111;&#107;&#105;&#110;&#103;&#32;&#102;&#111;&#114;&#32;&#116;&#119;&#111;&#32;&#99;&#111;&#110;&#115;&#101;&#99;&#117;&#116;&#105;&#118;&#101;&#32;&#105;&#110;&#116;&#101;&#103;&#101;&#114;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#116;&#101;&#112;&#32;&#51;&#46;&#32;&#78;&#97;&#109;&#101;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#119;&#104;&#97;&#116;&#32;&#119;&#101;&#32;&#97;&#114;&#101;&#32;&#108;&#111;&#111;&#107;&#105;&#110;&#103;&#32;&#102;&#111;&#114;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#76;&#101;&#116;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#110;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#32;&#102;&#105;&#114;&#115;&#116;&#32;&#105;&#110;&#116;&#101;&#103;&#101;&#114;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#110;&#43;&#50;&#61;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#110;&#101;&#120;&#116;&#32;&#99;&#111;&#110;&#115;&#101;&#99;&#117;&#116;&#105;&#118;&#101;&#32;&#111;&#100;&#100;&#32;&#105;&#110;&#116;&#101;&#103;&#101;&#114;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#116;&#101;&#112;&#32;&#52;&#46;&#32;&#84;&#114;&#97;&#110;&#115;&#108;&#97;&#116;&#101;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#110;&#116;&#111;&#32;&#97;&#110;&#32;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#46;&#32;&#82;&#101;&#115;&#116;&#97;&#116;&#101;&#32;&#116;&#104;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#112;&#114;&#111;&#98;&#108;&#101;&#109;&#32;&#105;&#110;&#32;&#97;&#32;&#115;&#101;&#110;&#116;&#101;&#110;&#99;&#101;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#104;&#101;&#32;&#112;&#114;&#111;&#100;&#117;&#99;&#116;&#32;&#111;&#102;&#32;&#116;&#104;&#101;&#32;&#116;&#119;&#111;&#32;&#99;&#111;&#110;&#115;&#101;&#99;&#117;&#116;&#105;&#118;&#101;&#32;&#111;&#100;&#100;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#110;&#116;&#101;&#103;&#101;&#114;&#115;&#32;&#105;&#115;&#32;&#51;&#50;&#51;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#46;&#52;&#55;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#110;&#92;&#108;&#101;&#102;&#116;&#40;&#110;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#51;&#50;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#116;&#101;&#112;&#32;&#53;&#46;&#32;&#83;&#111;&#108;&#118;&#101;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#32;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#46;&#53;&#55;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#110;&#61;&#51;&#50;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#66;&#114;&#105;&#110;&#103;&#32;&#97;&#108;&#108;&#32;&#116;&#104;&#101;&#32;&#116;&#101;&#114;&#109;&#115;&#32;&#116;&#111;&#32;&#111;&#110;&#101;&#32;&#115;&#105;&#100;&#101;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#56;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#110;&#45;&#51;&#50;&#51;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#70;&#97;&#99;&#116;&#111;&#114;&#32;&#116;&#104;&#101;&#32;&#116;&#114;&#105;&#110;&#111;&#109;&#105;&#97;&#108;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#110;&#45;&#49;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#110;&#43;&#49;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#85;&#115;&#101;&#32;&#116;&#104;&#101;&#32;&#90;&#101;&#114;&#111;&#32;&#80;&#114;&#111;&#100;&#117;&#99;&#116;&#32;&#80;&#114;&#111;&#112;&#101;&#114;&#116;&#121;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#111;&#108;&#118;&#101;&#32;&#116;&#104;&#101;&#32;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#110;&#45;&#49;&#55;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#110;&#43;&#49;&#57;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#110;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#49;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#110;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#49;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"258\" width=\"805\" style=\"vertical-align: -124px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p> There are two values for <em data-effect=\"italics\">n<\/em> that are solutions to this problem. So there are two sets of consecutive odd integers that will work.<\/p>\n<div data-type=\"newline\"><\/div>\n<p> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a48394af6e18c400dae3af922e6603c7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#73;&#102;&#32;&#116;&#104;&#101;&#32;&#102;&#105;&#114;&#115;&#116;&#32;&#105;&#110;&#116;&#101;&#103;&#101;&#114;&#32;&#105;&#115;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#110;&#61;&#49;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#73;&#102;&#32;&#116;&#104;&#101;&#32;&#102;&#105;&#114;&#115;&#116;&#32;&#105;&#110;&#116;&#101;&#103;&#101;&#114;&#32;&#105;&#115;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#110;&#61;&#45;&#49;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#110;&#32;&#116;&#104;&#101;&#32;&#110;&#101;&#120;&#116;&#32;&#111;&#100;&#100;&#32;&#105;&#110;&#116;&#101;&#103;&#101;&#114;&#32;&#105;&#115;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#110;&#32;&#116;&#104;&#101;&#32;&#110;&#101;&#120;&#116;&#32;&#111;&#100;&#100;&#32;&#105;&#110;&#116;&#101;&#103;&#101;&#114;&#32;&#105;&#115;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#110;&#43;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#110;&#43;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#55;&#43;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#48;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#49;&#57;&#43;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#49;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#55;&#44;&#49;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#48;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#49;&#55;&#44;&#45;&#49;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#116;&#101;&#112;&#32;&#54;&#46;&#32;&#67;&#104;&#101;&#99;&#107;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#32;&#97;&#110;&#115;&#119;&#101;&#114;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#104;&#101;&#32;&#114;&#101;&#115;&#117;&#108;&#116;&#115;&#32;&#97;&#114;&#101;&#32;&#99;&#111;&#110;&#115;&#101;&#99;&#117;&#116;&#105;&#118;&#101;&#32;&#111;&#100;&#100;&#32;&#105;&#110;&#116;&#101;&#103;&#101;&#114;&#115;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#49;&#55;&#44;&#49;&#57;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#49;&#57;&#44;&#45;&#49;&#55;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#49;&#55;&middot;&#49;&#57;&#61;&#51;&#50;&#51;&#10003;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#49;&#57;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#49;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#51;&#50;&#51;&#10003;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#66;&#111;&#116;&#104;&#32;&#112;&#97;&#105;&#114;&#115;&#32;&#111;&#102;&#32;&#99;&#111;&#110;&#115;&#101;&#99;&#117;&#116;&#105;&#118;&#101;&#32;&#105;&#110;&#116;&#101;&#103;&#101;&#114;&#115;&#32;&#97;&#114;&#101;&#32;&#115;&#111;&#108;&#117;&#116;&#105;&#111;&#110;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#116;&#101;&#112;&#32;&#55;&#46;&#32;&#65;&#110;&#115;&#119;&#101;&#114;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#32;&#113;&#117;&#101;&#115;&#116;&#105;&#111;&#110;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#104;&#101;&#32;&#99;&#111;&#110;&#115;&#101;&#99;&#117;&#116;&#105;&#118;&#101;&#32;&#105;&#110;&#116;&#101;&#103;&#101;&#114;&#115;&#32;&#97;&#114;&#101;&#32;&#49;&#55;&#44;&#32;&#49;&#57;&#32;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#49;&#57;&#44;&#45;&#49;&#55;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"259\" width=\"822\" style=\"vertical-align: -125px;\" \/><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167829809872\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836509306\">\n<div data-type=\"problem\" id=\"fs-id1167833024146\">\n<p id=\"fs-id1167833224626\">The product of two consecutive odd integers is 255. Find the integers.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836529154\">\n<p id=\"fs-id1167836520704\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8bd48ec68bf94de12af16be4f5359836_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#53;&#44;&#45;&#49;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"70\" style=\"vertical-align: -4px;\" \/> and 15, 17<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836620394\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836660223\">\n<div data-type=\"problem\" id=\"fs-id1167836417134\">\n<p id=\"fs-id1167829683765\">The product of two consecutive odd integers is 483 Find the integers.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836728273\">\n<p id=\"fs-id1167836624591\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-eb142cf5fc39ef20c0faea27e4204354_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#51;&#44;&#45;&#50;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"69\" style=\"vertical-align: -4px;\" \/> and 21, 23<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836447980\">Were you surprised by the pair of negative integers that is one of the solutions to the previous example? The product of the two positive integers and the product of the two negative integers both give positive results.<\/p>\n<p id=\"fs-id1167836393036\">In some applications, negative solutions will result from the algebra, but will not be realistic for the situation.<\/p>\n<div data-type=\"example\" id=\"fs-id1167836492373\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167833086444\">\n<div data-type=\"problem\" id=\"fs-id1167836507849\">\n<p id=\"fs-id1167836309170\">A rectangular bedroom has an area 117 square feet. The length of the bedroom is four feet more than the width. Find the length and width of the bedroom.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833059883\">\n<table id=\"fs-id1167836283397\" class=\"unnumbered unstyled can-break\" summary=\"Step 1: Read the problem. In problems involving geometric figures, a sketch can help you visualize the situation. Make a rectangle with the shorter side equal to w units and the longer one equal to w plus 4 units. Step 2: Identify what you are looking for. We are looking for the length and width. Step 3: Name what you are looking for. The length is four feet more than the width. Let w be the width of the bedroom and w plus 4 be the length of the garden. Step 4: Translate into an equation. Restate the important information in a sentence. The area of the bedroom is 117square feet. Use the formula for the area of a rectangle. A is L times W. Substitute in the variables. 117 equals open parentheses w plus 4 close parentheses w. Step 5: Solve the equation Distribute first. 117 equals w squared plus 4w. Get zero on one side. 0 equals w squared plus 4w minus 117. Factor the trinomial. 0 equals w plus 13, w minus 9. Use the Zero Product Property. 0 equals w plus 13 and 0 equals w plus 9. Solve each equation. Minus 13 equals w or 9 equals w. Since w is the width of the bedroom, it does not make sense for it to be negative. We eliminate that value for w. So w equals 9 and width is 9 feet. Find the value of the length. Length is w plus 4 equals 13. Step 6: Check the answer. Does the answer make sense? Yes, this makes sense. Step 7: Answer the question. The width of the bedroom is 9 feet and the length is 13 feet.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 1. Read<\/strong> the problem. In problems involving<\/p>\n<div data-type=\"newline\"><\/div>\n<p>geometric figures, a sketch can help you visualize<\/p>\n<div data-type=\"newline\"><\/div>\n<p>the situation.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836691642\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_05_003b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 2. Identify<\/strong> what you are looking for.<\/td>\n<td data-valign=\"top\" data-align=\"left\">We are looking for the length and width.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 3. Name<\/strong> what you are looking for.<\/td>\n<td data-valign=\"top\" data-align=\"left\">Let <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cba9af47bcc105aa4e2879c231fa76d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#119;&#61;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"31\" style=\"vertical-align: 0px;\" \/> the width of the bedroom.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">The length is four feet more than the width.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-286b11fbda60118343a4b14e06b10f68_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#119;&#43;&#52;&#61;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"62\" style=\"vertical-align: -2px;\" \/> the length of the garden<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 4. Translate<\/strong> into an equation.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Restate the important information in a sentence.<\/td>\n<td data-valign=\"top\" data-align=\"left\">The area of the bedroom is 117 square feet.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Use the formula for the area of a rectangle.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5730f1b3825ab511c6f7d858e8cf3e7f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;&#61;&#108;&middot;&#119;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"56\" style=\"vertical-align: 0px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Substitute in the variables.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-acc3439a5521ca7c317330958df3d2a8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#49;&#55;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#119;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#119;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"122\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 5. Solve<\/strong> the equation Distribute first.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3867e9344fcc786e4f5f6ce540042398_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#49;&#55;&#61;&#123;&#119;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#119;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"114\" style=\"vertical-align: -2px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Get zero on one side.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3867e9344fcc786e4f5f6ce540042398_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#49;&#55;&#61;&#123;&#119;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#119;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"114\" style=\"vertical-align: -2px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Factor the trinomial.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-696b68a472dfc24ed27f778cb9cb8bbd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#48;&#61;&#123;&#119;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#119;&#45;&#49;&#49;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"146\" style=\"vertical-align: -2px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Use the Zero Product Property.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-42fc4a6a1f4d764dce7b2cdaa346b298_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#46;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#48;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#119;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#119;&#45;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"169\" style=\"vertical-align: -7px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Solve each equation.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2e698590fbe064f3b72dab9fe9b46553_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#53;&#46;&#54;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#48;&#61;&#119;&#43;&#49;&#51;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#48;&#61;&#119;&#45;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"188\" style=\"vertical-align: -2px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Since <em data-effect=\"italics\">w<\/em> is the width of the bedroom, it does not<\/p>\n<div data-type=\"newline\"><\/div>\n<p>make sense for it to be negative. We eliminate that value for <em data-effect=\"italics\">w<\/em>.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ec206f979fc7fdbaa47d4b6b8c059843_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#111;&#118;&#101;&#114;&#108;&#105;&#110;&#101;&#123;&#41;&#45;&#49;&#51;&#61;&#119;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#57;&#61;&#119;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"191\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td><\/td>\n<td data-valign=\"top\" data-align=\"right\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a8eda7c05ef6483c613d60205042ba9d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#119;&#61;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"46\" style=\"vertical-align: 0px;\" \/>\u2003Width is 9 feet.&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Find the value of the length.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e7fb004991c1720e1799044d71897435_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#119;&#43;&#52;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"44\" style=\"vertical-align: -2px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1b9925b0fdd8269feef2d341ef73f068_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#43;&#52;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"40\" style=\"vertical-align: -2px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3e782f9461221a9c1c9a35590299529d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#55;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"1\" width=\"1\" style=\"vertical-align: 0px;\" \/>13\u2003\u2003Length is 13 feet.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 6. Check<\/strong> the answer.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>Does the answer make sense?<\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167836326705\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_05_003a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p>Yes, this makes sense.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 7. Answer<\/strong> the question.<\/td>\n<td data-valign=\"top\" data-align=\"left\">The width of the bedroom is 9 feet and<\/p>\n<div data-type=\"newline\"><\/div>\n<p>the length is 13 feet.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167829716003\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836693448\">\n<div data-type=\"problem\" id=\"fs-id1167836528456\">\n<p id=\"fs-id1167829748017\">A rectangular sign has area 30 square feet. The length of the sign is one foot more than the width. Find the length and width of the sign.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833047687\">\n<p id=\"fs-id1167829879335\">The width is 5 feet and length is 6 feet.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836619400\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836416135\">\n<div data-type=\"problem\" id=\"fs-id1167829689459\">\n<p id=\"fs-id1167836697766\">A rectangular patio has area 180 square feet. The width of the patio is three feet less than the length. Find the length and width of the patio.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829691910\">\n<p id=\"fs-id1167836550460\">The length of the patio is 12 feet and the width 15 feet.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836283441\">In the next example, we will use the Pythagorean Theorem <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-da1f4306b356ff47ad4b6a5a2fad45f0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#61;&#123;&#99;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"114\" style=\"vertical-align: -7px;\" \/> This formula gives the relation between the legs and the hypotenuse of a right triangle.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167833227285\" data-alt=\"Figure shows a right triangle with the shortest side being a, the second side being b and the hypotenuse being c.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_05_004_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure shows a right triangle with the shortest side being a, the second side being b and the hypotenuse being c.\" \/><\/span><\/p>\n<p id=\"fs-id1167836289692\">We will use this formula to in the next example.<\/p>\n<div data-type=\"example\" id=\"fs-id1167836558176\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167836547336\">\n<div data-type=\"problem\" id=\"fs-id1167836529160\">\n<p id=\"fs-id1167836392939\">A boat\u2019s sail is in the shape of a right triangle as shown. The hypotenuse will be 17 feet long. The length of one side will be 7 feet less than the length of the other side. Find the lengths of the sides of the sail.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167836688021\" data-alt=\"Figure shows a right triangle with the shortest side being x, the second side being x minus 7 and the hypotenuse being 17.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_05_005_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure shows a right triangle with the shortest side being x, the second side being x minus 7 and the hypotenuse being 17.\" \/><\/span><\/div>\n<div data-type=\"solution\" id=\"fs-id1167833025641\">\n<table id=\"fs-id1167829651452\" class=\"unnumbered unstyled can-break\" summary=\"Step 1: Read the problem. Step 2: Identify what you are looking for. We are looking for the lengths of the sides of the sail. Step 3: Name what you are looking for. One side is 7 less than the other. Let x be length of a side of the sail and x minus 7 be length of other side. Step 4: Translate into an equation. Since this is a right triangle we can use the Pythagorean Theorem a squared plus b squared equals c squared. Substitute in the variables: x squared plus open parentheses x minus 7 close parentheses squared equals 17 squared. Step 5: Solve the equation. Simplify: x squared plus x squared minus 14x plus 49 is 289. 