{"id":3817,"date":"2018-12-11T13:58:15","date_gmt":"2018-12-11T18:58:15","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/solve-quadratic-equations-by-completing-the-square\/"},"modified":"2018-12-11T13:58:15","modified_gmt":"2018-12-11T18:58:15","slug":"solve-quadratic-equations-by-completing-the-square","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/solve-quadratic-equations-by-completing-the-square\/","title":{"raw":"Solve Quadratic Equations by Completing the Square","rendered":"Solve Quadratic Equations by Completing the Square"},"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>Complete the square of a binomial expression<\/li><li>Solve quadratic equations of the form \\({x}^{2}+bx+c=0\\) by completing the square<\/li><li>Solve quadratic equations of the form \\({ax}^{2}+bx+c=0\\) by completing the square<\/li><\/ul><\/div><div data-type=\"note\" id=\"fs-id1167829746843\" class=\"be-prepared\"><p id=\"fs-id1167824737670\">Before you get started, take this readiness quiz.<\/p><ol id=\"fs-id1167836575945\" type=\"1\"><li>Expand: \\({\\left(x+9\\right)}^{2}.\\)<div data-type=\"newline\"><br><\/div> If you missed this problem, review <a href=\"\/contents\/0b9be1db-21c4-4bd0-8f8e-d809f6ff7c8c#fs-id1167836392219\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><li>Factor \\({y}^{2}-14y+49.\\)<div data-type=\"newline\"><br><\/div> If you missed this problem, review <a href=\"\/contents\/edd9c403-9825-4cf2-b237-2e05552ea3ec#fs-id1167836732680\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><li>Factor \\(5{n}^{2}+40n+80.\\)<div data-type=\"newline\"><br><\/div> If you missed this problem, review <a href=\"\/contents\/edd9c403-9825-4cf2-b237-2e05552ea3ec#fs-id1167836415146\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><\/ol><\/div><p id=\"fs-id1167833025786\">So far we have solved quadratic equations by factoring and using the Square Root Property. In this section, we will solve quadratic equations by a process called <span data-type=\"term\">completing the square<\/span>, which is important for our work on conics later.<\/p><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167829811989\"><h3 data-type=\"title\">Complete the Square of a Binomial Expression<\/h3><p id=\"fs-id1167829714915\">In the last section, we were able to use the <span data-type=\"term\" class=\"no-emphasis\">Square Root Property<\/span> to solve the equation (<em data-effect=\"italics\">y<\/em> \u2212 7)<sup>2<\/sup> = 12 because the left side was a perfect square.<\/p><div data-type=\"equation\" id=\"fs-id1167829908200\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{ccc}\\hfill {\\left(y-7\\right)}^{2}&amp; =\\hfill &amp; 12\\hfill \\\\ \\hfill y-7&amp; =\\hfill &amp; \u00b1\\sqrt{12}\\hfill \\\\ \\hfill y-7&amp; =\\hfill &amp; \u00b12\\sqrt{3}\\hfill \\\\ \\hfill y&amp; =\\hfill &amp; 7\u00b12\\sqrt{3}\\hfill \\end{array}\\)<\/div><p id=\"fs-id1167833311254\">We also solved an equation in which the left side was a perfect square trinomial, but we had to rewrite it the form \\({\\left(x-k\\right)}^{2}\\) in order to use the Square Root Property.<\/p><div data-type=\"equation\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{ccc}\\hfill {x}^{2}-10x+25&amp; =\\hfill &amp; 18\\hfill \\\\ \\hfill {\\left(x-5\\right)}^{2}&amp; =\\hfill &amp; 18\\hfill \\end{array}\\)<\/div><p id=\"fs-id1167824754928\">What happens if the variable is not part of a perfect square? Can we use algebra to make a perfect square?<\/p><p id=\"fs-id1167832925970\">Let\u2019s look at two examples to help us recognize the patterns.