{"id":3959,"date":"2018-12-11T13:58:38","date_gmt":"2018-12-11T18:58:38","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/solve-quadratic-equations-in-quadratic-form\/"},"modified":"2018-12-11T13:58:38","modified_gmt":"2018-12-11T18:58:38","slug":"solve-quadratic-equations-in-quadratic-form","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/solve-quadratic-equations-in-quadratic-form\/","title":{"raw":"Solve Quadratic Equations in Quadratic Form","rendered":"Solve Quadratic Equations in Quadratic Form"},"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>Solve equations in quadratic form<\/li><\/ul><\/div><div data-type=\"note\" class=\"be-prepared\"><p id=\"fs-id1167836611861\">Before you get started, take this readiness quiz.<\/p><ol id=\"fs-id1167833066309\" type=\"1\"><li>Factor by substitution: \\({y}^{4}-{y}^{2}-20.\\)<div data-type=\"newline\"><br><\/div> If you missed this problem, review <a href=\"\/contents\/edd9c403-9825-4cf2-b237-2e05552ea3ec#fs-id1167829908071\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><li>Factor by substitution: \\({\\left(y-4\\right)}^{2}+8\\left(y-4\\right)+15.\\)<div data-type=\"newline\"><br><\/div> If you missed this problem, review <a href=\"\/contents\/edd9c403-9825-4cf2-b237-2e05552ea3ec#fs-id1167836320647\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><li>Simplify: <span class=\"token\">\u24d0<\/span> \\({x}^{\\frac{1}{2}}\u00b7{x}^{\\frac{1}{4}}\\) <span class=\"token\">\u24d1<\/span> \\({\\left({x}^{\\frac{1}{3}}\\right)}^{2}\\) <span class=\"token\">\u24d2<\/span> \\({\\left({x}^{-1}\\right)}^{2}.\\)<div data-type=\"newline\"><br><\/div> If you missed this problem, review <a href=\"\/contents\/eb676a52-0094-4ffb-95d3-c8eb0596e397#fs-id1169147740829\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><\/ol><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167829717106\"><h3 data-type=\"title\">Solve Equations in Quadratic Form<\/h3><p id=\"fs-id1167826131862\">Sometimes when we factored trinomials, the trinomial did not appear to be in the <em data-effect=\"italics\">ax<\/em><sup>2<\/sup> + <em data-effect=\"italics\">bx<\/em> + <em data-effect=\"italics\">c<\/em> form. So we factored by substitution allowing us to make it fit the <em data-effect=\"italics\">ax<\/em><sup>2<\/sup> + <em data-effect=\"italics\">bx<\/em> + <em data-effect=\"italics\">c<\/em> form. We used the standard \\(u\\) for the substitution.<\/p><p id=\"fs-id1167826130467\">To factor the expression <em data-effect=\"italics\">x<\/em><sup>4<\/sup> \u2212 4<em data-effect=\"italics\">x<\/em><sup>2<\/sup> \u2212 5, we noticed the variable part of the middle term is <em data-effect=\"italics\">x<\/em><sup>2<\/sup> and its square, <em data-effect=\"italics\">x<\/em><sup>4<\/sup>, is the variable part of the first term. (We know \\({\\left({x}^{2}\\right)}^{2}={x}^{4}.\\)) So we let <em data-effect=\"italics\">u<\/em> = <em data-effect=\"italics\">x<\/em><sup>2<\/sup> and factored.<\/p><table id=\"fs-id1167824774147\" class=\"unnumbered unstyled\" summary=\"Start with the expression x to the fourth power minus 4 x squared minus 5. Rewrite the expression as the square of x squared minus 4 times x squared minus 5. Let u equal x squared and substitute u into the expression to yield u squared minus 4 u minus 5. Factor the trinomial as the product of u plus 1 and u minus 5. Replace u with x squared to get the product of x squared plus 1 and x squared minus 5.\" data-label=\"\"><tbody><tr><td><\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836444997\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_001a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td><\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836294689\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_001b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Let \\(u={x}^{2}\\) and substitute.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836440485\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_001c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Factor the trinomial.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167833096382\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_001d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Replace <em data-effect=\"italics\">u<\/em> with \\({x}^{2}\\).<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167829747325\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_001e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><p id=\"fs-id1167829694860\">Similarly, sometimes an equation is not in the <em data-effect=\"italics\">ax<\/em><sup>2<\/sup> + <em data-effect=\"italics\">bx<\/em> + <em data-effect=\"italics\">c<\/em> = 0 form but looks much like a quadratic equation. Then, we can often make a thoughtful substitution that will allow us to make it fit the <em data-effect=\"italics\">ax<\/em><sup>2<\/sup> + <em data-effect=\"italics\">bx<\/em> + <em data-effect=\"italics\">c<\/em> = 0 form. If we can make it fit the form, we can then use all of our methods to solve quadratic equations.<\/p><p id=\"fs-id1167833345138\">Notice that in the quadratic equation <em data-effect=\"italics\">ax<\/em><sup>2<\/sup> + <em data-effect=\"italics\">bx<\/em> + <em data-effect=\"italics\">c<\/em> = 0, the middle term has a variable, <em data-effect=\"italics\">x<\/em>, and its square, <em data-effect=\"italics\">x<\/em><sup>2<\/sup>, is the variable part of the first term. Look for this relationship as you try to find a substitution.<\/p><p id=\"fs-id1167826169624\">Again, we will use the standard <em data-effect=\"italics\">u<\/em> to make a substitution that will put the equation in quadratic form. If the substitution gives us an equation of the form <em data-effect=\"italics\">ax<\/em><sup>2<\/sup> + <em data-effect=\"italics\">bx<\/em> + <em data-effect=\"italics\">c<\/em> = 0, we say the original equation was of <span data-type=\"term\">quadratic form<\/span>.<\/p><p>The next example shows the steps for solving an equation in quadratic form.<\/p><div data-type=\"example\" id=\"fs-id1167836561327\" class=\"textbox textbox--examples\"><div data-type=\"title\">How to Solve Equations in Quadratic Form<\/div><div data-type=\"exercise\" id=\"fs-id1167836519219\"><div data-type=\"problem\" id=\"fs-id1167836690217\"><p id=\"fs-id1167836495586\">Solve: \\(6{x}^{4}-7{x}^{2}+2=0\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829850393\"><span data-type=\"media\" id=\"fs-id1167832971240\" data-alt=\"Step 1 is to identify a substitution that will put the equation in quadratic form. Look at the equation 6 x to the fourth power minus 7 x squared plus 2 equals 0. Since the square of x squared equals x to the fourth, let u equals x squared.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_002a_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 1 is to identify a substitution that will put the equation in quadratic form. Look at the equation 6 x to the fourth power minus 7 x squared plus 2 equals 0. Since the square of x squared equals x to the fourth, let u equals x squared.\"><\/span><span data-type=\"media\" data-alt=\"Step 2 is to rewrite the equation with the substitution to put it in quadratic form. Rewrite the equation to prepare for the substitution to show 6 times the square of x squared minus 7 times x squared plus 2 equals 0. Substitute u equals x squared to get the new equation 6 times u squared minus 7 u plus 2 equals 0.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_002b_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2 is to rewrite the equation with the substitution to put it in quadratic form. Rewrite the equation to prepare for the substitution to show 6 times the square of x squared minus 7 times x squared plus 2 equals 0. Substitute u equals x squared to get the new equation 6 times u squared minus 7 u plus 2 equals 0.\"><\/span><span data-type=\"media\" id=\"fs-id1167836363445\" data-alt=\"Step 3 is to solve the quadratic equation for u. We can solve by factoring, so rewrite the equation as the product of 2 u minus 1 and 3 u minus 2 equals 0. Use the Zero Product Property to create 2 equations. If 2 u minus 1 equals 0, then 2u equals one, so u equals one half. If 3 u minus 2 equals 0, then 3 u equals 2 and u equals two thirds.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_002c_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3 is to solve the quadratic equation for u. We can solve by factoring, so rewrite the equation as the product of 2 u minus 1 and 3 u minus 2 equals 0. Use the Zero Product Property to create 2 equations. If 2 u minus 1 equals 0, then 2u equals one, so u equals one half. If 3 u minus 2 equals 0, then 3 u equals 2 and u equals two thirds.\"><\/span><span data-type=\"media\" id=\"fs-id1167829578559\" data-alt=\"In step 4, substitute the original variable back into the results. In this case replace u with x squared. So u equals one half becomes x squared equals one half and u equals two thirds becomes x squared equals two thirds.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_002d_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"In step 4, substitute the original variable back into the results. In this case replace u with x squared. So u equals one half becomes x squared equals one half and u equals two thirds becomes x squared equals two thirds.\"><\/span><span data-type=\"media\" data-alt=\"Step 5 is to solve for the original variable, so use the Square Root Property to solve for x. If x squared equals one half, then x equals the positive or negative square root of one half. Rationalize the denominator to see that x equals the positive or negative square root of 2 divided by 2. If x squared equals two thirds, then x equals the positive or negative square root of two thirds. Rationalize the denominator to see that x equals the positive or negative square root of 6 divided by 3.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_002e_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 5 is to solve for the original variable, so use the Square Root Property to solve for x. If x squared equals one half, then x equals the positive or negative square root of one half. Rationalize the denominator to see that x equals the positive or negative square root of 2 divided by 2. If x squared equals two thirds, then x equals the positive or negative square root of two thirds. Rationalize the denominator to see that x equals the positive or negative square root of 6 divided by 3.\"><\/span><span data-type=\"media\" id=\"fs-id1167832999026\" data-alt=\"In step 6, check your solutions. We will show one check here, x equals square root 2 divided by 2. Substitute this value into the original equation. 6 times the fourth power of the quotient square root 2 divided by 2 minus 7 times the square of the quotient square root of 2 divided by 2 plus 2. Does this expression equal 0? Simplify the powers. 6 times four sixteenths minus 7 times two fourths plus 2. Simplify terms. Three halves minus seven halves plus four halves equals zero. Square root 2 divided by 2 is a solution. We leave the other checks to you!\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_002f_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"In step 6, check your solutions. We will show one check here, x equals square root 2 divided by 2. Substitute this value into the original equation. 6 times the fourth power of the quotient square root 2 divided by 2 minus 7 times the square of the quotient square root of 2 divided by 2 plus 2. Does this expression equal 0? Simplify the powers. 6 times four sixteenths minus 7 times two fourths plus 2. Simplify terms. Three halves minus seven halves plus four halves equals zero. Square root 2 divided by 2 is a solution. We leave the other checks to you!\"><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167826128131\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167829931396\"><div data-type=\"problem\" id=\"fs-id1167833381523\"><p id=\"fs-id1167824652260\">Solve: \\({x}^{4}-6{x}^{2}+8=0\\).<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833066687\"><p>\\(x=\\sqrt{2},x=\\text{\u2212}\\sqrt{2},x=2,x=-2\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167832999129\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167824585372\"><div data-type=\"problem\" id=\"fs-id1167829745832\"><p id=\"fs-id1167836495425\">Solve: \\({x}^{4}-11{x}^{2}+28=0\\).<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833020666\"><p id=\"fs-id1167824735327\">\\(x=\\sqrt{7},x=\\text{\u2212}\\sqrt{7},x=2,x=-2\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1167829843991\">We summarize the steps to solve an equation in quadratic form.<\/p><div data-type=\"note\" id=\"fs-id1167824755292\" class=\"howto\"><div data-type=\"title\">Solve equations in quadratic form.<\/div><ol id=\"fs-id1167836609968\" type=\"1\" class=\"stepwise\"><li>Identify a substitution that will put the equation in quadratic form.<\/li><li>Rewrite the equation with the substitution to put it in quadratic form.<\/li><li>Solve the quadratic equation for <em data-effect=\"italics\">u<\/em>.<\/li><li>Substitute the original variable back into the results, using the substitution.<\/li><li>Solve for the original variable.<\/li><li>Check the solutions.<\/li><\/ol><\/div><p id=\"fs-id1167836282817\">In the next example, the binomial in the middle term, (<em data-effect=\"italics\">x<\/em> \u2212 2) is squared in the first term. If we let <em data-effect=\"italics\">u<\/em> = <em data-effect=\"italics\">x<\/em> \u2212 2 and substitute, our trinomial will be in <em data-effect=\"italics\">ax<\/em><sup>2<\/sup> + <em data-effect=\"italics\">bx<\/em> + <em data-effect=\"italics\">c<\/em> form.<\/p><div data-type=\"example\" id=\"fs-id1167836333710\" class=\"textbox textbox--examples\"><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167826131376\"><p id=\"fs-id1167829747005\">Solve: \\({\\left(x-2\\right)}^{2}+7\\left(x-2\\right)+12=0.