2 x squared minus 14x plus 49 equals 289. It is a quadratic equation, so get zero on one side: 2 x squared minus 14x minus 240 equals 0. Factor the greatest common factor. Factor the trinomial. 2 open parentheses x minus 15 close parentheses open parentheses x plus 8 close parentheses. Use the Zero Product Property: 2 is not equal to 0, x minus 15 equals 0 or x plus 8 equal 0. Solve: x is 15 or x is minus 8. Since x is a side of the triangle, x equals minus 8 does not make sense. Find the length of the other side. If the length of one side is 15, then the length of the other side is 15 minus 7 equals 8. Step 6: Check the answer in the problem. Do these numbers make sense? Step 7: Answer the question: The sides of the sail are 8, 15 and 17 feet.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 1. Read<\/strong> the problem<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 2. Identify<\/strong> what you are looking for.<\/td>\n<td data-valign=\"top\" data-align=\"left\">We are looking for the lengths of the<\/p>\n<div data-type=\"newline\"><\/div>\n<p>sides of the sail.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 3. Name<\/strong> what you are looking for.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>One side is 7 less than the other.<\/td>\n<td data-valign=\"top\" data-align=\"left\">Let <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e7bbcde7229c9d7d6f7f2b6793961e97_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"28\" style=\"vertical-align: 0px;\" \/> length of a side of the sail.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-55597f973bde553251bd5802b5e0609f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#55;&#61;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"59\" style=\"vertical-align: 0px;\" \/> length of other side<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 4. Translate<\/strong> into an equation. Since this is a<\/p>\n<div data-type=\"newline\"><\/div>\n<p>right triangle we can use the Pythagorean Theorem.<\/td>\n<td data-valign=\"bottom\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3f8b2f080c7b832e2facdebb35ec8cd5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#46;&#56;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#61;&#123;&#99;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"93\" style=\"vertical-align: -2px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Substitute in the variables.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d653b6cc2b119b73c71cd671ecbb3dae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#123;&#49;&#55;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"150\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 5. Solve<\/strong> the equation<\/p>\n<div data-type=\"newline\"><\/div>\n<p>Simplify.<\/td>\n<td data-valign=\"bottom\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-19048c71cbf0db8e48326f129b272a64_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#52;&#120;&#43;&#52;&#57;&#61;&#50;&#56;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"197\" style=\"vertical-align: -2px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-869129fbf0f473c491efe146e165f8da_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#52;&#120;&#43;&#52;&#57;&#61;&#50;&#56;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"166\" style=\"vertical-align: -2px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">It is a quadratic equation, so get zero on one side.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-71e0c01b53124f5eca5a53c55766dc23_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#52;&#120;&#45;&#50;&#52;&#48;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"157\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Factor the greatest common factor.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-96b25c1e45d012edc13ad5b64399a30b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#56;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#55;&#120;&#45;&#49;&#50;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"168\" style=\"vertical-align: -7px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Factor the trinomial.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ca79fbfd61b3335f2c656052efc061f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#49;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"165\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Use the Zero Product Property.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fb9b90f8d568d1eea768d5175efd7ad1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#92;&#110;&#101;&#32;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#45;&#49;&#53;&#61;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#54;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#43;&#56;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"234\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Solve.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b9869b46ff998f919b97d07b069eb2bc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#92;&#110;&#101;&#32;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#61;&#49;&#53;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#61;&#45;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"212\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Since <em data-effect=\"italics\">x<\/em> is a side of the triangle, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7852d0704798164bd2ce5b572ac50cd9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"57\" style=\"vertical-align: 0px;\" \/> does not<\/p>\n<div data-type=\"newline\"><\/div>\n<p>make sense.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ccf580d59ae7c64362566e86546a94c4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#92;&#110;&#101;&#32;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#61;&#49;&#53;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#111;&#118;&#101;&#114;&#108;&#105;&#110;&#101;&#123;&#41;&#120;&#61;&#45;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"219\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Find the length of the other side.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">\u2003\u2003\u2003If the length of one side is<\/p>\n<div data-type=\"newline\"><\/div>\n<p>\u2003\u2003\u2003then the length of the other side is<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832981684\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_05_006l_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167836485100\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_05_006m_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167836545662\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_05_006n_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p>8 is the length of the other side.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 6. Check<\/strong> the answer in the problem<\/p>\n<div data-type=\"newline\"><\/div>\n<p>Do these numbers make sense?<\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167836551721\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_05_006a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 7. Answer<\/strong> the question<\/td>\n<td data-valign=\"top\" data-align=\"left\">The sides of the sail are 8, 15 and 17 feet.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836623889\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836450411\">\n<div data-type=\"problem\" id=\"fs-id1167836294961\">\n<p id=\"fs-id1167836560114\">Justine wants to put a deck in the corner of her backyard in the shape of a right triangle. The length of one side of the deck is 7 feet more than the other side. The hypotenuse is 13. Find the lengths of the two sides of the deck.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833381531\">\n<p>5 feet and 12 feet<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836377845\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167829747291\">\n<div data-type=\"problem\" id=\"fs-id1167836312153\">\n<p id=\"fs-id1167836534023\">A meditation garden is in the shape of a right triangle, with one leg 7 feet. The length of the hypotenuse is one more than the length of the other leg. Find the lengths of the hypotenuse and the other leg.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836628723\">\n<p id=\"fs-id1167836447576\">24 feet and 25 feet<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1171792850594\">The next example uses the function that gives the height of an object as a function of time when it is thrown from 80 feet above the ground.<\/p>\n<div data-type=\"example\" id=\"fs-id1167833022018\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167836503972\">\n<div data-type=\"problem\" id=\"fs-id1167829748344\">\n<p id=\"fs-id1167836330850\">Dennis is going to throw his rubber band ball upward from the top of a campus building. When he throws the rubber band ball from 80 feet above the ground, the function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-507333757ea1da4ee539d12d1f4dc085_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;&#92;&#108;&#101;&#102;&#116;&#40;&#116;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#49;&#54;&#123;&#116;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#52;&#116;&#43;&#56;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"188\" style=\"vertical-align: -4px;\" \/> models the height, <em data-effect=\"italics\">h<\/em>, of the ball above the ground as a function of time, <em data-effect=\"italics\">t.<\/em> Find:<\/p>\n<p id=\"fs-id1167836526114\"><span class=\"token\">\u24d0<\/span> the zeros of this function which tell us when the ball hits the ground<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> when the ball will be 80 feet above the ground<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span> the height of the ball at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-eec54adef7198c4decb9169a2b472009_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"38\" style=\"vertical-align: 0px;\" \/> seconds.<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836612483\">\n<p id=\"fs-id1167836576454\"><span class=\"token\">\u24d0<\/span> The zeros of this function are found by solving <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4b582fa44ca02eccb70ab4ebc54d0a16_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;&#92;&#108;&#101;&#102;&#116;&#40;&#116;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"70\" style=\"vertical-align: -4px;\" \/> This will tell us when the ball will hit the ground.<\/p>\n<div data-type=\"newline\"><\/div>\n<p> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-05a9caea8dd7ed593947fe29c1333fee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#55;&#46;&#52;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#104;&#92;&#108;&#101;&#102;&#116;&#40;&#116;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#117;&#98;&#115;&#116;&#105;&#116;&#117;&#116;&#101;&#32;&#105;&#110;&#32;&#116;&#104;&#101;&#32;&#112;&#111;&#108;&#121;&#110;&#111;&#109;&#105;&#97;&#108;&#32;&#102;&#111;&#114;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#104;&#92;&#108;&#101;&#102;&#116;&#40;&#116;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#46;&#48;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#49;&#54;&#123;&#116;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#52;&#116;&#43;&#56;&#48;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#70;&#97;&#99;&#116;&#111;&#114;&#32;&#116;&#104;&#101;&#32;&#71;&#67;&#70;&#44;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#49;&#54;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#46;&#51;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#49;&#54;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#116;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#116;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#70;&#97;&#99;&#116;&#111;&#114;&#32;&#116;&#104;&#101;&#32;&#116;&#114;&#105;&#110;&#111;&#109;&#105;&#97;&#108;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#46;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#49;&#54;&#92;&#108;&#101;&#102;&#116;&#40;&#116;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#116;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#85;&#115;&#101;&#32;&#116;&#104;&#101;&#32;&#90;&#101;&#114;&#111;&#32;&#80;&#114;&#111;&#100;&#117;&#99;&#116;&#32;&#80;&#114;&#111;&#112;&#101;&#114;&#116;&#121;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#111;&#108;&#118;&#101;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#116;&#45;&#53;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#116;&#43;&#49;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#116;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#116;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"125\" width=\"713\" style=\"vertical-align: -56px;\" \/><\/p>\n<p id=\"fs-id1167836334175\">The result <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4291011c7cdcdfeece3f6896faea35e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#61;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"38\" style=\"vertical-align: 0px;\" \/> tells us the ball will hit the ground 5 seconds after it is thrown. Since time cannot be negative, the result <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-35d11e298e261d1ee0e702c094322281_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#61;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"52\" style=\"vertical-align: -1px;\" \/> is discarded.<\/p>\n<p id=\"fs-id1167836282342\"><span class=\"token\">\u24d1<\/span> The ball will be 80 feet above the ground when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4f86349221fd02748244458d6cd3e3aa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;&#92;&#108;&#101;&#102;&#116;&#40;&#116;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#56;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"79\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9833d1523bae7945066658c8ae0e8652_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#55;&#46;&#53;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#104;&#92;&#108;&#101;&#102;&#116;&#40;&#116;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#56;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#117;&#98;&#115;&#116;&#105;&#116;&#117;&#116;&#101;&#32;&#105;&#110;&#32;&#116;&#104;&#101;&#32;&#112;&#111;&#108;&#121;&#110;&#111;&#109;&#105;&#97;&#108;&#32;&#102;&#111;&#114;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#104;&#92;&#108;&#101;&#102;&#116;&#40;&#116;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#46;&#48;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#49;&#54;&#123;&#116;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#52;&#116;&#43;&#56;&#48;&#61;&#56;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#117;&#98;&#116;&#114;&#97;&#99;&#116;&#32;&#56;&#48;&#32;&#102;&#114;&#111;&#109;&#32;&#98;&#111;&#116;&#104;&#32;&#115;&#105;&#100;&#101;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#49;&#54;&#123;&#116;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#52;&#116;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#70;&#97;&#99;&#116;&#111;&#114;&#32;&#116;&#104;&#101;&#32;&#71;&#67;&#70;&#44;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#49;&#54;&#116;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#49;&#54;&#116;&#92;&#108;&#101;&#102;&#116;&#40;&#116;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#85;&#115;&#101;&#32;&#116;&#104;&#101;&#32;&#90;&#101;&#114;&#111;&#32;&#80;&#114;&#111;&#100;&#117;&#99;&#116;&#32;&#80;&#114;&#111;&#112;&#101;&#114;&#116;&#121;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#111;&#108;&#118;&#101;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#54;&#116;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#116;&#45;&#52;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#116;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#116;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#104;&#101;&#32;&#98;&#97;&#108;&#108;&#32;&#119;&#105;&#108;&#108;&#32;&#98;&#101;&#32;&#97;&#116;&#32;&#56;&#48;&#32;&#102;&#101;&#101;&#116;&#32;&#116;&#104;&#101;&#32;&#109;&#111;&#109;&#101;&#110;&#116;&#32;&#68;&#101;&#110;&#110;&#105;&#115;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#111;&#115;&#115;&#101;&#115;&#32;&#116;&#104;&#101;&#32;&#98;&#97;&#108;&#108;&#32;&#97;&#110;&#100;&#32;&#116;&#104;&#101;&#110;&#32;&#52;&#32;&#115;&#101;&#99;&#111;&#110;&#100;&#115;&#32;&#108;&#97;&#116;&#101;&#114;&#44;&#32;&#119;&#104;&#101;&#110;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#32;&#98;&#97;&#108;&#108;&#32;&#105;&#115;&#32;&#102;&#97;&#108;&#108;&#105;&#110;&#103;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"193\" width=\"745\" style=\"vertical-align: -91px;\" \/><\/p>\n<p><span class=\"token\">\u24d2<\/span> To find the height ball at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-eec54adef7198c4decb9169a2b472009_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"38\" style=\"vertical-align: 0px;\" \/> seconds we find <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e94802f980d372b79768e10ac74a0e00_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"43\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-934b1c8aaaa1631b755e7726ca7d1b78_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#104;&#92;&#108;&#101;&#102;&#116;&#40;&#116;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#49;&#54;&#123;&#116;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#52;&#116;&#43;&#56;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#111;&#32;&#102;&#105;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#104;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#117;&#98;&#115;&#116;&#105;&#116;&#117;&#116;&#101;&#32;&#50;&#32;&#102;&#111;&#114;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#116;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#104;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#49;&#54;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#52;&middot;&#50;&#43;&#56;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#105;&#109;&#112;&#108;&#105;&#102;&#121;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#104;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#49;&#52;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#102;&#116;&#101;&#114;&#32;&#50;&#32;&#115;&#101;&#99;&#111;&#110;&#100;&#115;&#44;&#32;&#116;&#104;&#101;&#32;&#98;&#97;&#108;&#108;&#32;&#119;&#105;&#108;&#108;&#32;&#98;&#101;&#32;&#97;&#116;&#32;&#49;&#52;&#52;&#32;&#102;&#101;&#101;&#116;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"709\" style=\"vertical-align: -37px;\" \/><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836289305\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836706000\">\n<div data-type=\"problem\" id=\"fs-id1167836706002\">\n<p id=\"fs-id1167836530068\">Genevieve is going to throw a rock from the top a trail overlooking the ocean. When she throws the rock upward from 160 feet above the ocean, the function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-46591bde8ef77b6bbacff35f773c641d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;&#92;&#108;&#101;&#102;&#116;&#40;&#116;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#49;&#54;&#123;&#116;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#56;&#116;&#43;&#49;&#54;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"197\" style=\"vertical-align: -4px;\" \/> models the height, <em data-effect=\"italics\">h<\/em>, of the rock above the ocean as a function of time, <em data-effect=\"italics\">t<\/em>. Find:<\/p>\n<p id=\"fs-id1167836525352\"><span class=\"token\">\u24d0<\/span> the zeros of this function which tell us when the rock will hit the ocean<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> when the rock will be 160 feet above the ocean.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span> the height of the rock at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-17565a4a588b04248f6d1458aca7e316_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#61;&#49;&#46;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"52\" style=\"vertical-align: -1px;\" \/> seconds.<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836585148\">\n<p id=\"fs-id1167829596335\"><span class=\"token\">\u24d0<\/span> 5 <span class=\"token\">\u24d1<\/span> 0;3 <span class=\"token\">\u24d2<\/span> 196<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836550916\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167829744740\">\n<div data-type=\"problem\" id=\"fs-id1167836684102\">\n<p id=\"fs-id1167836297652\">Calib is going to throw his lucky penny from his balcony on a cruise ship. When he throws the penny upward from 128 feet above the ground, the function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2d4497e8db2957f038443846de763ae2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;&#92;&#108;&#101;&#102;&#116;&#40;&#116;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#49;&#54;&#123;&#116;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#50;&#116;&#43;&#49;&#50;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"197\" style=\"vertical-align: -4px;\" \/> models the height, <em data-effect=\"italics\">h<\/em>, of the penny above the ocean as a function of time, <em data-effect=\"italics\">t<\/em>. Find:<\/p>\n<p id=\"fs-id1167836545880\"><span class=\"token\">\u24d0<\/span> the zeros of this function which is when the penny will hit the ocean<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> when the penny will be 128 feet above the ocean.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span> the height the penny will be at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-abf8f2abfdf1913d009a51ad64786690_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"38\" style=\"vertical-align: -1px;\" \/> seconds which is when the penny will be at its highest point.<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836531169\">\n<p id=\"fs-id1167836531171\"><span class=\"token\">\u24d0<\/span> 4 <span class=\"token\">\u24d1<\/span> 0;2 <span class=\"token\">\u24d2<\/span> 144<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167833060072\" class=\"media-2\">\n<p id=\"fs-id1167836698870\">Access this online resource for additional instruction and practice with quadratic equations.<\/p>\n<ul id=\"fs-id1167836630386\" data-display=\"block\">\n<li><a href=\"https:\/\/openstax.org\/l\/37ZeroProperty\">Beginning Algebra &amp; Solving Quadratics with the Zero Property<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167829711704\">\n<h3 data-type=\"title\">Key Concepts<\/h3>\n<ul id=\"fs-id1167836356276\" data-bullet-style=\"bullet\">\n<li><strong data-effect=\"bold\">Polynomial Equation:<\/strong> A polynomial equation is an equation that contains a polynomial expression. The degree of the polynomial equation is the degree of the polynomial.<\/li>\n<li><strong data-effect=\"bold\">Quadratic Equation:<\/strong> An equation of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-29c8b40df2f348455a0b8aec28190877_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#120;&#43;&#99;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"129\" style=\"vertical-align: -2px;\" \/> is called a quadratic equation.\n<div data-type=\"newline\"><\/div>\n<div data-type=\"equation\" id=\"fs-id1167836518426\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e73ae5631fa44c8f99aec4ace13623d1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#44;&#98;&#44;&#99;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#114;&#101;&#32;&#114;&#101;&#97;&#108;&#32;&#110;&#117;&#109;&#98;&#101;&#114;&#115;&#32;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#97;&#92;&#110;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"254\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/li>\n<li><strong data-effect=\"bold\">Zero Product Property:<\/strong> If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1178b2b1f3897a309978aa3009660bb1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&middot;&#98;&#61;&#48;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"54\" style=\"vertical-align: -4px;\" \/> then either <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-072bed0ebf929f9d7c14da365c8512a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\" \/> or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-597ae6c01abdebf6816c77f3ce1fd9de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"40\" style=\"vertical-align: 0px;\" \/> or both.<\/li>\n<li><strong data-effect=\"bold\">How to use the Zero Product Property<\/strong>\n<ol type=\"1\" class=\"stepwise\">\n<li>Set each factor equal to zero.<\/li>\n<li>Solve the linear equations.<\/li>\n<li>Check.<\/li>\n<\/ol>\n<\/li>\n<li><strong data-effect=\"bold\">How to solve a quadratic equation by factoring.<\/strong>\n<ol id=\"fs-id1167836543767\" type=\"1\" class=\"stepwise\">\n<li>Write the quadratic equation in standard form, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ff55a4ef4f0f0f08cb24335fede5832e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#120;&#43;&#99;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"133\" style=\"vertical-align: -2px;\" \/><\/li>\n<li>Factor the quadratic expression.<\/li>\n<li>Use the Zero Product Property.\n<div data-type=\"newline\"><\/div>\n<\/li>\n<li>Solve the linear equations.<\/li>\n<li>Check. Substitute each solution separately into the original equation.<\/li>\n<\/ol>\n<\/li>\n<li><strong data-effect=\"bold\">Zero of a Function:<\/strong> For any function <em data-effect=\"italics\">f<\/em>, if <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4b6a350c5088c4d60bfbe05674203c9c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#48;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"74\" style=\"vertical-align: -4px;\" \/> then <em data-effect=\"italics\">x<\/em> is a zero of the function.<\/li>\n<li><strong data-effect=\"bold\">How to use a problem solving strategy to solve word problems.<\/strong>\n<ol id=\"fs-id1167836549434\" type=\"1\" class=\"stepwise\">\n<li><strong data-effect=\"bold\">Read<\/strong> the problem. Make sure all the words and ideas are understood.<\/li>\n<li><strong data-effect=\"bold\">Identify<\/strong> what we are looking for.<\/li>\n<li><strong data-effect=\"bold\">Name<\/strong> what we are looking for. Choose a variable to represent that quantity.<\/li>\n<li><strong data-effect=\"bold\">Translate<\/strong> into an equation. It may be helpful to restate the problem in one sentence with all the important information. Then, translate the English sentence into an algebraic equation.<\/li>\n<li><strong data-effect=\"bold\">Solve<\/strong> the equation using appropriate algebra techniques.<\/li>\n<li><strong data-effect=\"bold\">Check<\/strong> the answer in the problem and make sure it makes sense.<\/li>\n<li><strong data-effect=\"bold\">Answer<\/strong> the question with a complete sentence.<\/li>\n<\/ol>\n<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167829685381\">\n<h3 data-type=\"title\">Section Exercises<\/h3>\n<div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1167829656544\">\n<h4 data-type=\"title\">Practice Makes Perfect<\/h4>\n<p id=\"fs-id1167833018910\"><strong data-effect=\"bold\">Use the Zero Product Property<\/strong><\/p>\n<p id=\"fs-id1167836434570\">In the following exercises, solve.<\/p>\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167836391400\">\n<p id=\"fs-id1167829624077\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-93a0f3054bc08b40659e005300861f2e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#97;&#45;&#49;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#97;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"169\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829752365\">\n<p id=\"fs-id1167836545075\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-05fe5e91530935e0d22c85e9960c6e0a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&#49;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#47;&#125;&#51;&#44;&#97;&#61;&#55;&#92;&#116;&#101;&#120;&#116;&#123;&#47;&#125;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"135\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833024298\">\n<div data-type=\"problem\" id=\"fs-id1167829589900\">\n<p id=\"fs-id1167829589902\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-29d75120c2e1dd4cc1f66d4f797ebfb0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#98;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#98;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"156\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836623729\">\n<div data-type=\"problem\" id=\"fs-id1167836623731\">\n<p id=\"fs-id1167836798088\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-70c7d61a4eabd5bbaf6321eac8fb6e3c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#109;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#50;&#109;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"138\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836625460\">\n<p id=\"fs-id1167836525375\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a4a624b992c8a892ba78934f9978fcc6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#48;&#44;&#109;&#61;&#53;&#92;&#116;&#101;&#120;&#116;&#123;&#47;&#125;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"130\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836504078\">\n<div data-type=\"problem\" id=\"fs-id1167836504080\">\n<p id=\"fs-id1167836626599\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1f9e3c75347c389154ca807e763586d8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#120;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#120;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"118\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833082106\">\n<div data-type=\"problem\" id=\"fs-id1167833082108\">\n<p id=\"fs-id1167832984060\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ec654acc58870cfd0cc3c84c31af393f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#120;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"103\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8000322cad1feecda9d4763ad83b7161_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#47;&#125;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"59\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829741712\">\n<div data-type=\"problem\" id=\"fs-id1167829741714\">\n<p id=\"fs-id1167836620973\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d775dbd82cd40ad1fa2bc785e3e4ace0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#121;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"102\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167829627758\"><strong data-effect=\"bold\">Solve Quadratic Equations by Factoring<\/strong><\/p>\n<p id=\"fs-id1167836288545\">In the following exercises, solve.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167836529760\">\n<div data-type=\"problem\" id=\"fs-id1167836529763\">\n<p id=\"fs-id1167824585340\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d0cc5ef42ca7a4c548bbb1708a2ea514_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#54;&#97;&#61;&#50;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"116\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836449436\">\n<p id=\"fs-id1167829579647\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9f5b0433fed22e9a24a7a148c149d309_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&#45;&#53;&#92;&#116;&#101;&#120;&#116;&#123;&#47;&#125;&#52;&#44;&#97;&#61;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"123\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836635232\">\n<div data-type=\"problem\" id=\"fs-id1167836511402\">\n<p id=\"fs-id1167836511404\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3594ad7fd63ecbea04e76f1882d67da9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#43;&#55;&#98;&#61;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"109\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829717788\">\n<div data-type=\"problem\" id=\"fs-id1167836408890\">\n<p id=\"fs-id1167836549429\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b98077a64ff228d60a03df44dd2913ff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#55;&#109;&#45;&#49;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"128\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167824732278\">\n<p id=\"fs-id1167829715419\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bff3afc54f8667f290fe804c08a7ec4c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#53;&#92;&#116;&#101;&#120;&#116;&#123;&#47;&#125;&#52;&#44;&#109;&#61;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"122\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836318326\">\n<div data-type=\"problem\" id=\"fs-id1167836318328\">\n<p id=\"fs-id1167833381336\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-16d2a0edd8c4eefdf1d4d4b26ef7bb38_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#61;&#53;&#45;&#54;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"92\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829715838\">\n<div data-type=\"problem\" id=\"fs-id1167836667079\">\n<p id=\"fs-id1167836667081\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c1ed3b816d385cfe504fef1708f0eaf1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#52;&#97;&#61;&#55;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"116\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836662585\">\n<p id=\"fs-id1167836503843\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-719fec2e92d4ad88c8a1879f954e2943_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&#45;&#49;&#44;&#97;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"106\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829599736\">\n<div data-type=\"problem\" id=\"fs-id1167829599738\">\n<p id=\"fs-id1167829930662\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9d7cad00610103a341d954133b818cbe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#50;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#53;&#98;&#61;&#45;&#57;&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"134\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833386007\">\n<div data-type=\"problem\" id=\"fs-id1167833386009\">\n<p id=\"fs-id1167833022584\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-95b8d2b9adcbd92d8ef0b098822b1e9f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#57;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#52;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"92\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167829747887\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4f5794996b90bc25054371640930cdd0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#49;&#50;&#92;&#116;&#101;&#120;&#116;&#123;&#47;&#125;&#55;&#44;&#109;&#61;&#45;&#49;&#50;&#92;&#116;&#101;&#120;&#116;&#123;&#47;&#125;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"171\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836296256\">\n<div data-type=\"problem\" id=\"fs-id1167836296259\">\n<p id=\"fs-id1167829744476\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-73410aa4fdd1a9ddf4f68a0b7304906b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#50;&#53;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"67\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836543476\">\n<div data-type=\"problem\" id=\"fs-id1167836543478\">\n<p id=\"fs-id1167836616320\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bb72ff4b8afc88ed5846a13b27068bc6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#54;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#56;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"74\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836546088\">\n<p id=\"fs-id1167836546090\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7f59f99d42c678bc45078d1da184f847_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#57;&#92;&#116;&#101;&#120;&#116;&#123;&#47;&#125;&#52;&#44;&#121;&#61;&#57;&#92;&#116;&#101;&#120;&#116;&#123;&#47;&#125;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"141\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836526027\">\n<div data-type=\"problem\" id=\"fs-id1167829693190\">\n<p id=\"fs-id1167829693192\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f54fa58acae4c90e2ecfac9f1a734496_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#52;&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#61;&#50;&#50;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"84\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836386707\">\n<div data-type=\"problem\" id=\"fs-id1167836386709\">\n<p id=\"fs-id1167833017831\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-76ba031ab0aad0ebd2115099eb9260fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#50;&#49;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#61;&#51;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"86\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829748023\">\n<p id=\"fs-id1167829748025\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fd916dbf03a024b73671b26d3d9c9d66_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#61;&#45;&#54;&#92;&#116;&#101;&#120;&#116;&#123;&#47;&#125;&#49;&#49;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#110;&#125;&#61;&#54;&#92;&#116;&#101;&#120;&#116;&#123;&#47;&#125;&#49;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"160\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167836524654\">\n<p id=\"fs-id1167836524656\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2d4c107248b87fa695248bf9b1de88a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#48;&#48;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"75\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833385717\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167836597807\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-28c60870b6139bcd3ae17a0b531554ba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"157\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836611503\">\n<p id=\"fs-id1167833022610\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-38adf61b9b13f505340ff41243097c91_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#50;&#44;&#120;&#61;&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"106\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829832924\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167836292046\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-91a65ade9d3f673d50a86c93e852f29a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#112;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#112;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"155\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829739370\">\n<div data-type=\"problem\" id=\"fs-id1167833057313\">\n<p id=\"fs-id1167833057315\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-32cc0eed735b0e3845db366be026af2f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#120;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#52;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"176\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836487200\">\n<p id=\"fs-id1167836487202\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9842b2cf64f098f5c4be7e869535effb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#51;&#92;&#116;&#101;&#120;&#116;&#123;&#47;&#125;&#50;&#44;&#120;&#61;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"124\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833181233\">\n<div data-type=\"problem\" id=\"fs-id1167836299581\">\n<p id=\"fs-id1167836299583\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c03d251f07574cb36f1d22924185c2b8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#52;&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"151\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836706420\">\n<div data-type=\"problem\" id=\"fs-id1167833085272\">\n<p id=\"fs-id1167833085274\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4acf85714dba264425670c969df01cd0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#120;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#49;&#50;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"171\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836357235\">\n<p id=\"fs-id1167836357237\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9842b2cf64f098f5c4be7e869535effb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#51;&#92;&#116;&#101;&#120;&#116;&#123;&#47;&#125;&#50;&#44;&#120;&#61;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"124\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836362558\">\n<div data-type=\"problem\" id=\"fs-id1167836704677\">\n<p id=\"fs-id1167836704679\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-32bc7d0af8859de156d0346544dec274_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#121;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#121;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#56;&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"169\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833046960\">\n<div data-type=\"problem\" id=\"fs-id1167836626547\">\n<p id=\"fs-id1167836626549\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-139a10fd7795c15bfb63f5cb9e443592_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#48;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#48;&#120;&#61;&#45;&#52;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"140\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836504055\">\n<p id=\"fs-id1167836504057\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1743f7050a414dac26a15b842a22aee8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#50;&#92;&#116;&#101;&#120;&#116;&#123;&#47;&#125;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"74\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836450605\">\n<div data-type=\"problem\" id=\"fs-id1167836340606\">\n<p id=\"fs-id1167836340608\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ab89c9f38d694606807617785c241543_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#56;&#121;&#61;&#45;&#50;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"130\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836493013\">\n<div data-type=\"problem\" id=\"fs-id1167829789955\">\n<p id=\"fs-id1167829789957\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a1bdc2ec3852941fe0b757376f24d718_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#53;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#48;&#120;&#61;&#52;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"126\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836529136\">\n<p id=\"fs-id1167836529138\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-79bd9f03309e3eca3c227eb5f4dbe5a8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#50;&#44;&#120;&#61;&#45;&#52;&#92;&#116;&#101;&#120;&#116;&#123;&#47;&#125;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"125\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836704609\">\n<div data-type=\"problem\" id=\"fs-id1167833349767\">\n<p