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836537015\"><table id=\"fs-id1167824735246\" class=\"unnumbered unstyled\" summary=\"Write the equation. The square of x minus 2 plus 7 times the expression x minus 2 plus 12 equals zero. Prepare for the substitution by viewing x \u2212 2 as a variable. Let u equal x minus 2 and substitute. The equation becomes u squared plus 7 u plus 12 equals 0. Factor the expression on the left side of the equation. The product of u plus 3 and u plus 4 equals 0. By the Zero Product Property, u plus 3 equals 0 or u plus 4 equals 0. So u equals negative 3 or u equals negative 4. Replace u with x minus 2 and solve for x. If u equals negative 3, then x minus 2 equals negative 3 and x equals negative 1. If u equals negative 4, then x minus 2 equals negative 4 and x equals negative 2. Check the solutions. Start with x equals negative one. Substitute negative one for x in the left-hand side of the original equation to get square of negative one minus two plus seven times the expression negative one minus two plus twelve. We need to show that this expression equals zero. Simplify to yield the square of negative three plus seven times negative three plus twelve, or nine minus twenty-one plus twelve which equals zero. X equals negative one is a solution. Next, check x equals negative two. Substitute negative two for x in the left-hand side of the original equation to get square of negative two minus two plus seven times the expression negative two minus two plus twelve. We need to show that this expression equals zero. Simplify to yield the square of negative four plus seven times negative four plus twelve, or sixteen minus twenty-eight plus twelve which equals zero. X equals negative two is a solution.\" data-label=\"\"><tbody><tr><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829749329\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_003b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Prepare for the substitution.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167824652294\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_003c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Let \\(u=x-2\\) and substitute.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_003d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Solve by factoring.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836791325\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_003e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><div data-type=\"newline\"><br><\/div><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_003f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167832935517\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_003g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Replace \\(u\\) with \\(x-2.\\)<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829849398\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_003h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Solve for \\(x.\\)<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167824734382\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_003i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Check:<div data-type=\"newline\"><br><\/div><div data-type=\"newline\"><br><\/div> <span data-type=\"media\" id=\"fs-id1167833046823\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_003a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span> <\/td><td><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836729121\" class=\"try\"><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167836686218\"><p id=\"fs-id1167836792975\">Solve: \\({\\left(x-5\\right)}^{2}+6\\left(x-5\\right)+8=0.\\)<\/p><\/div><div data-type=\"solution\"><p id=\"fs-id1167829593750\">\\(x=3,x=1\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167833186416\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167833017969\"><div data-type=\"problem\"><p>Solve: \\({\\left(y-4\\right)}^{2}+8\\left(y-4\\right)+15=0.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836542459\"><p id=\"fs-id1167829620974\">\\(y=-1,y=1\\)<\/p><\/div><\/div><\/div><p>In the next example, we notice that \\({\\left(\\sqrt{x}\\right)}^{2}=x.\\) Also, remember that when we square both sides of an equation, we may introduce extraneous roots. Be sure to check your answers!<\/p><div data-type=\"example\" id=\"fs-id1167836791712\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167836705086\"><div data-type=\"problem\" id=\"fs-id1167836729261\"><p>Solve: \\(x-3\\sqrt{x}+2=0.\\)<\/p><\/div><div data-type=\"solution\"><p id=\"fs-id1167836573149\">The \\(\\sqrt{x}\\) in the middle term, is squared in the first term \\({\\left(\\sqrt{x}\\right)}^{2}=x.\\) If we let \\(u=\\sqrt{x}\\) and substitute, our trinomial will be in <em data-effect=\"italics\">ax<\/em><sup>2<\/sup> + <em data-effect=\"italics\">bx<\/em> + <em data-effect=\"italics\">c<\/em> = 0 form.<\/p><table id=\"fs-id1167829596626\" class=\"unnumbered unstyled\" summary=\"Write the equation. The x minus 3 times the square root of x plus 2 equals zero. Rewrite the x as the square of square root x to prepare for the substitution. Let u equal square root x and substitute. The equation becomes u squared minus 3 u plus 2 equals 0. Factor the expression on the left side of the equation. The product of u minus 3 and u minus 1 equals 0. By the Zero Product Property, u minus 2 equals 0 or u minus 1 equals 0. So u equals 1 or u equals 1. Replace u with square root x and solve for x by squaring both sides. If u equals 2, then square root x equals 2 and x equals 4. If u equals 1, then square root x equals 1 and x equals 1. Check the solutions. Start with x equals 4. Substitute 4 for x in the left-hand side of the original equation to get 4 minus 3 times square root 4 plus 2. We need to show that this expression equals zero. Simplify to yield 4 minus 6 plus 2 which equals zero. X equals 4 is a solution. Next, check x equals 1. Substitute 1 for x in the left-hand side of the original equation to get 1 minus 3 times square root 1 plus 2. We need to show that this expression equals zero. Simplify to yield 1 minus 3 plus 2 which equals zero. X equals 1 is a solution.\" data-label=\"\"><tbody><tr><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_004b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Rewrite the trinomial to prepare for the substitution.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833257740\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_004c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Let \\(u=\\sqrt{x}\\) and substitute.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_004d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Solve by factoring.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829751580\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_004e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167836666693\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_004f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167824617277\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_004g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Replace <em data-effect=\"italics\">u<\/em> with \\(\\sqrt{x}.\\)<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829696931\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_004h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Solve for <em data-effect=\"italics\">x<\/em>, by squaring both sides.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_004i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Check:<div data-type=\"newline\"><br><\/div><div data-type=\"newline\"><br><\/div> <span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_004a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span> <\/td><td><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167829738128\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167826152334\"><div data-type=\"problem\" id=\"fs-id1167833350833\"><p id=\"fs-id1167836614269\">Solve: \\(x-7\\sqrt{x}+12=0.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836534734\"><p id=\"fs-id1167836362655\">\\(x=9,x=16\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836448140\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836775158\"><div data-type=\"problem\" id=\"fs-id1167829712846\"><p>Solve: \\(x-6\\sqrt{x}+8=0.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836497713\"><p>\\(x=4,x=16\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1167829809763\">Substitutions for rational exponents can also help us solve an equation in quadratic form. Think of the properties of exponents as you begin the next example.<\/p><div data-type=\"example\" id=\"fs-id1167833397010\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167836515790\"><div data-type=\"problem\"><p id=\"fs-id1167829850601\">Solve: \\({x}^{\\frac{2}{3}}-2{x}^{\\frac{1}{3}}-24=0.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836326033\"><p id=\"fs-id1167824584008\">The \\({x}^{\\frac{1}{3}}\\) in the middle term is squared in the first term \\({\\left({x}^{\\frac{1}{3}}\\right)}^{2}={x}^{\\frac{2}{3}}.\\) If we let \\(u={x}^{\\frac{1}{3}}\\) and substitute, our trinomial will be in <em data-effect=\"italics\">ax<\/em><sup>2<\/sup> + <em data-effect=\"italics\">bx<\/em> + <em data-effect=\"italics\">c<\/em> = 0 form.<\/p><table id=\"fs-id1167833007178\" class=\"unnumbered unstyled can-break\" summary=\"Write the equation. X to the two-thirds power minus 2 times x to the one-third power minus 24 equals zero. Rewrite x to the two-thirds power as the square of x to the one-third power to prepare for the substitution. Let u equal the one-third power of x and substitute. The equation becomes u squared minus 2 u minus 24 equals 0. Factor the expression on the left side of the equation. The product of u minus 6 and u plus 4 equals 0. By the Zero Product Property, u minus 6 equals 0 or u plus 4 equals 0. So u equals 6 or u equals negative 4. Replace u with x to the one-third power and solve for x by cubing both sides. If u equals 6, then x to the one-third power equals 6, so x equals 6 cubed, or 216. If u equals negative 4, then x to the one-third power equals negative 4, so x equals the cube of negative 4, or negative 64. Check the solutions. Start with x equals 216. Substitute 216 for x in the left-hand side of the original equation to get 216 to the two-thirds power minus 2 times 216 to the one-third power minus 24. We need to show that this expression equals zero. Simplify to yield 36 minus 12 minus 24 which equals zero. X equals 216 is a solution. Next, check x equals negative 64. Substitute negative 64 for x in the left-hand side of the original equation to get negative 4 to the two-thirds power minus 2 times negative 4 to the one-third power minus 24. We need to show that this expression equals zero. Simplify to yield 16 plus 8 minus 24 which equals zero. X equals negative 64 is a solution.\" data-label=\"\"><tbody><tr><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836627734\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_005b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Rewrite the trinomial to prepare for the substitution.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_005c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Let \\(u={x}^{\\frac{1}{3}}\\) and substitute.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167824694795\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_005d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Solve by factoring.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829748215\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_005e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167836560607\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_005f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167829645110\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_005g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"middle\" data-align=\"left\">Replace <em data-effect=\"italics\">u<\/em> with \\({x}^{\\frac{1}{3}}.\\)<\/td><td data-valign=\"middle\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_005h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Solve for \\(x\\) by cubing both sides.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832980949\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_005i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167836415994\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_005j_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Check:<div data-type=\"newline\"><br><\/div><div data-type=\"newline\"><br><\/div> <span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_005a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span> <\/td><td><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167824748831\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167829599191\"><div data-type=\"problem\"><p>Solve: \\({x}^{\\frac{2}{3}}-5{x}^{\\frac{1}{3}}-14=0.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836506554\"><p id=\"fs-id1167836399398\">\\(x=-8,x=343\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167824765299\"><div data-type=\"problem\" id=\"fs-id1167833053722\"><p>Solve: \\({x}^{\\frac{1}{2}}+8{x}^{\\frac{1}{4}}+15=0.\\)<\/p><\/div><div data-type=\"solution\"><p id=\"fs-id1167829748045\">\\(x=81,x=625\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1165926678285\">In the next example, we need to keep in mind the definition of a negative exponent as well as the properties of exponents.<\/p><div data-type=\"example\" id=\"fs-id1167829719262\" class=\"textbox textbox--examples\"><div data-type=\"exercise\"><div data-type=\"problem\"><p>Solve: \\(3{x}^{-2}-7{x}^{-1}+2=0.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836509720\"><p>The \\({x}^{-1}\\) in the middle term is squared in the first term \\({\\left({x}^{-1}\\right)}^{2}={x}^{-2}.\\) If we let \\(u={x}^{-1}\\) and substitute, our trinomial will be in <em data-effect=\"italics\">ax<\/em><sup>2<\/sup> + <em data-effect=\"italics\">bx<\/em> + <em data-effect=\"italics\">c<\/em> = 0 form.<\/p><table id=\"fs-id1167833365963\" class=\"unnumbered unstyled can-break\" summary=\"Write the equation. Three times x to the negative two power minus 7 times x to the negative one power plus 2 equals zero. Rewrite x to the negative two power as the square of x to the negative one power to prepare for the substitution. Let u equal the x to the negative one power and substitute. The equation becomes 3 u squared minus 7 u plus 2 equals 0. Factor the expression on the left side of the equation. The product of 3 u minus 1 and u minus 2 equals 0. By the Zero Product Property, 3 u minus 1 equals 0 or u minus 2 equals 0. So u equals one third or u equals 2. Replace u with x to the negative one power and solve for x by taking the reciprocal. If u equals one third, then x to the negative one power equals one third, so x equals 3. If u equals 2, then x to the negative one power equals 2, so x equals one half. Check the solutions. Start with x equals 3. Substitute 3 for x in the left-hand side of the original equation to get 3 times 3 to the negative 2 power minus 7 times 3 to the negative one power plus 2. We need to show that this expression equals zero. Simplify to yield 3 times one ninth minus 7 times one third plus 2 which equals one third minus seven thirds plus six thirds, or 0. X equals 3 is a solution. Next, check x equals one half. Substitute one half for x in the left-hand side of the original equation to get 3 times one half to the negative 2 power minus 7 times one half to the negative one power plus 2. We need to show that this expression equals zero. Simplify to yield 3 times 4 minus 7 times 2 plus 2 which equals 12 minus 14 plus 2, or 0. X equals one half is a solution.\" data-label=\"\"><tbody><tr><td><\/td><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829599635\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_006b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Rewrite the trinomial to prepare for the substitution.<\/td><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836684589\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_006c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"middle\" data-align=\"left\">Let \\(u={x}^{-1}\\) and substitute.<\/td><td><\/td><td data-valign=\"middle\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167824732359\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_006d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Solve by factoring.<\/td><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836729694\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_006e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\"><\/td><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1171791084816\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_006f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\"><\/td><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836626538\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_006g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Replace <em data-effect=\"italics\">u<\/em> with \\({x}^{-1}.\\)<\/td><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836561361\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_006h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Solve for \\(x\\) by taking the reciprocal since \\({x}^{-1}=\\frac{1}{x}.\\)<\/td><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832961279\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_006i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Check:<div data-type=\"newline\"><br><\/div><div data-type=\"newline\"><br><\/div> <span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_006a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span> <\/td><td><\/td><td><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836554174\"><div data-type=\"problem\"><p id=\"fs-id1167836609520\">Solve: \\(8{x}^{-2}-10{x}^{-1}+3=0.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836512182\"><p>\\(x=\\frac{4}{3}x=2\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836610369\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167833024591\"><div data-type=\"problem\"><p id=\"fs-id1167836557624\">Solve: \\(6{x}^{-2}-23{x}^{-1}+20=0.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833256113\"><p>\\(x=\\frac{2}{5},x=\\frac{3}{4}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836541382\" class=\"media-2\"><p id=\"fs-id1167836629298\">Access this online resource for additional instruction and practice with solving quadratic equations.<\/p><ul id=\"fs-id1167829936758\" data-display=\"block\"><li><a href=\"https:\/\/openstax.org\/l\/37QuadForm4\">Solving Equations in Quadratic Form<\/a><\/li><\/ul><\/div><\/div><div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167836513746\"><h3 data-type=\"title\">Key Concepts<\/h3><ul data-bullet-style=\"bullet\"><li>How to solve equations in quadratic form. <ol id=\"fs-id1167836477354\" type=\"1\" class=\"stepwise\"><li>Identify a substitution that will put the equation in quadratic form.<\/li><li>Rewrite the equation with the substitution to put it in quadratic form.<\/li><li>Solve the quadratic equation for <em data-effect=\"italics\">u<\/em>.<\/li><li>Substitute the original variable back into the results, using the substitution.<\/li><li>Solve for the original variable.<\/li><li>Check the solutions.<\/li><\/ol><\/li><\/ul><\/div><div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167836477488\"><div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1167836390473\"><h4 data-type=\"title\">Practice Makes Perfect<\/h4><p><strong data-effect=\"bold\">Solve Equations in Quadratic Form<\/strong><\/p><p>In the following exercises, solve.<\/p><div data-type=\"exercise\" id=\"fs-id1167833048634\"><div data-type=\"problem\"><p>\\({x}^{4}-7{x}^{2}+12=0\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829740503\"><p id=\"fs-id1167836532767\">\\(x=\u00b1\\sqrt{3},x=\u00b12\\)<\/p><\/div><\/div><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167836283160\"><p>\\({x}^{4}-9{x}^{2}+18=0\\)<\/p><\/div><\/div><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167829685363\"><p id=\"fs-id1167836363985\">\\({x}^{4}-13{x}^{2}-30=0\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836417856\"><p id=\"fs-id1167825830254\">\\(x=\u00b1\\sqrt{15},x=\u00b1\\sqrt{2}i\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836652640\"><div data-type=\"problem\"><p id=\"fs-id1167833047402\">\\({x}^{4}+5{x}^{2}-36=0\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836379312\"><div data-type=\"problem\" id=\"fs-id1167836627066\"><p id=\"fs-id1167836629903\">\\(2{x}^{4}-5{x}^{2}+3=0\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836687994\"><p id=\"fs-id1167836619855\">\\(x=\u00b11,x=\\frac{\u00b1\\sqrt{6}}{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833269880\"><div data-type=\"problem\" id=\"fs-id1167836596280\"><p>\\(4{x}^{4}-5{x}^{2}+1=0\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829750536\"><div data-type=\"problem\" id=\"fs-id1167836305911\"><p id=\"fs-id1167836341518\">\\(2{x}^{4}-7{x}^{2}+3=0\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167824774181\"><p>\\(x=\u00b1\\sqrt{3},x=\u00b1\\frac{\\sqrt{2}}{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836408950\"><div data-type=\"problem\" id=\"fs-id1167836579268\"><p>\\(3{x}^{4}-14{x}^{2}+8=0\\)<\/p><\/div><\/div><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167836731868\"><p>\\({\\left(x-3\\right)}^{2}-5\\left(x-3\\right)-36=0\\)<\/p><\/div><div data-type=\"solution\"><p id=\"fs-id1167823013241\">\\(x=-1,x=12\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836620823\"><div data-type=\"problem\"><p id=\"fs-id1167829809683\">\\({\\left(x+2\\right)}^{2}-3\\left(x+2\\right)-54=0\\)<\/p><\/div><\/div><div data-type=\"exercise\"><div data-type=\"problem\"><p id=\"fs-id1167832977403\">\\({\\left(3y+2\\right)}^{2}+\\left(3y+2\\right)-6=0\\)<\/p><\/div><div data-type=\"solution\"><p>\\(x=-\\frac{5}{3},x=0\\)<\/p><\/div><\/div><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167836649268\"><p id=\"fs-id1167833239945\">\\({\\left(5y-1\\right)}^{2}+3\\left(5y-1\\right)-28=0\\)<\/p><\/div><\/div><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167829737953\"><p>\\({\\left({x}^{2}+1\\right)}^{2}-5\\left({x}^{2}+1\\right)+4=0\\)<\/p><\/div><div data-type=\"solution\"><p id=\"fs-id1167829742998\">\\(x=0,x=\u00b1\\sqrt{3}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829942584\"><div data-type=\"problem\"><p id=\"fs-id1167833036689\">\\({\\left({x}^{2}-4\\right)}^{2}-4\\left({x}^{2}-4\\right)+3=0\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829826691\"><div data-type=\"problem\" id=\"fs-id1167836553331\"><p>\\(2{\\left({x}^{2}-5\\right)}^{2}-5\\left({x}^{2}-5\\right)+2=0\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836707184\"><p>\\(x=\u00b1\\frac{11}{2},x=\u00b1\\frac{\\sqrt{22}}{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833022079\"><div data-type=\"problem\" id=\"fs-id1167836299364\"><p id=\"fs-id1167836375964\">\\(2{\\left({x}^{2}-5\\right)}^{2}-7\\left({x}^{2}-5\\right)+6=0\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836552088\"><div data-type=\"problem\" id=\"fs-id1167833058861\"><p id=\"fs-id1167830123856\">\\(x-\\sqrt{x}-20=0\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829790434\"><p id=\"fs-id1167836625623\">\\(x=25\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833059781\"><div data-type=\"problem\" id=\"fs-id1167829787104\"><p id=\"fs-id1167833049962\">\\(x-8\\sqrt{x}+15=0\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836329465\"><div data-type=\"problem\"><p id=\"fs-id1167836491957\">\\(x+6\\sqrt{x}-16=0\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833051614\"><p id=\"fs-id1167833158318\">\\(x=4\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829744349\"><div data-type=\"problem\" id=\"fs-id1167836522515\"><p id=\"fs-id1167836306229\">\\(x+4\\sqrt{x}-21=0\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836391104\"><div data-type=\"problem\" id=\"fs-id1167836630489\"><p id=\"fs-id1167830093104\">\\(6x+\\sqrt{x}-2=0\\)<\/p><\/div><div data-type=\"solution\"><p id=\"fs-id1167836537491\">\\(x=\\frac{1}{4}\\)<\/p><\/div><\/div><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167836600003\"><p>\\(6x+\\sqrt{x}-1=0\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836323284\"><div data-type=\"problem\" id=\"fs-id1167833379433\"><p id=\"fs-id1167836683538\">\\(10x-17\\sqrt{x}+3=0\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829720454\"><p id=\"fs-id1167833054312\">\\(x=\\frac{1}{25},x=\\frac{9}{4}\\)<\/p><\/div><\/div><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167836328808\"><p>\\(12x+5\\sqrt{x}-3=0\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836492727\"><div data-type=\"problem\" id=\"fs-id1167826206258\"><p id=\"fs-id1167833054408\">\\({x}^{\\frac{2}{3}}+9{x}^{\\frac{1}{3}}+8=0\\)<\/p><\/div><div data-type=\"solution\"><p>\\(x=-1,x=-512\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836612020\"><div data-type=\"problem\"><p id=\"fs-id1167836361636\">\\({x}^{\\frac{2}{3}}-3{x}^{\\frac{1}{3}}=28\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836756609\"><div data-type=\"problem\" id=\"fs-id1167836629341\"><p>\\({x}^{\\frac{2}{3}}+4{x}^{\\frac{1}{3}}=12\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829742759\"><p id=\"fs-id1167836798074\">\\(x=8,x=-216\\)<\/p><\/div><\/div><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167829753596\"><p id=\"fs-id1167836729573\">\\({x}^{\\frac{2}{3}}-11{x}^{\\frac{1}{3}}+30=0\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829745221\"><div data-type=\"problem\" id=\"fs-id1167836448360\"><p id=\"fs-id1167836379404\">\\(6{x}^{\\frac{2}{3}}-{x}^{\\frac{1}{3}}=12\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836538104\"><p id=\"fs-id1167836433089\">\\(x=\\frac{27}{8},x=-\\frac{64}{27}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836522254\"><div data-type=\"problem\" id=\"fs-id1167825791292\"><p id=\"fs-id1167833142430\">\\(3{x}^{\\frac{2}{3}}-10{x}^{\\frac{1}{3}}=8\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829748135\"><div data-type=\"problem\" id=\"fs-id1167833396850\"><p id=\"fs-id1167833139611\">\\(8{x}^{\\frac{2}{3}}-43{x}^{\\frac{1}{3}}+15=0\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836691776\"><p id=\"fs-id1167836608352\">\\(x=27,x=64,000\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836480304\"><div data-type=\"problem\" id=\"fs-id1167829651072\"><p id=\"fs-id1167825949274\">\\(20{x}^{\\frac{2}{3}}-23{x}^{\\frac{1}{3}}+6=0\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826131782\"><div data-type=\"problem\" id=\"fs-id1167836296736\"><p>\\(x-8{x}^{\\frac{1}{2}}+7=0\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836390041\"><p id=\"fs-id1167829747750\">\\(x=1,x=49\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836553976\"><div data-type=\"problem\" id=\"fs-id1167836570969\"><p id=\"fs-id1167836607787\">\\(2x-7{x}^{\\frac{1}{2}}=15\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829721215\"><div data-type=\"problem\" id=\"fs-id1167836524620\"><p id=\"fs-id1167836492217\">\\(6{x}^{-2}+13{x}^{-1}+5=0\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836729031\"><p id=\"fs-id1167836399348\">\\(x=-2,x=-\\frac{3}{5}\\)<\/p><\/div><\/div><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167829619888\"><p>\\(15{x}^{-2}-26{x}^{-1}+8=0\\)<\/p><\/div><\/div><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167824754866\"><p>\\(8{x}^{-2}-2{x}^{-1}-3=0\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836440443\"><p>\\(x=-2,x=\\frac{4}{3}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836367159\"><div data-type=\"problem\" id=\"fs-id1167829860664\"><p>\\(15{x}^{-2}-4{x}^{-1}-4=0\\)<\/p><\/div><\/div><\/div><div class=\"writing\" data-depth=\"2\" id=\"fs-id1167833408043\"><h4 data-type=\"title\">Writing Exercises<\/h4><div data-type=\"exercise\" id=\"fs-id1167829750381\"><div data-type=\"problem\" id=\"fs-id1167836417261\"><p id=\"fs-id1167836352160\">Explain how to recognize an equation in quadratic form.