id=\"fs-id1167833349769\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e280761c6f94b960ea82741e085345c7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#52;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#55;&#55;&#121;&#61;&#45;&#51;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"137\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836407331\">\n<div data-type=\"problem\" id=\"fs-id1167836319739\">\n<p id=\"fs-id1167836319741\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e859a0853dfd4257a039172115ae9a33_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#56;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#57;&#61;&#45;&#50;&#49;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"131\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836424061\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d09600bbfc732239cba44b8c647fd3da_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#51;&#92;&#116;&#101;&#120;&#116;&#123;&#47;&#125;&#50;&#44;&#120;&#61;&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#47;&#125;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"143\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167836312383\">\n<p id=\"fs-id1167836312385\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2c60f908a2326b15e44855a9af98d307_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#54;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#50;&#61;&#45;&#51;&#50;&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"138\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836286867\">\n<div data-type=\"problem\" id=\"fs-id1167836329708\">\n<p id=\"fs-id1167836329710\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f682063ff603c0cc9f70fae77e6d1944_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#54;&#123;&#112;&#125;&#94;&#123;&#51;&#125;&#61;&#50;&#52;&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#43;&#57;&#112;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"131\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836493553\">\n<p id=\"fs-id1167829715616\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c3096beec16f3ee106c43c538b0e59c4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#112;&#61;&#48;&#44;&#112;&#61;&frac34;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"77\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829595273\">\n<div data-type=\"problem\" id=\"fs-id1167829595275\">\n<p id=\"fs-id1167836298187\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3ffab47f0b8489a27103e931bafe7e66_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#109;&#125;&#94;&#123;&#51;&#125;&#45;&#50;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#109;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"116\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829790573\">\n<div data-type=\"problem\" id=\"fs-id1167836329438\">\n<p id=\"fs-id1167836329440\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2908b068d199578ad6059a8b74bd3430_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#43;&#55;&#50;&#120;&#61;&#50;&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"135\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836574901\">\n<p id=\"fs-id1167836295294\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7a770283c17b31f4bdfecd22f6612324_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#48;&#44;&#120;&#61;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"93\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836300282\">\n<div data-type=\"problem\" id=\"fs-id1167836300285\">\n<p id=\"fs-id1167836515605\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e4783cd84f4196d2a63c293a1e401b41_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#123;&#121;&#125;&#94;&#123;&#51;&#125;&#43;&#52;&#56;&#121;&#61;&#50;&#52;&#123;&#121;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"132\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836439638\">\n<div data-type=\"problem\" id=\"fs-id1167836439640\">\n<p id=\"fs-id1167829811262\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-37a643214e37cace7707c53b9dbf3204_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#54;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#43;&#50;&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#61;&#45;&#52;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"149\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167824732569\">\n<p id=\"fs-id1167824732571\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1a56f201d06d51d600e4c9182334c3f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#48;&#44;&#120;&#61;&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#47;&#125;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"111\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836529024\">\n<div data-type=\"problem\" id=\"fs-id1167829695577\">\n<p id=\"fs-id1167829695579\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8445397665f48c8322f8bd57cda62f6a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#123;&#121;&#125;&#94;&#123;&#51;&#125;&#43;&#50;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#50;&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"124\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836728266\"><strong data-effect=\"bold\">Solve Equations with Polynomial Functions<\/strong><\/p>\n<p id=\"fs-id1167836377496\">In the following exercises, solve.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167836377499\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167833053850\">\n<p id=\"fs-id1167833053852\">For the function, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cda46df3162520b0d26555db62a4b4df_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#56;&#120;&#43;&#56;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"154\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d0<\/span> find when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8b079a0bd8cb258dd6194cee203d2f7c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"84\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d1<\/span> Use this information to find two points that lie on the graph of the function.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836433034\">\n<p id=\"fs-id1167829744445\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c657687cbbf5ea9a7545edb42190e592_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\" \/> or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-116dfacb4c2a32bdfb5ab9e852f29986_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6d5b80bc57d91ba9e2936cc72d75618b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-eb87bfe9ba69a89c4ac12d88180c12e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#44;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836311497\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836310449\">\n<p id=\"fs-id1167836310451\">For the function, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-29f8a3cd1d805ed1cc6c5bc3b475a8b8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#49;&#120;&#43;&#50;&#48;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"172\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d0<\/span> find when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7d91c5d57a1f82043378cd4488c85297_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"84\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d1<\/span> Use this information to find two points that lie on the graph of the function.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836535278\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836282991\">\n<p id=\"fs-id1167836282993\">For the function, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0ae60d3c0f7b96a00109add017d2080a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#56;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#56;&#120;&#43;&#53;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"172\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d0<\/span> find when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8b079a0bd8cb258dd6194cee203d2f7c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"84\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d1<\/span> Use this information to find two points that lie on the graph of the function.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836444808\">\n<p id=\"fs-id1167836444810\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-82aa8812cbd6ad536dcb3c4fddfef6d6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"43\" style=\"vertical-align: -6px;\" \/> or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c079d7a942761124394a871098a2020e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"43\" style=\"vertical-align: -6px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5732036deb7b9637cbfd6e4cea855a4a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#125;&#44;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"54\" style=\"vertical-align: -7px;\" \/> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-edccb8e845935c6e561918c0e1e004ec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#44;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"54\" style=\"vertical-align: -7px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836455830\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836455832\">\n<p id=\"fs-id1167836526816\">For the function, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-eeabb6690bdbc57f99d686d3daf02a99_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#49;&#56;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#53;&#120;&#45;&#49;&#48;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"190\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d0<\/span> find when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0344008d128bee8f68e8461047546011_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#49;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"78\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d1<\/span> Use this information to find two points that lie on the graph of the function.<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836417930\">In the following exercises, for each function, find: <span class=\"token\">\u24d0<\/span> the zeros of the function <span class=\"token\">\u24d1<\/span> the <em data-effect=\"italics\">x<\/em>-intercepts of the graph of the function <span class=\"token\">\u24d2<\/span> the <em data-effect=\"italics\">y<\/em>-intercept of the graph of the function.<\/p>\n<div data-type=\"exercise\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836595837\">\n<p id=\"fs-id1167836595839\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8de5c7100dcd78daef0a47682e264169_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#57;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"118\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836717499\">\n<p id=\"fs-id1167836717501\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dd606513e0b1ba31937cdc32d1e43458_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"43\" style=\"vertical-align: -6px;\" \/> or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9af98521b89333b949a5950d447fdf84_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"56\" style=\"vertical-align: -6px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-de702e3f7fd4f26121cfe64713c191f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"40\" style=\"vertical-align: -7px;\" \/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ba45454908ae59d684870b218400858d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"54\" style=\"vertical-align: -7px;\" \/> <span class=\"token\">\u24d2<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-42f3c0bd5adc0ec0fa1707d7989e5c00_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829739808\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829739810\">\n<p id=\"fs-id1167829739812\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9b9e26cff522ed7bef03fdd30a6a487d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#53;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"136\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836407732\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836407734\">\n<p id=\"fs-id1167829692537\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-72f7187d20d3e0d154efec2486a413c7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#55;&#120;&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"158\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829596938\">\n<p id=\"fs-id1167833263514\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-902243c7ff3206df7fb360eec5379208_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"43\" style=\"vertical-align: -6px;\" \/> or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fd2171ab5e610de7079ae5a4090bee64_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"56\" style=\"vertical-align: -6px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a20b456d688b34512410b1d22e49d50e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#51;&#125;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"40\" style=\"vertical-align: -7px;\" \/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a8c70614d783d7b286ce8e07160a47f2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"54\" style=\"vertical-align: -7px;\" \/> <span class=\"token\">\u24d2<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5f6ab1a0ed415088c10eaaa3977a4992_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836524185\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836524188\">\n<p id=\"fs-id1167836524190\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-44f3405c76630a16506df210b7dcbc2f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#49;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#49;&#120;&#43;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"176\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167829984109\"><strong data-effect=\"bold\">Solve Applications Modeled by Quadratic Equations<\/strong><\/p>\n<p id=\"fs-id1167833086901\">In the following exercises, solve.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167829908175\" class=\"material-set-2\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167836536765\">The product of two consecutive odd integers is 143. Find the integers.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836542378\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-18b0b51936cf47501a8f90715c37d208_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#51;&#44;&#45;&#49;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"69\" style=\"vertical-align: -4px;\" \/> and 11, 13<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833057832\" class=\"material-set-2\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167833057836\">The product of two consecutive odd integers is 195. Find the integers.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829712386\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167826171959\">\n<p id=\"fs-id1167826171961\">The product of two consecutive even integers is 168. Find the integers.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829586760\">\n<p id=\"fs-id1167829586762\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-72d348c53c5a4f5914f094d1b018ccaa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#52;&#44;&#45;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"69\" style=\"vertical-align: -4px;\" \/> and 12, 14<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832940587\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167832940589\">\n<p id=\"fs-id1167836625236\">The product of two consecutive even integers is 288. Find the integers.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833311290\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829579684\">\n<p id=\"fs-id1167829579686\">The area of a rectangular carpet is 28 square feet. The length is three feet more than the width. Find the length and the width of the carpet.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833086705\">\n<p id=\"fs-id1167833086707\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-00b9cce9021441b203ec0271d72e6ba2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"22\" style=\"vertical-align: -1px;\" \/> and 7<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836288744\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836288746\">\n<p id=\"fs-id1167836288748\">A rectangular retaining wall has area 15 square feet. The height of the wall is two feet less than its length. Find the height and the length of the wall.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836432337\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836311198\">\n<p id=\"fs-id1167836311201\">The area of a bulletin board is 55 square feet. The length is four feet less than three times the width. Find the length and the width of the a bulletin board.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833381374\">\n<p id=\"fs-id1167833381376\">5, 11<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836294735\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836294737\">\n<p id=\"fs-id1167833050836\">A rectangular carport has area 150 square feet. The height of the carport is five feet less than twice its length. Find the height and the length of the carport.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833050541\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167833050543\">\n<p id=\"fs-id1167836282646\">A pennant is shaped like a right triangle, with hypotenuse 10 feet. The length of one side of the pennant is two feet longer than the length of the other side. Find the length of the two sides of the pennant.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829746946\">\n<p id=\"fs-id1167829746948\">6, 8<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836683998\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836684001\">\n<p id=\"fs-id1167836684003\">A stained glass window is shaped like a right triangle. The hypotenuse is 15 feet. One leg is three more than the other. Find the lengths of the legs.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829589871\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829589873\">\n<p id=\"fs-id1167829589875\">A reflecting pool is shaped like a right triangle, with one leg along the wall of a building. The hypotenuse is 9 feet longer than the side along the building. The third side is 7 feet longer than the side along the building. Find the lengths of all three sides of the reflecting pool.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836322358\">\n<p id=\"fs-id1167836706550\">8, 15, 17<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833024496\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167833024499\">\n<p id=\"fs-id1167833024501\">A goat enclosure is in the shape of a right triangle. One leg of the enclosure is built against the side of the barn. The other leg is 4 feet more than the leg against the barn. The hypotenuse is 8 feet more than the leg along the barn. Find the three sides of the goat enclosure.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836610606\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836610609\">\n<p id=\"fs-id1167836610611\">Juli is going to launch a model rocket in her back yard. When she launches the rocket, the function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7d372928bc966e44378f09f1ea47abae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;&#92;&#108;&#101;&#102;&#116;&#40;&#116;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#49;&#54;&#123;&#116;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#50;&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"148\" style=\"vertical-align: -4px;\" \/> models the height, <em data-effect=\"italics\">h<\/em>, of the rocket above the ground as a function of time, <em data-effect=\"italics\">t<\/em>. Find:<\/p>\n<p id=\"fs-id1167829716526\"><span class=\"token\">\u24d0<\/span> the zeros of this function which tells us when the rocket will hit the ground. <span class=\"token\">\u24d1<\/span> the time the rocket will be 16 feet above the ground.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836433972\">\n<p id=\"fs-id1167836543376\"><span class=\"token\">\u24d0<\/span> 0, 2 <span class=\"token\">\u24d1<\/span> 1<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829747436\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829747439\">\n<p id=\"fs-id1167829747441\">Gianna is going to throw a ball from the top floor of her middle school. When she throws the ball from 48 feet above the ground, the function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d644d0f66d185f7a0e2920e7e7f46ca1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;&#92;&#108;&#101;&#102;&#116;&#40;&#116;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#49;&#54;&#123;&#116;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#50;&#116;&#43;&#52;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"188\" style=\"vertical-align: -4px;\" \/> models the height, <em data-effect=\"italics\">h<\/em>, of the ball above the ground as a function of time, <em data-effect=\"italics\">t<\/em>. Find:<\/p>\n<p id=\"fs-id1167836700379\"><span class=\"token\">\u24d0<\/span> the zeros of this function which tells us when the ball will hit the ground. <span class=\"token\">\u24d1<\/span> the time(s) the ball will be 48 feet above the ground. <span class=\"token\">\u24d2<\/span> the height the ball will be at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-abf8f2abfdf1913d009a51ad64786690_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"38\" style=\"vertical-align: -1px;\" \/> seconds which is when the ball will be at its highest point.