<\/p><\/div><div data-type=\"solution\"><p id=\"fs-id1167836415834\">Answers will vary.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833328888\"><div data-type=\"problem\" id=\"fs-id1167824584365\"><p id=\"fs-id1167829596848\">Explain the procedure for solving an equation in quadratic form.<\/p><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167833128991\"><h4 data-type=\"title\">Self Check<\/h4><p id=\"fs-id1167829860685\"><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\" data-alt=\"This table provides a checklist to evaluate mastery of the objectives of this section. Choose how would you respond to the statement \u201cI can solve equations in quadratic form.\u201d \u201cConfidently,\u201d \u201cwith some help,\u201d or \u201cNo, I don\u2019t get it.\u201d\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_201_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This table provides a checklist to evaluate mastery of the objectives of this section. Choose how would you respond to the statement \u201cI can solve equations in quadratic form.\u201d \u201cConfidently,\u201d \u201cwith some help,\u201d or \u201cNo, I don\u2019t get it.\u201d\"><\/span><p id=\"fs-id1167836705311\"><span class=\"token\">\u24d1<\/span> On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?<\/p><\/div><\/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>Solve equations in quadratic form<\/li>\n<\/ul>\n<\/div>\n<div data-type=\"note\" class=\"be-prepared\">\n<p id=\"fs-id1167836611861\">Before you get started, take this readiness quiz.<\/p>\n<ol id=\"fs-id1167833066309\" type=\"1\">\n<li>Factor by substitution: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-af902c9066e39f3d60ab4ab4de067ad4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#94;&#123;&#52;&#125;&#45;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"99\" style=\"vertical-align: -4px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p> If you missed this problem, review <a href=\"\/contents\/edd9c403-9825-4cf2-b237-2e05552ea3ec#fs-id1167829908071\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<li>Factor by substitution: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e29d4e524672e1d7409abbfe0746748d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#56;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#49;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"191\" style=\"vertical-align: -4px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p> If you missed this problem, review <a href=\"\/contents\/edd9c403-9825-4cf2-b237-2e05552ea3ec#fs-id1167836320647\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<li>Simplify: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9da62efc7b0a9e8b3b0b584073ac33d2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#125;&middot;&#123;&#120;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"39\" 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-6aaf97df9d7afc1b95def8582212a03e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"46\" style=\"vertical-align: -12px;\" \/> <span class=\"token\">\u24d2<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fe530be2f4ef3e33b63a69267fddc04c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#94;&#123;&#45;&#49;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"54\" style=\"vertical-align: -7px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p> If you missed this problem, review <a href=\"\/contents\/eb676a52-0094-4ffb-95d3-c8eb0596e397#fs-id1169147740829\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<\/ol>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167829717106\">\n<h3 data-type=\"title\">Solve Equations in Quadratic Form<\/h3>\n<p id=\"fs-id1167826131862\">Sometimes when we factored trinomials, the trinomial did not appear to be in the <em data-effect=\"italics\">ax<\/em><sup>2<\/sup> + <em data-effect=\"italics\">bx<\/em> + <em data-effect=\"italics\">c<\/em> form. So we factored by substitution allowing us to make it fit the <em data-effect=\"italics\">ax<\/em><sup>2<\/sup> + <em data-effect=\"italics\">bx<\/em> + <em data-effect=\"italics\">c<\/em> form. We used the standard <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-43fe27dc3e528266a619764d90fce60b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/> for the substitution.<\/p>\n<p id=\"fs-id1167826130467\">To factor the expression <em data-effect=\"italics\">x<\/em><sup>4<\/sup> \u2212 4<em data-effect=\"italics\">x<\/em><sup>2<\/sup> \u2212 5, we noticed the variable part of the middle term is <em data-effect=\"italics\">x<\/em><sup>2<\/sup> and its square, <em data-effect=\"italics\">x<\/em><sup>4<\/sup>, is the variable part of the first term. (We know <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8f16a2c0a03e4cc2c1d9d3f939289485_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"85\" style=\"vertical-align: -7px;\" \/>) So we let <em data-effect=\"italics\">u<\/em> = <em data-effect=\"italics\">x<\/em><sup>2<\/sup> and factored.<\/p>\n<table id=\"fs-id1167824774147\" class=\"unnumbered unstyled\" summary=\"Start with the expression x to the fourth power minus 4 x squared minus 5. Rewrite the expression as the square of x squared minus 4 times x squared minus 5. Let u equal x squared and substitute u into the expression to yield u squared minus 4 u minus 5. Factor the trinomial as the product of u plus 1 and u minus 5. Replace u with x squared to get the product of x squared plus 1 and x squared minus 5.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836444997\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_001a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836294689\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_001b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\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-506ad41ae004b491ff353f08fd6eed0e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"51\" style=\"vertical-align: 0px;\" \/> and substitute.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836440485\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_001c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Factor the trinomial.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167833096382\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_001d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Replace <em data-effect=\"italics\">u<\/em> with <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b40448f90dbf1bf9cce1035e2f3b1120_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"17\" style=\"vertical-align: 0px;\" \/>.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167829747325\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_001e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167829694860\">Similarly, sometimes an equation is not in the <em data-effect=\"italics\">ax<\/em><sup>2<\/sup> + <em data-effect=\"italics\">bx<\/em> + <em data-effect=\"italics\">c<\/em> = 0 form but looks much like a quadratic equation. Then, we can often make a thoughtful substitution that will allow us to make it fit the <em data-effect=\"italics\">ax<\/em><sup>2<\/sup> + <em data-effect=\"italics\">bx<\/em> + <em data-effect=\"italics\">c<\/em> = 0 form. If we can make it fit the form, we can then use all of our methods to solve quadratic equations.<\/p>\n<p id=\"fs-id1167833345138\">Notice that in the quadratic equation <em data-effect=\"italics\">ax<\/em><sup>2<\/sup> + <em data-effect=\"italics\">bx<\/em> + <em data-effect=\"italics\">c<\/em> = 0, the middle term has a variable, <em data-effect=\"italics\">x<\/em>, and its square, <em data-effect=\"italics\">x<\/em><sup>2<\/sup>, is the variable part of the first term. Look for this relationship as you try to find a substitution.<\/p>\n<p id=\"fs-id1167826169624\">Again, we will use the standard <em data-effect=\"italics\">u<\/em> to make a substitution that will put the equation in quadratic form. If the substitution gives us an equation of the form <em data-effect=\"italics\">ax<\/em><sup>2<\/sup> + <em data-effect=\"italics\">bx<\/em> + <em data-effect=\"italics\">c<\/em> = 0, we say the original equation was of <span data-type=\"term\">quadratic form<\/span>.<\/p>\n<p>The next example shows the steps for solving an equation in quadratic form.<\/p>\n<div data-type=\"example\" id=\"fs-id1167836561327\" class=\"textbox textbox--examples\">\n<div data-type=\"title\">How to Solve Equations in Quadratic Form<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836519219\">\n<div data-type=\"problem\" id=\"fs-id1167836690217\">\n<p id=\"fs-id1167836495586\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f7d8b6039afdbf4a842af52273ad9ba1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#45;&#55;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"138\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829850393\"><span data-type=\"media\" id=\"fs-id1167832971240\" data-alt=\"Step 1 is to identify a substitution that will put the equation in quadratic form. Look at the equation 6 x to the fourth power minus 7 x squared plus 2 equals 0. Since the square of x squared equals x to the fourth, let u equals x squared.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_002a_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 1 is to identify a substitution that will put the equation in quadratic form. Look at the equation 6 x to the fourth power minus 7 x squared plus 2 equals 0. Since the square of x squared equals x to the fourth, let u equals x squared.\" \/><\/span><span data-type=\"media\" data-alt=\"Step 2 is to rewrite the equation with the substitution to put it in quadratic form. Rewrite the equation to prepare for the substitution to show 6 times the square of x squared minus 7 times x squared plus 2 equals 0. Substitute u equals x squared to get the new equation 6 times u squared minus 7 u plus 2 equals 0.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_002b_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2 is to rewrite the equation with the substitution to put it in quadratic form. Rewrite the equation to prepare for the substitution to show 6 times the square of x squared minus 7 times x squared plus 2 equals 0. Substitute u equals x squared to get the new equation 6 times u squared minus 7 u plus 2 equals 0.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167836363445\" data-alt=\"Step 3 is to solve the quadratic equation for u. We can solve by factoring, so rewrite the equation as the product of 2 u minus 1 and 3 u minus 2 equals 0. Use the Zero Product Property to create 2 equations. If 2 u minus 1 equals 0, then 2u equals one, so u equals one half. If 3 u minus 2 equals 0, then 3 u equals 2 and u equals two thirds.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_002c_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3 is to solve the quadratic equation for u. We can solve by factoring, so rewrite the equation as the product of 2 u minus 1 and 3 u minus 2 equals 0. Use the Zero Product Property to create 2 equations. If 2 u minus 1 equals 0, then 2u equals one, so u equals one half. If 3 u minus 2 equals 0, then 3 u equals 2 and u equals two thirds.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167829578559\" data-alt=\"In step 4, substitute the original variable back into the results. In this case replace u with x squared. So u equals one half becomes x squared equals one half and u equals two thirds becomes x squared equals two thirds.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_002d_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"In step 4, substitute the original variable back into the results. In this case replace u with x squared. So u equals one half becomes x squared equals one half and u equals two thirds becomes x squared equals two thirds.\" \/><\/span><span data-type=\"media\" data-alt=\"Step 5 is to solve for the original variable, so use the Square Root Property to solve for x. If x squared equals one half, then x equals the positive or negative square root of one half. Rationalize the denominator to see that x equals the positive or negative square root of 2 divided by 2. If x squared equals two thirds, then x equals the positive or negative square root of two thirds. Rationalize the denominator to see that x equals the positive or negative square root of 6 divided by 3.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_002e_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 5 is to solve for the original variable, so use the Square Root Property to solve for x. If x squared equals one half, then x equals the positive or negative square root of one half. Rationalize the denominator to see that x equals the positive or negative square root of 2 divided by 2. If x squared equals two thirds, then x equals the positive or negative square root of two thirds. Rationalize the denominator to see that x equals the positive or negative square root of 6 divided by 3.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167832999026\" data-alt=\"In step 6, check your solutions. We will show one check here, x equals square root 2 divided by 2. Substitute this value into the original equation. 6 times the fourth power of the quotient square root 2 divided by 2 minus 7 times the square of the quotient square root of 2 divided by 2 plus 2. Does this expression equal 0? Simplify the powers. 6 times four sixteenths minus 7 times two fourths plus 2. Simplify terms. Three halves minus seven halves plus four halves equals zero. Square root 2 divided by 2 is a solution. We leave the other checks to you!\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_002f_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"In step 6, check your solutions. We will show one check here, x equals square root 2 divided by 2. Substitute this value into the original equation. 6 times the fourth power of the quotient square root 2 divided by 2 minus 7 times the square of the quotient square root of 2 divided by 2 plus 2. Does this expression equal 0? Simplify the powers. 6 times four sixteenths minus 7 times two fourths plus 2. Simplify terms. Three halves minus seven halves plus four halves equals zero. Square root 2 divided by 2 is a solution. We leave the other checks to you!\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167826128131\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167829931396\">\n<div data-type=\"problem\" id=\"fs-id1167833381523\">\n<p id=\"fs-id1167824652260\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e071e6d24c9b59c90466978bc7c247dc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#45;&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#56;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"129\" style=\"vertical-align: -2px;\" \/>.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833066687\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3d339ee1f4419e2620c5c74c9dd9f144_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#125;&#44;&#120;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#125;&#44;&#120;&#61;&#50;&#44;&#120;&#61;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"237\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167832999129\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167824585372\">\n<div data-type=\"problem\" id=\"fs-id1167829745832\">\n<p id=\"fs-id1167836495425\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2533df7d72bc3da4c88c03b1da632962_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#45;&#49;&#49;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#56;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"147\" style=\"vertical-align: -2px;\" \/>.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833020666\">\n<p id=\"fs-id1167824735327\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d78323170ba5a9644cd9576345d72c8e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#55;&#125;&#44;&#120;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#55;&#125;&#44;&#120;&#61;&#50;&#44;&#120;&#61;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"237\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167829843991\">We summarize the steps to solve an equation in quadratic form.<\/p>\n<div data-type=\"note\" id=\"fs-id1167824755292\" class=\"howto\">\n<div data-type=\"title\">Solve equations in quadratic form.<\/div>\n<ol id=\"fs-id1167836609968\" type=\"1\" class=\"stepwise\">\n<li>Identify a substitution that will put the equation in quadratic form.<\/li>\n<li>Rewrite the equation with the substitution to put it in quadratic form.<\/li>\n<li>Solve the quadratic equation for <em data-effect=\"italics\">u<\/em>.<\/li>\n<li>Substitute the original variable back into the results, using the substitution.<\/li>\n<li>Solve for the original variable.<\/li>\n<li>Check the solutions.<\/li>\n<\/ol>\n<\/div>\n<p id=\"fs-id1167836282817\">In the next example, the binomial in the middle term, (<em data-effect=\"italics\">x<\/em> \u2212 2) is squared in the first term. If we let <em data-effect=\"italics\">u<\/em> = <em data-effect=\"italics\">x<\/em> \u2212 2 and substitute, our trinomial will be in <em data-effect=\"italics\">ax<\/em><sup>2<\/sup> + <em data-effect=\"italics\">bx<\/em> + <em data-effect=\"italics\">c<\/em> form.<\/p>\n<div data-type=\"example\" id=\"fs-id1167836333710\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167826131376\">\n<p id=\"fs-id1167829747005\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f304ac72140b43100ceca78783cec44e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#55;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#49;&#50;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"225\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836537015\">\n<table id=\"fs-id1167824735246\" class=\"unnumbered unstyled\" summary=\"Write the equation. The square of x minus 2 plus 7 times the expression x minus 2 plus 12 equals zero. Prepare for the substitution by viewing x \u2212 2 as a variable. Let u equal x minus 2 and substitute. The equation becomes u squared plus 7 u plus 12 equals 0. Factor the expression on the left side of the equation. The product of u plus 3 and u plus 4 equals 0. By the Zero Product Property, u plus 3 equals 0 or u plus 4 equals 0. So u equals negative 3 or u equals negative 4. Replace u with x minus 2 and solve for x. If u equals negative 3, then x minus 2 equals negative 3 and x equals negative 1. If u equals negative 4, then x minus 2 equals negative 4 and x equals negative 2. Check the solutions. Start with x equals negative one. Substitute negative one for x in the left-hand side of the original equation to get square of negative one minus two plus seven times the expression negative one minus two plus twelve. We need to show that this expression equals zero. Simplify to yield the square of negative three plus seven times negative three plus twelve, or nine minus twenty-one plus twelve which equals zero. X equals negative one is a solution. Next, check x equals negative two. Substitute negative two for x in the left-hand side of the original equation to get square of negative two minus two plus seven times the expression negative two minus two plus twelve. We need to show that this expression equals zero. Simplify to yield the square of negative four plus seven times negative four plus twelve, or sixteen minus twenty-eight plus twelve which equals zero. X equals negative two is a solution.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829749329\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_003b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Prepare for the substitution.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167824652294\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_003c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\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-ad9c7f486afd000c57fe590e0b1820c1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#61;&#120;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"74\" style=\"vertical-align: 0px;\" \/> and substitute.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_003d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Solve by factoring.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836791325\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_003e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_003f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167832935517\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_003g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Replace <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-43fe27dc3e528266a619764d90fce60b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/> with <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cfdc25fb64ffc9ad0325b336b33b7ea5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"45\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829849398\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_003h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Solve for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a9cc293b28f198c32e0356b52e2e23bd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"14\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167824734382\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_003i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Check:<\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p> <span data-type=\"media\" id=\"fs-id1167833046823\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_003a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span> <\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836729121\" class=\"try\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167836686218\">\n<p id=\"fs-id1167836792975\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5eab2dda4e5d34a9fb54437db092c34d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#56;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"216\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167829593750\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ff48c2020229518c105b85a282397842_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#51;&#44;&#120;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"92\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167833186416\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167833017969\">\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-96dfab59b19552448cfa27f74415ddbc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#56;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#49;&#53;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"224\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836542459\">\n<p id=\"fs-id1167829620974\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3e1ddcec664d80e8590ad8cfa48e3691_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#49;&#44;&#121;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"104\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the next example, we notice that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e81876aa7f734120327deaa415317a29_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#115;&#113;&#114;&#116;&#123;&#120;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#120;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"83\" style=\"vertical-align: -4px;\" \/> Also, remember that when we square both sides of an equation, we may introduce extraneous roots. Be sure to check your answers!<\/p>\n<div data-type=\"example\" id=\"fs-id1167836791712\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167836705086\">\n<div data-type=\"problem\" id=\"fs-id1167836729261\">\n<p>Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-06bdf611af2682353422f0f6b7c5e04c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#51;&#92;&#115;&#113;&#114;&#116;&#123;&#120;&#125;&#43;&#50;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"133\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167836573149\">The <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b9f7bc96a9743d361c5f388ee3a20b06_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"25\" style=\"vertical-align: -4px;\" \/> in the middle term, is squared in the first term <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e81876aa7f734120327deaa415317a29_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#115;&#113;&#114;&#116;&#123;&#120;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#120;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"83\" style=\"vertical-align: -4px;\" \/> If we let <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-57caf14e2162c2e9e17ad5a74d9aea75_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"60\" style=\"vertical-align: -4px;\" \/> and substitute, our trinomial will be in <em data-effect=\"italics\">ax<\/em><sup>2<\/sup> + <em data-effect=\"italics\">bx<\/em> + <em data-effect=\"italics\">c<\/em> = 0 form.<\/p>\n<table id=\"fs-id1167829596626\" class=\"unnumbered unstyled\" summary=\"Write the equation. The x minus 3 times the square root of x plus 2 equals zero. Rewrite the x as the square of square root x to prepare for the substitution. Let u equal square root x and substitute. The equation becomes u squared minus 3 u plus 2 equals 0. Factor the expression on the left side of the equation. The product of u minus 3 and u minus 1 equals 0. By the Zero Product Property, u minus 2 equals 0 or u minus 1 equals 0. So u equals 1 or u equals 1. Replace u with square root x and solve for x by squaring both sides. If u equals 2, then square root x equals 2 and x equals 4. If u equals 1, then square root x equals 1 and x equals 1. Check the solutions. Start with x equals 4. Substitute 4 for x in the left-hand side of the original equation to get 4 minus 3 times square root 4 plus 2. We need to show that this expression equals zero. Simplify to yield 4 minus 6 plus 2 which equals zero. X equals 4 is a solution. Next, check x equals 1. Substitute 1 for x in the left-hand side of the original equation to get 1 minus 3 times square root 1 plus 2. We need to show that this expression equals zero. Simplify to yield 1 minus 3 plus 2 which equals zero. X equals 1 is a solution.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_004b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Rewrite the trinomial to prepare for the substitution.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833257740\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_004c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\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-57caf14e2162c2e9e17ad5a74d9aea75_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"60\" style=\"vertical-align: -4px;\" \/> and substitute.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_004d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Solve by factoring.