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"writing\" data-depth=\"2\" id=\"fs-id1167836362679\">\n<h4 data-type=\"title\">Writing Exercises<\/h4>\n<div data-type=\"exercise\" id=\"fs-id1167836519198\">\n<div data-type=\"problem\" id=\"fs-id1167836519200\">\n<p id=\"fs-id1167833054701\">Explain how you solve a quadratic equation. How many answers do you expect to get for a quadratic equation?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833054706\">\n<p id=\"fs-id1167836706782\">Answers will vary.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836551135\">\n<div data-type=\"problem\" id=\"fs-id1167836551137\">\n<p id=\"fs-id1167836551139\">Give an example of a quadratic equation that has a GCF and none of the solutions to the equation is zero.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167836282836\">\n<h4 data-type=\"title\">Self Check<\/h4>\n<p id=\"fs-id1167836321166\"><span class=\"token\">\u24d0<\/span> After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167833272086\" data-alt=\"This table has 4 columns, 3 rows and a header row. The header row labels each column: I can, confidently, with some help and no, I don\u2019t get it. The first column has the following statements: solve quadratic equations by using the zero product property, solve quadratic equations by factoring and solve applications modeled by quadratic equations.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_05_201_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This table has 4 columns, 3 rows and a header row. The header row labels each column: I can, confidently, with some help and no, I don\u2019t get it. The first column has the following statements: solve quadratic equations by using the zero product property, solve quadratic equations by factoring and solve applications modeled by quadratic equations.\" \/><\/span><\/p>\n<p id=\"fs-id1167836612849\"><span class=\"token\">\u24d1<\/span> Overall, after looking at the checklist, do you think you are well-prepared for the next section? Why or why not?<\/p>\n<\/div>\n<\/div>\n<div class=\"review-exercises\" data-depth=\"1\" id=\"fs-id1167829614395\">\n<h3 data-type=\"title\">Chapter Review Exercises<\/h3>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167836662547\">\n<h4 data-type=\"title\"><a href=\"\/contents\/b380a9f9-f0f7-4fdf-9a79-5646a9f745f9\" class=\"target-chapter\">Greatest Common Factor and Factor by Grouping<\/a><\/h4>\n<p><strong data-effect=\"bold\">Find the Greatest Common Factor of Two or More Expressions<\/strong><\/p>\n<p id=\"fs-id1167833060062\">In the following exercises, find the greatest common factor.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167833060065\">\n<div data-type=\"problem\" id=\"fs-id1167833060067\">\n<p id=\"fs-id1167836521832\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6aa8cb210e1a834973589f6e8c0f9924_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#50;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#123;&#98;&#125;&#94;&#123;&#51;&#125;&#44;&#49;&#53;&#97;&#123;&#98;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"99\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167824617105\">\n<p id=\"fs-id1167836399826\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9cab041def1e31eae0acc93bc9b9f12f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#97;&#123;&#98;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"33\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836455876\">\n<div data-type=\"problem\" id=\"fs-id1167836349187\">\n<p id=\"fs-id1167836349189\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8aa5773520664e1e510c712a4782000d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#50;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#123;&#110;&#125;&#94;&#123;&#51;&#125;&#44;&#52;&#50;&#123;&#109;&#125;&#94;&#123;&#53;&#125;&#123;&#110;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"125\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836732282\">\n<div data-type=\"problem\" id=\"fs-id1167833339509\">\n<p id=\"fs-id1167833339511\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b66f90c2679756db48fc55fec41cfc47_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#53;&#123;&#121;&#125;&#94;&#123;&#51;&#125;&#44;&#50;&#49;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#44;&#51;&#48;&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"111\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829752740\">\n<p id=\"fs-id1167829752742\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f729160324276ebdb243789c8000f253_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"18\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836628598\">\n<div data-type=\"problem\" id=\"fs-id1167836628600\">\n<p id=\"fs-id1167836628602\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-687571866e92ec903bad4e9bd90d1f12_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#53;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#44;&#49;&#53;&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#121;&#44;&#49;&#48;&#123;&#120;&#125;&#94;&#123;&#53;&#125;&#123;&#121;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"165\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167829645054\"><strong data-effect=\"bold\">Factor the Greatest Common Factor from a Polynomial<\/strong><\/p>\n<p id=\"fs-id1167836321656\">In the following exercises, factor the greatest common factor from each polynomial.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167836321659\">\n<div data-type=\"problem\" id=\"fs-id1167829740287\">\n<p id=\"fs-id1167829740289\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-41b3927cf4f5edce5919e81706896630_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#53;&#121;&#43;&#56;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"67\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836529086\">\n<p id=\"fs-id1167836665130\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-107b1eb4652443cf1e4710d440ee1e61_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#121;&#43;&#49;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"83\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829715304\">\n<div data-type=\"problem\" id=\"fs-id1167829585662\">\n<p id=\"fs-id1167829585664\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a1d2bf0fadeeb2fba2a3f34863016d9a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#50;&#121;&#45;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"105\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836693737\">\n<div data-type=\"problem\" id=\"fs-id1167836693739\">\n<p id=\"fs-id1167836693741\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-563df2916de0352796f220fb122e567b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#56;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#45;&#49;&#53;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"84\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836613926\">\n<p id=\"fs-id1167824732294\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2a2446e6cf3b53a28012b6a82c8637c1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#120;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"93\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836525208\">\n<div data-type=\"problem\" id=\"fs-id1167832925972\">\n<p id=\"fs-id1167832925974\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-28e37c7c3026faa21159e419bd4ab42d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#53;&#123;&#109;&#125;&#94;&#123;&#52;&#125;&#43;&#54;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"105\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836532823\">\n<div data-type=\"problem\" id=\"fs-id1167829861895\">\n<p id=\"fs-id1167829861897\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-186fae493f753aff8d6690432fc19ada_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#45;&#49;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#54;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"134\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829620341\">\n<p id=\"fs-id1167829620344\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0e004cfc7757d6a4f13bf1ccc4cd15d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#120;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#120;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"125\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829738393\">\n<div data-type=\"problem\" id=\"fs-id1167829738395\">\n<p id=\"fs-id1167829720761\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ae5b5d25a96e3c88070d9481d7f3144f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#120;&#43;&#50;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"71\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829860032\">\n<div data-type=\"problem\" id=\"fs-id1167829860034\">\n<p id=\"fs-id1167829860036\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f398bf4495d5906402dd0fcafc794cb4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#43;&#50;&#55;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#50;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"146\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829718550\">\n<p id=\"fs-id1167829718552\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-008f637bbcd5b0de963023592ec17788_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#120;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#57;&#120;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"138\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836327058\">\n<div data-type=\"problem\" id=\"fs-id1167836389930\">\n<p id=\"fs-id1167836389932\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c0f927e9c8f981209191b8b398ed5273_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#120;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#53;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"167\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836620104\"><strong data-effect=\"bold\">Factor by Grouping<\/strong><\/p>\n<p id=\"fs-id1167836621172\">In the following exercises, factor by grouping.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167836621175\">\n<div data-type=\"problem\" id=\"fs-id1167829877871\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-813b59a641fa8e17e516b0274c54e640_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#120;&#45;&#97;&#121;&#43;&#98;&#120;&#45;&#98;&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"138\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836318397\">\n<p id=\"fs-id1167836318399\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7be948f966b1e39e222ef2ac16947cda_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"108\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836511579\">\n<div data-type=\"problem\" id=\"fs-id1167836511581\">\n<p id=\"fs-id1167836293668\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1e5a489232d3979c82dcde1d7eb18254_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#121;&#45;&#120;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#120;&#45;&#50;&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"156\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836544551\">\n<div data-type=\"problem\" id=\"fs-id1167836544553\">\n<p id=\"fs-id1167836544555\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9f8a88dd582c39022a5a51dfff8eda13_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#55;&#120;&#45;&#51;&#120;&#45;&#50;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"138\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836524290\">\n<p id=\"fs-id1167836524292\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2867eb8f9eba45ca69747b2ed11c8d5f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"110\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833059863\">\n<div data-type=\"problem\" id=\"fs-id1167833059865\">\n<p id=\"fs-id1167836547604\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f5e7cd5f60842125dd6c25cefb82fb5d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#54;&#120;&#43;&#51;&#120;&#45;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"156\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836554999\">\n<div data-type=\"problem\" id=\"fs-id1167836555001\">\n<p id=\"fs-id1167836511374\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-84f7d706404504cbc15f450c7b73e8ab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#109;&#125;&#94;&#123;&#51;&#125;&#43;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#43;&#109;&#43;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"135\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829717351\">\n<p id=\"fs-id1167829717353\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-15be27866b66ef72143a42f56ef4aa8d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#109;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"130\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836310151\">\n<div data-type=\"problem\" id=\"fs-id1167836310153\">\n<p id=\"fs-id1167836310155\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-38d6091fc8d1aba96bcb9e5654d74f24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#120;&#45;&#53;&#121;&#45;&#121;&#43;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"122\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167836299806\">\n<h4 data-type=\"title\"><a href=\"\/contents\/edd9c403-9825-4cf2-b237-2e05552ea3ec\" class=\"target-chapter\">Factor Trinomials<\/a><\/h4>\n<p id=\"fs-id1167836520946\"><strong data-effect=\"bold\">Factor Trinomials of the Form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f3995eb3274bb5bf3d3e8fb22d033b1f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#120;&#43;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"87\" style=\"vertical-align: -2px;\" \/><\/strong><\/p>\n<p id=\"fs-id1167829732082\">In the following exercises, factor each trinomial of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3dab64246626866d653558ab5ec3bb4d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#120;&#43;&#99;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"91\" style=\"vertical-align: -2px;\" \/><\/p>\n<div data-type=\"exercise\" id=\"fs-id1167836663209\">\n<div data-type=\"problem\" id=\"fs-id1167826131854\">\n<p id=\"fs-id1167826131856\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-de45c98cc9ab50ed65578e3ba9b7ccd9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#52;&#97;&#43;&#51;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"105\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829694283\">\n<p id=\"fs-id1167836322552\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-16ea3acabe5d9a5c001072d74656cb61_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#49;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"118\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833135499\">\n<div data-type=\"problem\" id=\"fs-id1167833135501\">\n<p id=\"fs-id1167833135503\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c755fb6ddb0766c283b6510b60768675_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#107;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#54;&#107;&#43;&#54;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"106\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836289089\">\n<div data-type=\"problem\" id=\"fs-id1167836289091\">\n<p id=\"fs-id1167836415377\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-935948871ec567d383a768a48a34caa9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#109;&#45;&#53;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"109\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836315108\">\n<p id=\"fs-id1167836315110\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-74ffe4670b811deb9ac419f7d38dcfce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#109;&#43;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#109;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"121\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829713135\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167829595235\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d4a27b86291db1cf42073498cd96deae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#120;&#45;&#49;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"98\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836419022\">In the following examples, factor each trinomial of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-601fada7dc0d9f3ff1ea382efa1a3a5a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#120;&#121;&#43;&#99;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"117\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"exercise\" id=\"fs-id1167836363319\">\n<div data-type=\"problem\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8ad90d6c5df44cec892bba6e91d3b27c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#50;&#120;&#121;&#43;&#51;&#53;&#123;&#121;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"132\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829599193\">\n<p id=\"fs-id1167829599195\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9e6d0095ab4333b79e6990a178bb3fd9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#53;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#55;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"129\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836481436\">\n<div data-type=\"problem\" id=\"fs-id1167836481439\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f8f815647c4de8458399eb0456a4122e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#114;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#114;&#115;&#45;&#50;&#56;&#123;&#115;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"118\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836660098\">\n<div data-type=\"problem\" id=\"fs-id1167836660100\">\n<p id=\"fs-id1167836660102\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-97a880f548a785f599a5c2f3e87ce9e9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#97;&#98;&#45;&#50;&#49;&#123;&#98;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"119\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829789646\">\n<p id=\"fs-id1167829789648\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-16e4309b13d07646308a6b32c8d1b6d5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#55;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#45;&#51;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"124\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836533387\">\n<div data-type=\"problem\" id=\"fs-id1167836533389\">\n<p id=\"fs-id1167833017936\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-69d164a4d074d1ca82601c38643b6c55_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#112;&#113;&#45;&#51;&#54;&#123;&#113;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"121\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833279905\">\n<div data-type=\"problem\" id=\"fs-id1167833279907\">\n<p id=\"fs-id1167833019257\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9b0c7adbc105b6cad71e461d82e7b9f2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#109;&#110;&#43;&#51;&#48;&#123;&#110;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"138\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167836387220\">Prime<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836387226\"><strong data-effect=\"bold\">Factor Trinomials of the Form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dd69fd4ca7cff6b9f4eee239f6d713d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#120;&#43;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"96\" style=\"vertical-align: -2px;\" \/> Using Trial and Error<\/strong><\/p>\n<p id=\"fs-id1167829720901\">In the following exercises, factor completely using trial and error.