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829751580\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_004e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167836666693\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_004f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167824617277\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_004g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Replace <em data-effect=\"italics\">u<\/em> with <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c82d569e8a0bbc225eee410893702d57_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#120;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"28\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829696931\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_004h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Solve for <em data-effect=\"italics\">x<\/em>, by squaring both sides.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_004i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Check:<\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p> <span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_004a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span> <\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167829738128\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167826152334\">\n<div data-type=\"problem\" id=\"fs-id1167833350833\">\n<p id=\"fs-id1167836614269\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8da61e63712c54790edda373760667e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#55;&#92;&#115;&#113;&#114;&#116;&#123;&#120;&#125;&#43;&#49;&#50;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"142\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836534734\">\n<p id=\"fs-id1167836362655\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cfe21dd1746888dbaafba529fa822664_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#57;&#44;&#120;&#61;&#49;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"102\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836448140\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836775158\">\n<div data-type=\"problem\" id=\"fs-id1167829712846\">\n<p>Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-53fc4f34c439675b17c8e139c9c99d58_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#54;&#92;&#115;&#113;&#114;&#116;&#123;&#120;&#125;&#43;&#56;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"133\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836497713\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e86e73ada74f8fd23506a10f30ec1a25_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#52;&#44;&#120;&#61;&#49;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"102\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167829809763\">Substitutions for rational exponents can also help us solve an equation in quadratic form. Think of the properties of exponents as you begin the next example.<\/p>\n<div data-type=\"example\" id=\"fs-id1167833397010\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167836515790\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167829850601\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-555c0897b31c97c92e5bdcce849a0602_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#125;&#45;&#50;&#123;&#120;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#125;&#45;&#50;&#52;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"148\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836326033\">\n<p id=\"fs-id1167824584008\">The <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0d036c63d34c43f458124788a788e509_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"18\" style=\"vertical-align: 0px;\" \/> in the middle term is squared in the first term <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d50351121bf39e86dcc9a77634336305_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#123;&#120;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"95\" style=\"vertical-align: -12px;\" \/> If we let <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-42ef41b41f0321b51f41ae2096d9c36e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#61;&#123;&#120;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: 0px;\" \/> and substitute, our trinomial will be in <em data-effect=\"italics\">ax<\/em><sup>2<\/sup> + <em data-effect=\"italics\">bx<\/em> + <em data-effect=\"italics\">c<\/em> = 0 form.<\/p>\n<table id=\"fs-id1167833007178\" class=\"unnumbered unstyled can-break\" summary=\"Write the equation. X to the two-thirds power minus 2 times x to the one-third power minus 24 equals zero. Rewrite x to the two-thirds power as the square of x to the one-third power to prepare for the substitution. Let u equal the one-third power of x and substitute. The equation becomes u squared minus 2 u minus 24 equals 0. Factor the expression on the left side of the equation. The product of u minus 6 and u plus 4 equals 0. By the Zero Product Property, u minus 6 equals 0 or u plus 4 equals 0. So u equals 6 or u equals negative 4. Replace u with x to the one-third power and solve for x by cubing both sides. If u equals 6, then x to the one-third power equals 6, so x equals 6 cubed, or 216. If u equals negative 4, then x to the one-third power equals negative 4, so x equals the cube of negative 4, or negative 64. Check the solutions. Start with x equals 216. Substitute 216 for x in the left-hand side of the original equation to get 216 to the two-thirds power minus 2 times 216 to the one-third power minus 24. We need to show that this expression equals zero. Simplify to yield 36 minus 12 minus 24 which equals zero. X equals 216 is a solution. Next, check x equals negative 64. Substitute negative 64 for x in the left-hand side of the original equation to get negative 4 to the two-thirds power minus 2 times negative 4 to the one-third power minus 24. We need to show that this expression equals zero. Simplify to yield 16 plus 8 minus 24 which equals zero. X equals negative 64 is a solution.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836627734\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_005b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Rewrite the trinomial to prepare for the substitution.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_005c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\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-42ef41b41f0321b51f41ae2096d9c36e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#61;&#123;&#120;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: 0px;\" \/> and substitute.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167824694795\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_005d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Solve by factoring.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829748215\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_005e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167836560607\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_005f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167829645110\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_005g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"middle\" data-align=\"left\">Replace <em data-effect=\"italics\">u<\/em> with <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-daacde955df0e95003dcb426c78d239e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"25\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-valign=\"middle\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_005h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Solve for <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;\" \/> by cubing both sides.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832980949\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_005i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167836415994\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_005j_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Check:<\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p> <span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_005a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span> <\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167824748831\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167829599191\">\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-1ad80fa4814684af8b1e03c98c8e5353_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#125;&#45;&#53;&#123;&#120;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#125;&#45;&#49;&#52;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"148\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836506554\">\n<p id=\"fs-id1167836399398\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bb5bd4e0d4d4fddeac8fe73bde56f8df_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#56;&#44;&#120;&#61;&#51;&#52;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"125\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167824765299\">\n<div data-type=\"problem\" id=\"fs-id1167833053722\">\n<p>Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d861b47423ce99ff78c1fb86c5fabbf9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#125;&#43;&#56;&#123;&#120;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#125;&#43;&#49;&#53;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"148\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167829748045\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-12a80927e24cc2b9d4a1661c879a9370_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#56;&#49;&#44;&#120;&#61;&#54;&#50;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"119\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1165926678285\">In the next example, we need to keep in mind the definition of a negative exponent as well as the properties of exponents.<\/p>\n<div data-type=\"example\" id=\"fs-id1167829719262\" class=\"textbox textbox--examples\">\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-f536894ccd04d88212cc2d35370fd5a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#123;&#120;&#125;&#94;&#123;&#45;&#50;&#125;&#45;&#55;&#123;&#120;&#125;&#94;&#123;&#45;&#49;&#125;&#43;&#50;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"164\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836509720\">\n<p>The <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4081ce39ebf1bedca684a78d5429b7b3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#45;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"27\" style=\"vertical-align: 0px;\" \/> in the middle term is squared in the first term <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bbac49275e238a6bdb0a688ea8097b35_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#94;&#123;&#45;&#49;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#123;&#120;&#125;&#94;&#123;&#45;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"107\" style=\"vertical-align: -7px;\" \/> If we let <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e6cc2328fc4955528162a515fdc170dc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#61;&#123;&#120;&#125;&#94;&#123;&#45;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"61\" style=\"vertical-align: 0px;\" \/> and substitute, our trinomial will be in <em data-effect=\"italics\">ax<\/em><sup>2<\/sup> + <em data-effect=\"italics\">bx<\/em> + <em data-effect=\"italics\">c<\/em> = 0 form.<\/p>\n<table id=\"fs-id1167833365963\" class=\"unnumbered unstyled can-break\" summary=\"Write the equation. Three times x to the negative two power minus 7 times x to the negative one power plus 2 equals zero. Rewrite x to the negative two power as the square of x to the negative one power to prepare for the substitution. Let u equal the x to the negative one power and substitute. The equation becomes 3 u squared minus 7 u plus 2 equals 0. Factor the expression on the left side of the equation. The product of 3 u minus 1 and u minus 2 equals 0. By the Zero Product Property, 3 u minus 1 equals 0 or u minus 2 equals 0. So u equals one third or u equals 2. Replace u with x to the negative one power and solve for x by taking the reciprocal. If u equals one third, then x to the negative one power equals one third, so x equals 3. If u equals 2, then x to the negative one power equals 2, so x equals one half. Check the solutions. Start with x equals 3. Substitute 3 for x in the left-hand side of the original equation to get 3 times 3 to the negative 2 power minus 7 times 3 to the negative one power plus 2. We need to show that this expression equals zero. Simplify to yield 3 times one ninth minus 7 times one third plus 2 which equals one third minus seven thirds plus six thirds, or 0. X equals 3 is a solution. Next, check x equals one half. Substitute one half for x in the left-hand side of the original equation to get 3 times one half to the negative 2 power minus 7 times one half to the negative one power plus 2. We need to show that this expression equals zero. Simplify to yield 3 times 4 minus 7 times 2 plus 2 which equals 12 minus 14 plus 2, or 0. X equals one half is a solution.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829599635\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_006b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Rewrite the trinomial to prepare for the substitution.<\/td>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836684589\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_006c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"middle\" data-align=\"left\">Let <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e6cc2328fc4955528162a515fdc170dc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#61;&#123;&#120;&#125;&#94;&#123;&#45;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"61\" style=\"vertical-align: 0px;\" \/> and substitute.<\/td>\n<td><\/td>\n<td data-valign=\"middle\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167824732359\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_006d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Solve by factoring.<\/td>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836729694\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_006e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1171791084816\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_006f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836626538\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_006g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Replace <em data-effect=\"italics\">u<\/em> with <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6ce61863b8120c1ba136fa2a3a3d1e44_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#45;&#49;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"33\" style=\"vertical-align: 0px;\" \/><\/td>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836561361\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_006h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Solve for <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;\" \/> by taking the reciprocal since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-73e1e8e19dbcb430e5643985d111d1ae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#45;&#49;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#120;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"68\" style=\"vertical-align: -6px;\" \/><\/td>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832961279\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_006i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Check:<\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p> <span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_006a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span> <\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836554174\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167836609520\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5ec2d9efc486abd34c24d9d48dc941a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#123;&#120;&#125;&#94;&#123;&#45;&#50;&#125;&#45;&#49;&#48;&#123;&#120;&#125;&#94;&#123;&#45;&#49;&#125;&#43;&#51;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"172\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836512182\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dfec4cecb79fa7e5023518f16093fd6d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#51;&#125;&#120;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"86\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836610369\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167833024591\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167836557624\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-db0485cd5cf5e2ebbcdc7e52f5c01265_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#123;&#120;&#125;&#94;&#123;&#45;&#50;&#125;&#45;&#50;&#51;&#123;&#120;&#125;&#94;&#123;&#45;&#49;&#125;&#43;&#50;&#48;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"181\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833256113\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f556018d8a906015ce9bf114d3e93bcc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#53;&#125;&#44;&#120;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"95\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836541382\" class=\"media-2\">\n<p id=\"fs-id1167836629298\">Access this online resource for additional instruction and practice with solving quadratic equations.<\/p>\n<ul id=\"fs-id1167829936758\" data-display=\"block\">\n<li><a href=\"https:\/\/openstax.org\/l\/37QuadForm4\">Solving Equations in Quadratic Form<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167836513746\">\n<h3 data-type=\"title\">Key Concepts<\/h3>\n<ul data-bullet-style=\"bullet\">\n<li>How to solve equations in quadratic form.\n<ol id=\"fs-id1167836477354\" type=\"1\" class=\"stepwise\">\n<li>Identify a substitution that will put the equation in quadratic form.<\/li>\n<li>Rewrite the equation with the substitution to put it in quadratic form.<\/li>\n<li>Solve the quadratic equation for <em data-effect=\"italics\">u<\/em>.<\/li>\n<li>Substitute the original variable back into the results, using the substitution.<\/li>\n<li>Solve for the original variable.<\/li>\n<li>Check the solutions.<\/li>\n<\/ol>\n<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167836477488\">\n<div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1167836390473\">\n<h4 data-type=\"title\">Practice Makes Perfect<\/h4>\n<p><strong data-effect=\"bold\">Solve Equations in Quadratic Form<\/strong><\/p>\n<p>In the following exercises, solve.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167833048634\">\n<div data-type=\"problem\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-85e69a3853552868717b3ae7e2a11ad8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#45;&#55;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#50;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"138\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829740503\">\n<p id=\"fs-id1167836532767\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-61f45033bf9c453e0aefdcef57c303a7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&plusmn;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#125;&#44;&#120;&#61;&plusmn;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"107\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167836283160\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dbeab9389f14092f4da12e3f9a6fa799_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#45;&#57;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#56;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"138\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167829685363\">\n<p id=\"fs-id1167836363985\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8d14ce63c93bc1a935114b3f32d7d619_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;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"147\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836417856\">\n<p id=\"fs-id1167825830254\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0acf288a531b6d0eeaecbd13ab18c24f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&plusmn;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#53;&#125;&#44;&#120;&#61;&plusmn;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#125;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"138\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836652640\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167833047402\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-00c9265a6be4103383855172c61fbe07_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#43;&#53;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#54;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"138\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836379312\">\n<div data-type=\"problem\" id=\"fs-id1167836627066\">\n<p id=\"fs-id1167836629903\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-48e792bd9ab98ed09f3eb00d9ec24368_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#45;&#53;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"138\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836687994\">\n<p id=\"fs-id1167836619855\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f23a50687c9ba78ed72d729dcf3f00b2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&plusmn;&#49;&#44;&#120;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&plusmn;&#92;&#115;&#113;&#114;&#116;&#123;&#54;&#125;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"105\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833269880\">\n<div data-type=\"problem\" id=\"fs-id1167836596280\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-32dbb7975812cd20031684a90ed82f13_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#45;&#53;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"138\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829750536\">\n<div data-type=\"problem\" id=\"fs-id1167836305911\">\n<p id=\"fs-id1167836341518\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-872d8d6d80e17ab636d53a6563bede75_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#45;&#55;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"138\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167824774181\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-47ed6aeff2aae03564df00514fbc0915_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&plusmn;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#125;&#44;&#120;&#61;&plusmn;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#125;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"120\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836408950\">\n<div data-type=\"problem\" id=\"fs-id1167836579268\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b68b0a9cec4260fb7da37a9912b2798d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#45;&#49;&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#56;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"147\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167836731868\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f4521db3afae4fc5d8a27837aaabca92_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;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"221\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167823013241\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0185ff9898d96165d4eadd0e8c94dae2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#49;&#44;&#120;&#61;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"115\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836620823\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167829809683\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f0b14f27d8d69f00a29be89aff5d1d36_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#53;&#52;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"221\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167832977403\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b99faebe0bcf60e7721e598f04965aca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#121;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#121;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#54;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"217\" 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-84540552153e07d99ed0fa22ad572699_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#51;&#125;&#44;&#120;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"109\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167836649268\">\n<p id=\"fs-id1167833239945\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cfc247fd17593bba2dfa8c027ea1792d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#121;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#121;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#50;&#56;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"237\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167829737953\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a6bb6d304fc4c353dc9ea842ae25ba21_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#52;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"232\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167829742998\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f08c988157839ce81c7647abb1499c4f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#48;&#44;&#120;&#61;&plusmn;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"108\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829942584\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167833036689\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-50b4c7d159f52be92337b476c86cf081_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#51;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"232\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829826691\">\n<div data-type=\"problem\" id=\"fs-id1167836553331\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-10106e544499fafdf76d7051bb08c182_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#50;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"242\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836707184\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6bb3860bed0910957935c65cb198e3c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&plusmn;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#49;&#125;&#123;&#50;&#125;&#44;&#120;&#61;&plusmn;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#50;&#125;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"121\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833022079\">\n<div data-type=\"problem\" id=\"fs-id1167836299364\">\n<p id=\"fs-id1167836375964\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f4a84bcec1d44057d689ebd9ee03383a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#45;&#55;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#54;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"242\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836552088\">\n<div data-type=\"problem\" id=\"fs-id1167833058861\">\n<p id=\"fs-id1167830123856\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0d14c008bb61c1f2c18471ddd3146567_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#120;&#125;&#45;&#50;&#48;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"129\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829790434\">\n<p id=\"fs-id1167836625623\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e51879a28c6b53e49edc5b5353715393_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#50;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"51\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833059781\">\n<div data-type=\"problem\" id=\"fs-id1167829787104\">\n<p