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167829720904\">\n<div data-type=\"problem\" id=\"fs-id1167829720906\">\n<p id=\"fs-id1167836513956\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6bab35c0f135b9831668774a624bc581_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#43;&#53;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#52;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"116\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829745235\">\n<div data-type=\"problem\" id=\"fs-id1167829745237\">\n<p id=\"fs-id1167829745239\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-aed4986b1ccf6a4c07e5bdeb561cb553_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#123;&#121;&#125;&#94;&#123;&#51;&#125;&#45;&#50;&#49;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#48;&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"131\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836640498\">\n<p id=\"fs-id1167836640500\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7f76f5823211701109b7f15efb32e15c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#121;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"131\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836355740\">\n<div data-type=\"problem\" id=\"fs-id1167836355743\">\n<p id=\"fs-id1167836355745\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-df23dccc5bda26658bbce0a16d22f2fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#43;&#49;&#48;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#45;&#55;&#53;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"141\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167824732243\">\n<div data-type=\"problem\" id=\"fs-id1167824732246\">\n<p id=\"fs-id1167836700642\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-06e378e69e6c00b3de630fedd3e7fe73_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#52;&#121;&#43;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"105\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832936381\">\n<p id=\"fs-id1167836613060\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6d58460255e36ee109ec07fb3f1cd5ba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#121;&#43;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"117\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829596060\">\n<div data-type=\"problem\" id=\"fs-id1167829596063\">\n<p id=\"fs-id1167829595434\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9035a9a67e26b3e83711c4e0d6c68edf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#53;&#120;&#43;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"107\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836282761\">\n<div data-type=\"problem\" id=\"fs-id1167836282763\">\n<p id=\"fs-id1167836652632\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8c1fe0859f95e07d147593f474f0d41a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#48;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#51;&#121;&#45;&#49;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"121\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836698812\">\n<p id=\"fs-id1167829740760\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-98a6fa2614628582dd13109aec52da24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#121;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#121;&#45;&#49;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"135\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829811185\">\n<div data-type=\"problem\" id=\"fs-id1167836294851\">\n<p id=\"fs-id1167836294853\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c6fe62b5a4e2bb2b44c5c2ddcb795ede_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#57;&#112;&#113;&#43;&#49;&#48;&#123;&#113;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"137\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829789212\">\n<div data-type=\"problem\" id=\"fs-id1167836597114\">\n<p id=\"fs-id1167836597116\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-26a517b9f4baae594ead76a247a8a9e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#56;&#49;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#53;&#51;&#97;&#43;&#49;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"145\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836513741\">\n<p id=\"fs-id1167836513743\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ae2fa1eef0d3d4ad8c93970b4ad746b8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#57;&#92;&#108;&#101;&#102;&#116;&#40;&#57;&#97;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"143\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167829695659\"><strong data-effect=\"bold\">Factor Trinomials of the Form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dd69fd4ca7cff6b9f4eee239f6d713d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#120;&#43;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"96\" style=\"vertical-align: -2px;\" \/> using the \u2018ac\u2019 Method<\/strong><\/p>\n<p id=\"fs-id1167836295302\">In the following exercises, factor.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167836295305\">\n<div data-type=\"problem\" id=\"fs-id1167836295606\">\n<p id=\"fs-id1167836295608\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3a3562d595ca0b4db2f9dfd6163e542f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#57;&#120;&#43;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"98\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836628901\">\n<div data-type=\"problem\" id=\"fs-id1167836314158\">\n<p id=\"fs-id1167836314160\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5693f9973935adff92e244b23e2824e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#56;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#57;&#97;&#43;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"103\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836512081\">\n<p id=\"fs-id1167836512084\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-87a6b0c06e42dbcfcdafb0e663896599_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#97;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#97;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"126\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836620731\">\n<div data-type=\"problem\" id=\"fs-id1167836620733\">\n<p id=\"fs-id1167836620735\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-481fc7fa76be9e8a6083ad56b9f50576_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#53;&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#112;&#45;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"103\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167829748808\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-87d6eb11c8f76b2170b3b0e92faf8a7b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#53;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#120;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"105\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836537191\">\n<p id=\"fs-id1167832945806\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-aa8ce3c0308815f63aa1661272c481b9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#120;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#120;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"128\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836688452\">\n<div data-type=\"problem\" id=\"fs-id1167836399698\">\n<p id=\"fs-id1167836399700\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-058ddfabe082e77d2c465fe3c4074989_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#50;&#97;&#43;&#50;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"114\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836297149\">\n<div data-type=\"problem\" id=\"fs-id1167836544134\">\n<p id=\"fs-id1167836544136\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b023d98303f873950212b58594efd262_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#120;&#45;&#51;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"107\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167833018721\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-073119c4bad51979ad5a620bbc1a38b3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"123\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836613589\">\n<div data-type=\"problem\" id=\"fs-id1167836509003\">\n<p id=\"fs-id1167836509005\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bb8f02373a70afc732b141ee826ce8eb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#56;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#50;&#121;&#45;&#51;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"123\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836510934\">\n<div data-type=\"problem\" id=\"fs-id1167836510936\">\n<p id=\"fs-id1167836510938\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5422fe1d54396c0db2012337a799e449_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#56;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#55;&#97;&#45;&#50;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"121\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833327203\">\n<p id=\"fs-id1167833327206\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5fbf8a65c84d0229040d86040197e860_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#97;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#97;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"139\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167829894397\">\n<p id=\"fs-id1167829894399\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fe5b10e9589f07a2c25b2db7f03b5a1e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#123;&#110;&#125;&#94;&#123;&#52;&#125;&#45;&#49;&#50;&#123;&#110;&#125;&#94;&#123;&#51;&#125;&#45;&#57;&#54;&#123;&#110;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"143\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836747825\"><strong data-effect=\"bold\">Factor using substitution<\/strong><\/p>\n<p id=\"fs-id1167829787750\">In the following exercises, factor using substitution.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167829787754\">\n<div data-type=\"problem\" id=\"fs-id1167829787756\">\n<p id=\"fs-id1167836485811\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f8471ef59e0eff9c69673dee06fc317c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#45;&#49;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"115\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836334488\">\n<p id=\"fs-id1167836334490\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-be2e6940f8411c7ec0a85641a6c6604e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"137\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833082476\">\n<div data-type=\"problem\" id=\"fs-id1167833082478\">\n<p id=\"fs-id1167833082480\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-717c8844b9e06bfcb6c79740df6727c9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#51;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"189\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167833055815\">\n<h4 data-type=\"title\"><a href=\"\/contents\/d844a3e4-0163-4936-91ca-a71142f07358\" class=\"target-chapter\">Factor Special Products<\/a><\/h4>\n<p id=\"fs-id1167836331150\"><strong data-effect=\"bold\">Factor Perfect Square Trinomials<\/strong><\/p>\n<p id=\"fs-id1167836701146\">In the following exercises, factor completely using the perfect square trinomials pattern.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167833059265\">\n<div data-type=\"problem\" id=\"fs-id1167833059267\">\n<p id=\"fs-id1167833059269\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b908fdb6d8ec379897589e90bd5cda34_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#53;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#48;&#120;&#43;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"116\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836539601\">\n<p id=\"fs-id1167836539603\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0391bd226dae2a9c1ab31c5ce309ed1a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#120;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"69\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833057727\">\n<div data-type=\"problem\" id=\"fs-id1167836697611\">\n<p id=\"fs-id1167836697613\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ff196ae11b8f015f218a30459b149125_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#54;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#56;&#52;&#97;&#98;&#43;&#52;&#57;&#123;&#98;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"146\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829743405\">\n<div data-type=\"problem\" id=\"fs-id1167829743407\">\n<p id=\"fs-id1167829743409\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1ce51bd688e0b126df5c7e0421379767_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#48;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#54;&#48;&#120;&#43;&#56;&#49;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"143\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836602252\">\n<p id=\"fs-id1167836602254\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-443304d44519d4a8ce5d5abf6642beca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#48;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#120;&#43;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"87\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836282423\">\n<div data-type=\"problem\" id=\"fs-id1167836282425\">\n<p id=\"fs-id1167836282427\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-867d8fbfd30e914ae93e6560118a214d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#123;&#107;&#125;&#94;&#123;&#51;&#125;&#45;&#55;&#48;&#123;&#107;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#52;&#53;&#107;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"141\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167836531526\">\n<p id=\"fs-id1167836531528\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-da5f8545865a58669df140937a3c5c42_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#53;&#123;&#117;&#125;&#94;&#123;&#52;&#125;&#45;&#51;&#48;&#123;&#117;&#125;&#94;&#123;&#51;&#125;&#118;&#43;&#51;&#123;&#117;&#125;&#94;&#123;&#50;&#125;&#123;&#118;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"167\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836551978\">\n<p id=\"fs-id1167836494113\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c0179b56932dc9b5462a4b1da59ab914_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#123;&#117;&#125;&#94;&#123;&#50;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#117;&#45;&#118;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"97\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167833086731\"><strong data-effect=\"bold\">Factor Differences of Squares<\/strong><\/p>\n<p id=\"fs-id1167833224512\">In the following exercises, factor completely using the difference of squares pattern, if possible.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167833224516\">\n<div data-type=\"problem\" id=\"fs-id1167836597153\">\n<p id=\"fs-id1167836597155\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e8914813db1aace70789e0c88117678b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#49;&#123;&#114;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"73\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833224386\">\n<div data-type=\"problem\" id=\"fs-id1167833224388\">\n<p id=\"fs-id1167833224390\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a276b7c3eef72977a829fdb1b4c79a4f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#54;&#57;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#45;&#123;&#110;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"89\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836357359\">\n<p id=\"fs-id1167836357362\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7b95dc848e91eec2898559166abf67f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#51;&#109;&#43;&#110;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#51;&#109;&#45;&#110;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"160\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833047487\">\n<div data-type=\"problem\" id=\"fs-id1167833047489\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b87e4262e3e02d0d265726502da5189a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#53;&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"64\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836546848\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167829719631\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3ec388124677bd69759520ec3ff0bc08_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#45;&#49;&#50;&#49;&#123;&#121;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"73\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829811950\">\n<p id=\"fs-id1167836556695\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-484201ff0d427110e3b6f571d9283313_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#43;&#49;&#49;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#45;&#49;&#49;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"144\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836492382\">\n<div data-type=\"problem\" id=\"fs-id1167829829336\">\n<p id=\"fs-id1167829829339\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d18db3d3b3072b462e5508dc026801d8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#48;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#50;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"83\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836543391\">\n<div data-type=\"problem\" id=\"fs-id1167836543393\">\n<p id=\"fs-id1167833009345\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3eb0a23337b28972b2ccc142dca46a4d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#54;&#57;&#123;&#110;&#125;&#94;&#123;&#51;&#125;&#45;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"77\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836321020\">\n<p id=\"fs-id1167836542091\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b79eaf8b24956d8dc74efa6a0abd8ae4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#51;&#110;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#51;&#110;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"161\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836548731\">\n<div data-type=\"problem\" id=\"fs-id1167836548733\">\n<p id=\"fs-id1167836548735\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6b4f2ca0d1abfbb93e419b260b65ec31_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#123;&#113;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#52;&#123;&#112;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"97\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836524842\">\n<div data-type=\"problem\" id=\"fs-id1167836524845\">\n<p id=\"fs-id1167836524847\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1ad9113284fb05fc84c3ab7fcb66c75b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#52;&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#43;&#53;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"74\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836625184\">\n<p id=\"fs-id1167836625186\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b826d5d64b9b53ac3db3be04831e3c96_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#43;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"82\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829721107\">\n<div data-type=\"problem\" id=\"fs-id1167829721109\">\n<p id=\"fs-id1167829721111\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1f2fcae1beef0d8f1c2aa7bcae323026_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#57;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#56;&#49;&#123;&#121;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"91\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836546672\">\n<div data-type=\"problem\" id=\"fs-id1167833059205\">\n<p id=\"fs-id1167833059208\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b441a1b616d40218a901d4a516457cf2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#54;&#123;&#122;&#125;&#94;&#123;&#52;&#125;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"63\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836533874\">\n<p id=\"fs-id1167836533876\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f4006d97849bade2da1461a808a65c7d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#122;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#122;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#123;&#122;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"200\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836309880\">\n<div data-type=\"problem\" id=\"fs-id1167836309882\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-69908679fa5cae3cef5b54b4cba9df3c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#56;&#123;&#109;&#125;&#94;&#123;&#52;&#125;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#52;&#51;&#123;&#110;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"126\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836548892\">\n<div data-type=\"problem\" id=\"fs-id1167836548895\">\n<p id=\"fs-id1167836548897\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-32f18cd39ab3d3fc26c8492f806641bd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#97;&#43;&#57;&#45;&#57;&#123;&#98;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"133\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833018367\">\n<p id=\"fs-id1167833018369\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-26d27d53b8dc797c4b58bf2fbc57b0a3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#51;&#45;&#51;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#51;&#43;&#51;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"185\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836688854\">\n<div data-type=\"problem\" id=\"fs-id1167836360981\">\n<p id=\"fs-id1167836360983\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b146ad6500c7a8950908804a21113775_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#54;&#120;&#43;&#54;&#52;&#45;&#123;&#121;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"145\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836545257\"><strong data-effect=\"bold\">Factor Sums and Differences of Cubes<\/strong><\/p>\n<p id=\"fs-id1167836547592\">In the following exercises, factor completely using the sums and differences of cubes pattern, if possible.