id=\"fs-id1167833049962\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f532ad76d3bd446c560118b263f05e5f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#56;&#92;&#115;&#113;&#114;&#116;&#123;&#120;&#125;&#43;&#49;&#53;&#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-id1167836329465\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167836491957\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ee2ce3240419d2a22b45ac2c2ae0d638_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#43;&#54;&#92;&#115;&#113;&#114;&#116;&#123;&#120;&#125;&#45;&#49;&#54;&#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-id1167833051614\">\n<p id=\"fs-id1167833158318\"><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;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829744349\">\n<div data-type=\"problem\" id=\"fs-id1167836522515\">\n<p id=\"fs-id1167836306229\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c29c7b8fa7d1466fc8bce45fd4a97e60_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#43;&#52;&#92;&#115;&#113;&#114;&#116;&#123;&#120;&#125;&#45;&#50;&#49;&#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-id1167836391104\">\n<div data-type=\"problem\" id=\"fs-id1167836630489\">\n<p id=\"fs-id1167830093104\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d0a29554b27f8490508cd4ad505049ff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#120;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#120;&#125;&#45;&#50;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"129\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167836537491\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c8359433f92a78fa959edae8995b107d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"43\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167836600003\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f59da538d0b1f6f64f569ae8cb6c6ec0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#120;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#120;&#125;&#45;&#49;&#61;&#48;\" 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-id1167836323284\">\n<div data-type=\"problem\" id=\"fs-id1167833379433\">\n<p id=\"fs-id1167836683538\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e2c7c72262fde35301b0bb68958c530e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#48;&#120;&#45;&#49;&#55;&#92;&#115;&#113;&#114;&#116;&#123;&#120;&#125;&#43;&#51;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"154\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829720454\">\n<p id=\"fs-id1167833054312\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4b545ca51d56d40c5b4cc34f7a9adecd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#53;&#125;&#44;&#120;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"102\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167836328808\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-00d59e1bb9952b73c6cf44e5d2b578b9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#50;&#120;&#43;&#53;&#92;&#115;&#113;&#114;&#116;&#123;&#120;&#125;&#45;&#51;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"145\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836492727\">\n<div data-type=\"problem\" id=\"fs-id1167826206258\">\n<p id=\"fs-id1167833054408\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d7bd1a99d9ddabce9e8681eb28a08629_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#125;&#43;&#57;&#123;&#120;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#125;&#43;&#56;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"135\" style=\"vertical-align: -2px;\" \/><\/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-e3d95e77ab3a6e7d178b3fccb274295b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#49;&#44;&#120;&#61;&#45;&#53;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"138\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836612020\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167836361636\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ade42e8492213a62003da82c964aff12_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#125;&#45;&#51;&#123;&#120;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#125;&#61;&#50;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"114\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836756609\">\n<div data-type=\"problem\" id=\"fs-id1167836629341\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9507f4ffbb989675d215f3818b3a4d82_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#125;&#43;&#52;&#123;&#120;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#125;&#61;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"113\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829742759\">\n<p id=\"fs-id1167836798074\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a2dc298300dfde19a92ddc80b6ba1113_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#56;&#44;&#120;&#61;&#45;&#50;&#49;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"125\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167829753596\">\n<p id=\"fs-id1167836729573\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9d1c65096c1907fb4cb53c82bc6dacfc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#125;&#45;&#49;&#49;&#123;&#120;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#125;&#43;&#51;&#48;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"153\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829745221\">\n<div data-type=\"problem\" id=\"fs-id1167836448360\">\n<p id=\"fs-id1167836379404\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f921a0619c1ffdafc42b4150974925cc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#123;&#120;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#125;&#45;&#123;&#120;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#125;&#61;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"113\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836538104\">\n<p id=\"fs-id1167836433089\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-17fa6924d448d72d752b2812d6e368e8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#55;&#125;&#123;&#56;&#125;&#44;&#120;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#52;&#125;&#123;&#50;&#55;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"123\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836522254\">\n<div data-type=\"problem\" id=\"fs-id1167825791292\">\n<p id=\"fs-id1167833142430\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f250adec714c1bfd03506e3ea63d0b7e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#123;&#120;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#125;&#45;&#49;&#48;&#123;&#120;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#125;&#61;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"122\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829748135\">\n<div data-type=\"problem\" id=\"fs-id1167833396850\">\n<p id=\"fs-id1167833139611\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-17d41b3cc2677f29e3ba4663926b2a50_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#123;&#120;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#125;&#45;&#52;&#51;&#123;&#120;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#125;&#43;&#49;&#53;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"162\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836691776\">\n<p id=\"fs-id1167836608352\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-89244abc7d1ba6482697ba579fd1f84d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#50;&#55;&#44;&#120;&#61;&#54;&#52;&#44;&#48;&#48;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"146\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836480304\">\n<div data-type=\"problem\" id=\"fs-id1167829651072\">\n<p id=\"fs-id1167825949274\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a8931cf5dd8f06f64dc80e6d49fe822a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#48;&#123;&#120;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#125;&#45;&#50;&#51;&#123;&#120;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#125;&#43;&#54;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"162\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826131782\">\n<div data-type=\"problem\" id=\"fs-id1167836296736\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1a1658acb8895e96479eb68cb66ffd3a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#56;&#123;&#120;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#125;&#43;&#55;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"125\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836390041\">\n<p id=\"fs-id1167829747750\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a653701451af747d878c5299a42ceb7b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#49;&#44;&#120;&#61;&#52;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"102\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836553976\">\n<div data-type=\"problem\" id=\"fs-id1167836570969\">\n<p id=\"fs-id1167836607787\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b64430e988e2781ba53b8f5068ce3889_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#120;&#45;&#55;&#123;&#120;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#125;&#61;&#49;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"111\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829721215\">\n<div data-type=\"problem\" id=\"fs-id1167836524620\">\n<p id=\"fs-id1167836492217\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c99d872e3d06caef702a1c21bd1dcfdf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#123;&#120;&#125;&#94;&#123;&#45;&#50;&#125;&#43;&#49;&#51;&#123;&#120;&#125;&#94;&#123;&#45;&#49;&#125;&#43;&#53;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"168\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836729031\">\n<p id=\"fs-id1167836399348\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a2f592bbcc0065b33190ec31f6c06bb8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#50;&#44;&#120;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"121\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167829619888\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3b6f7395bd3e831718db9fb58b5a07dd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#53;&#123;&#120;&#125;&#94;&#123;&#45;&#50;&#125;&#45;&#50;&#54;&#123;&#120;&#125;&#94;&#123;&#45;&#49;&#125;&#43;&#56;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"176\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167824754866\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e2018b793c6f89d2554bee3ceddbe4d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#123;&#120;&#125;&#94;&#123;&#45;&#50;&#125;&#45;&#50;&#123;&#120;&#125;&#94;&#123;&#45;&#49;&#125;&#45;&#51;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"160\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836440443\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c3a5bf3bd0efa2b1e3cabf2dc9dba920_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#50;&#44;&#120;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"107\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836367159\">\n<div data-type=\"problem\" id=\"fs-id1167829860664\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3d48bc6b70bb6939f18a6ca8699dc7ed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#53;&#123;&#120;&#125;&#94;&#123;&#45;&#50;&#125;&#45;&#52;&#123;&#120;&#125;&#94;&#123;&#45;&#49;&#125;&#45;&#52;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"167\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"writing\" data-depth=\"2\" id=\"fs-id1167833408043\">\n<h4 data-type=\"title\">Writing Exercises<\/h4>\n<div data-type=\"exercise\" id=\"fs-id1167829750381\">\n<div data-type=\"problem\" id=\"fs-id1167836417261\">\n<p id=\"fs-id1167836352160\">Explain how to recognize an equation in quadratic form.<\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167836415834\">Answers will vary.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833328888\">\n<div data-type=\"problem\" id=\"fs-id1167824584365\">\n<p id=\"fs-id1167829596848\">Explain the procedure for solving an equation in quadratic form.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167833128991\">\n<h4 data-type=\"title\">Self Check<\/h4>\n<p id=\"fs-id1167829860685\"><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\" data-alt=\"This table provides a checklist to evaluate mastery of the objectives of this section. Choose how would you respond to the statement \u201cI can solve equations in quadratic form.\u201d \u201cConfidently,\u201d \u201cwith some help,\u201d or \u201cNo, I don\u2019t get it.\u201d\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_04_201_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This table provides a checklist to evaluate mastery of the objectives of this section. Choose how would you respond to the statement \u201cI can solve equations in quadratic form.\u201d \u201cConfidently,\u201d \u201cwith some help,\u201d or \u201cNo, I don\u2019t get it.\u201d\" \/><\/span><\/p>\n<p id=\"fs-id1167836705311\"><span class=\"token\">\u24d1<\/span> On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?<\/p>\n<\/div>\n<\/div>\n","protected":false},"author":103,"menu_order":5,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-3959","chapter","type-chapter","status-publish","hentry"],"part":3677,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/3959","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":0,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/3959\/revisions"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/parts\/3677"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/3959\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/media?parent=3959"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapter-type?post=3959"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/contributor?post=3959"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/license?post=3959"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}