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167833191672\">\n<div data-type=\"problem\" id=\"fs-id1167833191674\">\n<p id=\"fs-id1167836322156\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9620feed5f9d7d05b9f89ad090b0daee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#94;&#123;&#51;&#125;&#45;&#49;&#50;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"65\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836574949\">\n<p id=\"fs-id1167836574951\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f340137f1e29ea20814576e7c08efb89_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#43;&#53;&#97;&#43;&#50;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"166\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836522701\">\n<div data-type=\"problem\" id=\"fs-id1167836522703\">\n<p id=\"fs-id1167836522705\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1b31cde55b7c9331ac9435d26558a5df_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#98;&#125;&#94;&#123;&#51;&#125;&#45;&#50;&#49;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"64\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836602902\">\n<div data-type=\"problem\" id=\"fs-id1167836602904\">\n<p id=\"fs-id1167836602906\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-16f87f171ba541f6ac4a4a427ec104b9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#123;&#109;&#125;&#94;&#123;&#51;&#125;&#43;&#53;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"72\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836613722\">\n<p id=\"fs-id1167836613724\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f965c357b95a479e4725a0520b1ac9e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#109;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#109;&#43;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"189\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836541992\">\n<div data-type=\"problem\" id=\"fs-id1167836541994\">\n<p id=\"fs-id1167836289221\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9305f6e09ea7c9fbc3f188eaaafb4586_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#49;&#123;&#109;&#125;&#94;&#123;&#51;&#125;&#43;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"72\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167836417911\">\n<h4 data-type=\"title\"><a href=\"\/contents\/93db7208-2f30-47df-91eb-33984d9fb8b7\" class=\"target-chapter\">General Strategy for Factoring Polynomials<\/a><\/h4>\n<p id=\"fs-id1167836578726\"><strong data-effect=\"bold\">Recognize and Use the Appropriate Method to Factor a Polynomial Completely<\/strong><\/p>\n<p id=\"fs-id1167836508117\">In the following exercises, factor completely.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167829594300\">\n<div data-type=\"problem\" id=\"fs-id1167829594303\">\n<p id=\"fs-id1167829594305\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-02e6299b57a8989bd6ac13f21d1d3fff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#52;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#43;&#52;&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"92\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833114725\">\n<p id=\"fs-id1167833114727\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dfdf8fed767a21e9b0a9c096b905f3ea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#120;&#43;&#49;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"101\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833349928\">\n<div data-type=\"problem\" id=\"fs-id1167833349930\">\n<p id=\"fs-id1167833349932\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6466c67521ff55e504afefb4030e62be_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#52;&#123;&#97;&#125;&#94;&#123;&#52;&#125;&#45;&#57;&#123;&#97;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"81\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829893749\">\n<div data-type=\"problem\" id=\"fs-id1167829893751\">\n<p id=\"fs-id1167829893753\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-52e34b617788508adbfef8f2da339064_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#54;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#54;&#109;&#110;&#43;&#52;&#57;&#123;&#109;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"163\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836732073\">\n<p id=\"fs-id1167836445000\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d9d37a285f65b87fabbfea791c6db24a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#110;&#45;&#55;&#109;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"85\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833356111\">\n<div data-type=\"problem\" id=\"fs-id1167833356113\">\n<p id=\"fs-id1167836558483\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c95cb87b144455618083a92a2732c81f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#53;&#97;&#45;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"105\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829693743\">\n<div data-type=\"problem\" id=\"fs-id1167829717356\">\n<p id=\"fs-id1167829717358\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c530d1fd188df4d74b87f65785321779_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#123;&#117;&#125;&#94;&#123;&#52;&#125;&#45;&#52;&#53;&#123;&#117;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"83\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836283643\">\n<p id=\"fs-id1167833386290\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6dade81f96818b450ca1cad274fc9f86_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#123;&#117;&#125;&#94;&#123;&#50;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#117;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#117;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"141\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829694732\">\n<div data-type=\"problem\" id=\"fs-id1167829694734\">\n<p id=\"fs-id1167829586698\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-90bf88b900bb3c689534a8bae8fdef56_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#110;&#125;&#94;&#123;&#52;&#125;&#45;&#56;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"57\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833030958\">\n<div data-type=\"problem\" id=\"fs-id1167833030960\">\n<p id=\"fs-id1167833030963\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5e3005bafacd9326c4793a2310fd4d24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#52;&#123;&#106;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#50;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"81\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p>prime<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167829690364\">\n<p id=\"fs-id1167829690366\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ca5b861646ce4f988a7162e2461cd25c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#53;&#120;&#45;&#54;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"107\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829742656\">\n<div data-type=\"problem\" id=\"fs-id1167829786977\">\n<p id=\"fs-id1167829786979\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8e28a47523abfed12390501cc4b7bc8f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#98;&#125;&#94;&#123;&#51;&#125;&#45;&#54;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"55\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167829732135\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-17432669580b51e9863b3a1307c0e3bd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#98;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#98;&#43;&#49;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"161\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836340027\">\n<div data-type=\"problem\" id=\"fs-id1167836340029\">\n<p id=\"fs-id1167836520449\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-973d9f308a936cb4e59859068790cd63_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#109;&#125;&#94;&#123;&#51;&#125;&#43;&#49;&#50;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"71\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833009195\">\n<div data-type=\"problem\" id=\"fs-id1167833009197\">\n<p id=\"fs-id1167833009200\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1ffde12c035457f07afa98c9b931057e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#98;&#99;&#43;&#53;&#99;&#98;&#45;&#53;&#123;&#99;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"161\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836609387\">\n<p id=\"fs-id1167836510088\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f46aa18844e91fe14920015d85b72506_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#98;&#43;&#53;&#99;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#98;&#45;&#99;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"121\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836299787\">\n<div data-type=\"problem\" id=\"fs-id1167836283360\">\n<p id=\"fs-id1167836283362\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3dd76e00a15cf1893a94e96a47435eca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#56;&#123;&#120;&#125;&#94;&#123;&#53;&#125;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#52;&#51;&#120;&#123;&#121;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"127\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836325869\">\n<div data-type=\"problem\" id=\"fs-id1167836325871\">\n<p id=\"fs-id1167836551985\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-314ba5dda091ce4863ab537bd8f94332_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#123;&#113;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#53;&#113;&#45;&#57;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"113\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836559750\">\n<p id=\"fs-id1167836520643\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-769540f2044fb41bf8162e1390bfb71a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#92;&#108;&#101;&#102;&#116;&#40;&#113;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#113;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"120\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836629042\">\n<div data-type=\"problem\" id=\"fs-id1167836629045\">\n<p id=\"fs-id1167833018074\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-652b480f5804116142d238b928bfe33c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#117;&#125;&#94;&#123;&#53;&#125;&#118;&#43;&#52;&#123;&#117;&#125;&#94;&#123;&#50;&#125;&#123;&#118;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"101\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832989135\">\n<div data-type=\"problem\" id=\"fs-id1167832989138\">\n<p id=\"fs-id1167832989140\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-06a660630505017d187eb1daaa5f9012_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#48;&#123;&#109;&#125;&#94;&#123;&#52;&#125;&#45;&#54;&#50;&#53;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"98\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833061000\">\n<p id=\"fs-id1167833061002\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7609b8c5d06054bcdf8e0516475622c1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#48;&#92;&#108;&#101;&#102;&#116;&#40;&#109;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#109;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"222\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836612462\">\n<div data-type=\"problem\" id=\"fs-id1167836580067\">\n<p id=\"fs-id1167836580069\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-742702686ea33d725e270aaf504194ba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#48;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#121;&#45;&#55;&#53;&#120;&#121;&#43;&#51;&#48;&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"152\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829924607\">\n<div data-type=\"problem\" id=\"fs-id1167829924610\">\n<p id=\"fs-id1167829924612\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e530e4be5bd4fb41a13b24eb61343431_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#52;&#120;&#121;&#43;&#57;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"181\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836737880\">\n<p id=\"fs-id1167836737882\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cb7b82eda144737780ed263429227184_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#120;&#45;&#51;&#121;&#43;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#120;&#45;&#51;&#121;&#45;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"208\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167836522886\">\n<h4 data-type=\"title\"><a href=\"\/contents\/da8478b4-93bc-4919-81a1-5e3267050e7e\" class=\"target-chapter\">Polynomial Equations<\/a><\/h4>\n<p id=\"fs-id1167836399885\"><strong data-effect=\"bold\">Use the Zero Product Property<\/strong><\/p>\n<p id=\"fs-id1167832945786\">In the following exercises, solve.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167832945789\">\n<div data-type=\"problem\" id=\"fs-id1167833059297\">\n<p id=\"fs-id1167833059299\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b6068dee9bbb1c4281c887c2c7d51568_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"142\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829721041\">\n<div data-type=\"problem\" id=\"fs-id1167836493342\">\n<p id=\"fs-id1167836493344\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-29d75120c2e1dd4cc1f66d4f797ebfb0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#98;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#98;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"156\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836635208\">\n<p id=\"fs-id1167833338482\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f5bdc9100dfbc96ab05e1bf71b6d953d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;&#61;&#45;&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#47;&#125;&#53;&#44;&#98;&#61;&#45;&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#47;&#125;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"151\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832927596\">\n<div data-type=\"problem\" id=\"fs-id1167833224761\">\n<p id=\"fs-id1167833224763\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-70c7d61a4eabd5bbaf6321eac8fb6e3c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#109;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#50;&#109;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"138\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836558785\">\n<div data-type=\"problem\" id=\"fs-id1167836558787\">\n<p id=\"fs-id1167836558790\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ec654acc58870cfd0cc3c84c31af393f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#120;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"103\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829893802\">\n<p id=\"fs-id1167836434155\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8000322cad1feecda9d4763ad83b7161_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#47;&#125;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"59\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833197270\">\n<div data-type=\"problem\" id=\"fs-id1167836525640\">\n<p id=\"fs-id1167836525642\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-61fc757258d5ce485dc8766eb2a85db2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#109;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#109;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#109;&#43;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"192\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836665477\"><strong data-effect=\"bold\">Solve Quadratic Equations by Factoring<\/strong><\/p>\n<p id=\"fs-id1167829597317\">In the following exercises, solve.<\/p>\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167829627721\">\n<p id=\"fs-id1167829627723\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9d2f9681f86a0a2d2d1bf721a56c112f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#57;&#120;&#43;&#50;&#48;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"131\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829614600\">\n<p id=\"fs-id1167829755864\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8e655b3c798ab3a18cdd261c0ffb1ba8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#52;&#44;&#120;&#61;&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"120\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836417814\">\n<div data-type=\"problem\" id=\"fs-id1167836417816\">\n<p id=\"fs-id1167836439970\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cd9201a36a5d921542500ff14e7f5539_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#121;&#45;&#55;&#50;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"120\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836550168\">\n<div data-type=\"problem\" id=\"fs-id1167836550171\">\n<p id=\"fs-id1167836620378\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-34fadd93d4d81fea93ea20d745f09b75_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#49;&#112;&#61;&#52;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"115\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836378798\">\n<p id=\"fs-id1167829693685\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7cb20dbcee3cb0e1de5b6a442b477425_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#112;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#50;&#125;&#44;&#112;&#61;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"108\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829624398\">\n<div data-type=\"problem\" id=\"fs-id1167829624400\">\n<p id=\"fs-id1167836318300\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ece9bd7497111e8b5d536eb372dc94a0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#113;&#125;&#94;&#123;&#51;&#125;&#43;&#51;&#123;&#113;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#113;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"135\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829850992\">\n<div data-type=\"problem\" id=\"fs-id1167829645169\">\n<p id=\"fs-id1167829645171\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1187d14626023d52e40aa8ed237c7ed7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#52;&#52;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#53;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"121\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167836521550\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6ee6d09fe4ff7c1075993fdf6032238a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#49;&#50;&#125;&#44;&#109;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#49;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"134\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836703866\">\n<div data-type=\"problem\" id=\"fs-id1167836703868\">\n<p id=\"fs-id1167836703870\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7d61f80ddbf6b9ec7879ce8f13ff7430_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#61;&#51;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"69\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836292807\">\n<div data-type=\"problem\" id=\"fs-id1167836292809\">\n<p id=\"fs-id1167836292811\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-28c60870b6139bcd3ae17a0b531554ba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"157\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836282386\">\n<p id=\"fs-id1167836282388\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-38adf61b9b13f505340ff41243097c91_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#50;&#44;&#120;&#61;&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"106\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836293165\">\n<div data-type=\"problem\" id=\"fs-id1167836293167\">\n<p id=\"fs-id1167836293169\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4acf85714dba264425670c969df01cd0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#120;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#49;&#50;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"171\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836519229\">\n<div data-type=\"problem\" id=\"fs-id1167836519231\">\n<p id=\"fs-id1167836519233\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f682063ff603c0cc9f70fae77e6d1944_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#54;&#123;&#112;&#125;&#94;&#123;&#51;&#125;&#61;&#50;&#52;&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#43;&#57;&#112;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"131\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836507317\">\n<p id=\"fs-id1167836507319\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c3096beec16f3ee106c43c538b0e59c4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#112;&#61;&#48;&#44;&#112;&#61;&frac34;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"77\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833059796\">\n<div data-type=\"problem\" id=\"fs-id1167833059798\">\n<p id=\"fs-id1167833059801\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8445397665f48c8322f8bd57cda62f6a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#123;&#121;&#125;&#94;&#123;&#51;&#125;&#43;&#50;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#50;&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"124\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836532602\"><strong data-effect=\"bold\">Solve Equations with Polynomial Functions<\/strong><\/p>\n<p id=\"fs-id1167829579731\">In the following exercises, solve.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167829579735\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836544735\">\n<p id=\"fs-id1167836544737\">For the function, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-29f8a3cd1d805ed1cc6c5bc3b475a8b8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#49;&#120;&#43;&#50;&#48;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"172\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d0<\/span> find when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7d91c5d57a1f82043378cd4488c85297_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"84\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d1<\/span> Use this information to find two points that lie on the graph of the function.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829586777\">\n<p id=\"fs-id1167829586779\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-48ebdcda08cf6d53794431f5b209e4f4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"57\" style=\"vertical-align: 0px;\" \/> or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b3dc975a98ccada6f136856736d7df06_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"57\" style=\"vertical-align: -1px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d62d4701922c914ba4d246ac8667573b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#55;&#44;&#45;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"65\" style=\"vertical-align: -4px;\" \/> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-65f32a4571b9803eb38b1b691d82144c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#44;&#45;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"65\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836323177\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836323179\">\n<p id=\"fs-id1167836323181\">For the function, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2e2c91c57c435a5f37345f123ba0a1ee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#57;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#56;&#120;&#43;&#53;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"172\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d0<\/span> find when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-88b398517a8fd92e3e380be44e696ed2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"84\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d1<\/span> Use this information to find two points that lie on the graph of the function.<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836548728\">In each function, find: <span class=\"token\">\u24d0<\/span> the zeros of the function <span class=\"token\">\u24d1<\/span> the <em data-effect=\"italics\">x<\/em>-intercepts of the graph of the function <span class=\"token\">\u24d2<\/span> the <em data-effect=\"italics\">y<\/em>-intercept of the graph of the function.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167829716854\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829716856\">\n<p id=\"fs-id1167829689062\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1acd3e5bb83d0195680d511300b419ed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#54;&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"136\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836525231\">\n<p id=\"fs-id1167836525233\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1d0f4980ebd4e62078906c635e60f7c4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"43\" style=\"vertical-align: -6px;\" \/> or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6554a1fcb192bf6b7d717df683abcaf6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"56\" style=\"vertical-align: -6px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-de2e37d3a5203d50ed07c85b00315a63_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#56;&#125;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"49\" style=\"vertical-align: -7px;\" \/> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-233d01f5ad81f3120b3697ede2022c3d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#56;&#125;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"54\" style=\"vertical-align: -7px;\" \/> <span class=\"token\">\u24d2<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-946fcae4c0b6d86e2aff30b8ccbd5d49_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#45;&#52;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"61\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836690553\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836690555\">\n<p id=\"fs-id1167836529480\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8815295e57e967a1d5a2c80497919041_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#51;&#120;&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"167\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836700518\"><strong data-effect=\"bold\">Solve Applications Modeled by Quadratic Equations<\/strong><\/p>\n<p id=\"fs-id1167836704341\">In the following exercises, solve.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167836704344\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836481188\">\n<p id=\"fs-id1167836481190\">The product of two consecutive numbers is 399. Find the numbers.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836627120\">\n<p id=\"fs-id1167836627122\">The numbers are <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f2cae475c6cc4ade824f228f09d1e1ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"30\" style=\"vertical-align: -1px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0e22bf27371ae8e82f2cf9857cc59c29_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"31\" style=\"vertical-align: -1px;\" \/> or 19 and 21.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836551058\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836551060\">\n<p id=\"fs-id1167836551062\">The area of a rectangular shaped patio 432 square feet. The length of the patio is 6 feet more than its width. Find the length and width.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836392250\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829783696\">\n<p id=\"fs-id1167829783698\">A ladder leans against the wall of a building. The length of the ladder is 9 feet longer than the distance of the bottom of the ladder from the building. The distance of the top of the ladder reaches up the side of the building is 7 feet longer than the distance of the bottom of the ladder from the building. Find the lengths of all three sides of the triangle formed by the ladder leaning against the building.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836692413\">\n<p id=\"fs-id1167829783701\">The lengths are 8, 15, and 17 ft.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829621112\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836340204\">\n<p id=\"fs-id1167836340206\">Shruti is going to throw a ball from the top of a cliff. When she throws the ball from 80 feet above the ground, the function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-507333757ea1da4ee539d12d1f4dc085_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;&#92;&#108;&#101;&#102;&#116;&#40;&#116;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#49;&#54;&#123;&#116;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#52;&#116;&#43;&#56;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"188\" style=\"vertical-align: -4px;\" \/> models the height, <em data-effect=\"italics\">h<\/em>, of the ball above the ground as a function of time, <em data-effect=\"italics\">t<\/em>. Find: <span class=\"token\">\u24d0<\/span> the zeros of this function which tells us when the ball will hit the ground. <span class=\"token\">\u24d1<\/span> the time(s) the ball will be 80 feet above the ground. <span class=\"token\">\u24d2<\/span> the height the ball will be at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-eec54adef7198c4decb9169a2b472009_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"38\" style=\"vertical-align: 0px;\" \/> seconds which is when the ball will be at its highest point.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"practice-test\" data-depth=\"1\" id=\"fs-id1167833139379\">\n<h3 data-type=\"title\">Chapter Practice Test<\/h3>\n<p id=\"fs-id1167836486354\">In the following exercises, factor completely.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167829787150\">\n<div data-type=\"problem\" id=\"fs-id1167829787152\">\n<p id=\"fs-id1167836544360\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3cc6143022ebd7ce4c3b719f8ec38485_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#48;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#50;&#48;&#123;&#97;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"99\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829720971\">\n<p id=\"fs-id1167829720973\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f01c53d1c8636e435ffba33ee8748dab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#48;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#43;&#51;&#97;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"99\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836528609\">\n<div data-type=\"problem\" id=\"fs-id1167836300253\">\n<p id=\"fs-id1167836300255\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4e57388da727e6a578961669b82eac98_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#109;&#92;&#108;&#101;&#102;&#116;&#40;&#109;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#109;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"180\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836493210\">\n<div data-type=\"problem\" id=\"fs-id1167836493212\">\n<p id=\"fs-id1167836493214\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-557335874e3a27ee9a04250c61d8c4fd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#51;&#120;&#43;&#51;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"107\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836612999\">\n<p id=\"fs-id1167836613001\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-460d748739e93153063385b794c79566_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"110\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836608432\">\n<div data-type=\"problem\" id=\"fs-id1167829614267\">\n<p id=\"fs-id1167829614270\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-457e17cd89d063adf93aad024151ff1f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#43;&#112;&#113;&#45;&#49;&#50;&#123;&#113;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"112\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836424174\">\n<div data-type=\"problem\" id=\"fs-id1167836424176\">\n<p id=\"fs-id1167836424178\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-70b1f3e25dcc911d77bdd98f6a2642da_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#121;&#45;&#56;&#121;&#43;&#55;&#120;&#45;&#53;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"140\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836527655\">\n<p id=\"fs-id1167829651446\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f5bc2507ac92de19218dfec57967115a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"109\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836423912\">\n<div data-type=\"problem\" id=\"fs-id1167836423914\">\n<p id=\"fs-id1167836423916\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f7091bf949badd49c711adacf151eed1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#48;&#123;&#114;&#125;&#94;&#123;&#50;&#125;&#43;&#56;&#49;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"83\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833386900\">\n<div data-type=\"problem\" id=\"fs-id1167833386903\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a405fe05b16f8bd998f276f9c77f31a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#123;&#115;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#50;&#115;&#43;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"103\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836557699\">\n<p id=\"fs-id1167836557702\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d25dcff8917b41d22b919280da0ca6c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#115;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"67\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836688178\">\n<div data-type=\"problem\" id=\"fs-id1167836688180\">\n<p id=\"fs-id1167836688183\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-02ad486aa6c031781155d4f82d02be87_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#49;&#120;&#45;&#49;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"116\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836689546\">\n<div data-type=\"problem\" id=\"fs-id1167836689548\">\n<p id=\"fs-id1167836408252\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2af9bd442859e99cb03e14d9020f1326_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#55;&#53;&#123;&#121;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"82\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836282607\">\n<p id=\"fs-id1167836282609\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-70d4ed3d9d9f55dbad7e426c60c16ed0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#53;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#53;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"141\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836423849\">\n<div data-type=\"problem\" id=\"fs-id1167836408244\">\n<p id=\"fs-id1167836408246\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5d15ed2f26e894b6c6f3e61452c59c73_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#123;&#117;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#117;&#45;&#49;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"107\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836548214\">\n<div data-type=\"problem\" id=\"fs-id1167836548216\">\n<p id=\"fs-id1167836548218\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-505499bdaab8fc45d31d897055ee8f1b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#43;&#49;&#50;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"65\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836663062\">\n<p id=\"fs-id1167836663064\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-44398c41da41c70087dc98990b5fc0b9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#120;&#43;&#50;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"169\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836293549\">\n<div data-type=\"problem\" id=\"fs-id1167836293551\">\n<p id=\"fs-id1167836293553\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-aa3073d33ec43a227046d3b20c6fbaaf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#50;&#123;&#120;&#125;&#94;&#123;&#53;&#125;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#54;&#50;&#120;&#123;&#121;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"127\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833381407\">\n<div data-type=\"problem\" id=\"fs-id1167833381409\">\n<p id=\"fs-id1167829808159\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-aca5280ece3ce8de8c25f97f005bc07c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#45;&#49;&#57;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"123\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836621163\">\n<p id=\"fs-id1167836456503\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fe79260d75f259ba7642e43b4707fd74_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"146\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836701190\">\n<div data-type=\"problem\" id=\"fs-id1167836731147\">\n<p id=\"fs-id1167836731149\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d039ccd9817b206deeece45697cc428b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#45;&#51;&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#48;&#56;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"143\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836553477\">In the following exercises, solve<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167836553480\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836553482\">\n<p id=\"fs-id1167829853138\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0ceec724e7ba2d3b4191b008fb512455_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#54;&#97;&#61;&#50;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"116\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167836518301\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-733bff9f6df073c07a11d893575ef365_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&#52;&#92;&#116;&#101;&#120;&#116;&#123;&#47;&#125;&#53;&#44;&#97;&#61;&#45;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"123\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836502587\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836502590\">\n<p id=\"fs-id1167836502592\">The product of two consecutive integers is 156. Find the integers.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829740680\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829740682\">\n<p id=\"fs-id1167836407030\">The area of a rectangular place mat is 168 square inches. Its length is two inches longer than the width. Find the length and width of the placemat.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836407036\">\n<p id=\"fs-id1167836432100\">The width is 12 inches and the length is 14 inches.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836432105\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836540607\">\n<p id=\"fs-id1167836540609\">Jing is going to throw a ball from the balcony of her condo. When she throws the ball from 80 feet above the ground, the function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-507333757ea1da4ee539d12d1f4dc085_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;&#92;&#108;&#101;&#102;&#116;&#40;&#116;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#49;&#54;&#123;&#116;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#52;&#116;&#43;&#56;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"188\" style=\"vertical-align: -4px;\" \/> models the height, <em data-effect=\"italics\">h<\/em>, of the ball above the ground as a function of time, <em data-effect=\"italics\">t<\/em>. Find: <span class=\"token\">\u24d0<\/span> the zeros of this function which tells us when the ball will hit the ground. <span class=\"token\">\u24d1<\/span> the time(s) the ball will be 128 feet above the ground. <span class=\"token\">\u24d2<\/span> the height the ball will be at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-da12aae18022853a0285258f3e797890_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#61;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"39\" style=\"vertical-align: -1px;\" \/> seconds.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829861913\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836540808\">\n<p id=\"fs-id1167836540810\">For the function, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ab201466736537b3d4ca2aa4385571c5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#55;&#120;&#43;&#53;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"154\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d0<\/span> find when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-953a4085703c7d520f686a7acfea35b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"84\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d1<\/span> Use this information to find two points that lie on the graph of the function.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836697643\">\n<p id=\"fs-id1167833139186\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3573bf1ea4c223bb71878796b2106731_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\" \/> or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2145acc2878ed61214887e120f2485b7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"43\" style=\"vertical-align: -1px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-71a3917114bd7e1e3370ff870709e685_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f5437f64603e4a5ed63fa26e11681271_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#44;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829785512\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829785514\">\n<p id=\"fs-id1167836502746\">For the function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b0760d15003e2f6e9940a4f51c7011fc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#53;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#56;&#49;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"140\" style=\"vertical-align: -4px;\" \/> find: <span class=\"token\">\u24d0<\/span> the zeros of the function <span class=\"token\">\u24d1<\/span> the <em data-effect=\"italics\">x<\/em>-intercepts of the graph of the function <span class=\"token\">\u24d2<\/span> the <em data-effect=\"italics\">y<\/em>-intercept of the graph of the function.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"glossary\" class=\"textbox shaded\">\n<h3 data-type=\"glossary-title\">Glossary<\/h3>\n<dl id=\"fs-id1167836610089\">\n<dt>degree of the polynomial equation<\/dt>\n<dd id=\"fs-id1167836610094\">The degree of the polynomial equation is the degree of the polynomial.<\/dd>\n<\/dl>\n<dl id=\"fs-id1167836699095\">\n<dt>polynomial equation<\/dt>\n<dd id=\"fs-id1167836699100\">A polynomial equation is an equation that contains a polynomial expression.<\/dd>\n<\/dl>\n<dl id=\"fs-id1167833047228\">\n<dt>quadratic equation<\/dt>\n<dd id=\"fs-id1167833047234\">Polynomial equations of degree two are called quadratic equations.<\/dd>\n<\/dl>\n<dl id=\"fs-id1167833082445\">\n<dt>zero of the function<\/dt>\n<dd>A value of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/> where the function is 0, is called a zero of the function.<\/dd>\n<\/dl>\n<dl id=\"fs-id1167836516138\">\n<dt>Zero Product Property<\/dt>\n<dd id=\"fs-id1167836516143\">The Zero Product Property says that if the product of two quantities is zero, then at least one of the quantities is zero.<\/dd>\n<\/dl>\n<\/div>\n","protected":false},"author":103,"menu_order":6,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-3129","chapter","type-chapter","status-publish","hentry"],"part":2962,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/3129","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/users\/103"}],"version-history":[{"count":1,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/3129\/revisions"}],"predecessor-version":[{"id":15247,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/3129\/revisions\/15247"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/parts\/2962"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/3129\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/media?parent=3129"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapter-type?post=3129"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/contributor?post=3129"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/license?post=3129"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}