<\/p><div data-type=\"equation\" id=\"fs-id1167836579092\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{cccc}\\hfill {\\left(x+9\\right)}^{2}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{3em}{0ex}}{\\left(y-7\\right)}^{2}\\hfill \\\\ \\hfill \\left(x+9\\right)\\left(x+9\\right)\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{3em}{0ex}}\\left(y-7\\right)\\left(y-7\\right)\\hfill \\\\ \\hfill {x}^{2}+9x+9x+81\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{3em}{0ex}}{y}^{2}-7y-7y+49\\hfill \\\\ \\hfill {x}^{2}+18x+81\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{3em}{0ex}}{y}^{2}-14y+49\\hfill \\end{array}\\)<\/div><p id=\"fs-id1167829688618\">We restate the patterns here for reference.<\/p><div data-type=\"note\" id=\"fs-id1167836363097\"><div data-type=\"title\">Binomial Squares Pattern<\/div><p id=\"fs-id1167826132392\">If <em data-effect=\"italics\">a<\/em> and <em data-effect=\"italics\">b<\/em> are real numbers,<\/p><span data-type=\"media\" id=\"fs-id1167829739760\" data-alt=\"Quantity a plus b squared equals a squared plus 2 a b plus b2 where the binomial squared equals the first term squared plus 2 times the product of terms plus the second term squared. Quantity a minus b squared equals a squared minus 2 a b plus b2 where the binomial squared equals the first term squared minus 2 times the product of terms plus the second term squared.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_019_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Quantity a plus b squared equals a squared plus 2 a b plus b2 where the binomial squared equals the first term squared plus 2 times the product of terms plus the second term squared. Quantity a minus b squared equals a squared minus 2 a b plus b2 where the binomial squared equals the first term squared minus 2 times the product of terms plus the second term squared.\"><\/span><\/div><p id=\"fs-id1167836526163\">We can use this pattern to \u201cmake\u201d a perfect square.<\/p><p id=\"fs-id1167836507775\">We will start with the expression <em data-effect=\"italics\">x<\/em><sup>2<\/sup> + 6<em data-effect=\"italics\">x<\/em>. Since there is a plus sign between the two terms, we will use the (<em data-effect=\"italics\">a<\/em> + <em data-effect=\"italics\">b<\/em>)<sup>2<\/sup> pattern, <em data-effect=\"italics\">a<\/em><sup>2<\/sup> + 2<em data-effect=\"italics\">ab<\/em> + <em data-effect=\"italics\">b<\/em><sup>2<\/sup> = (<em data-effect=\"italics\">a<\/em> + <em data-effect=\"italics\">b<\/em>)<sup>2<\/sup>.<\/p><span data-type=\"media\" id=\"fs-id1167833272243\" data-alt=\"The perfect square expression a squared plus 2 a b plus b squared is shown above the expression x squared plus 6x plus an unknown to allow a comparison of the corresponding terms of the expressions.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_001_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The perfect square expression a squared plus 2 a b plus b squared is shown above the expression x squared plus 6x plus an unknown to allow a comparison of the corresponding terms of the expressions.\"><\/span><p>We ultimately need to find the last term of this trinomial that will make it a perfect square trinomial. To do that we will need to find <em data-effect=\"italics\">b<\/em>. But first we start with determining <em data-effect=\"italics\">a<\/em>. Notice that the first term of <em data-effect=\"italics\">x<\/em><sup>2<\/sup> + 6<em data-effect=\"italics\">x<\/em> is a square, <em data-effect=\"italics\">x<\/em><sup>2<\/sup>. This tells us that <em data-effect=\"italics\">a<\/em> = <em data-effect=\"italics\">x<\/em>.<\/p><span data-type=\"media\" id=\"fs-id1167836448681\" data-alt=\"The perfect square expression a squared plus 2 a b plus b squared is shown above the expression x squared plus 2 x b + b squared. Note that x has been substituted for a in the second equation and compare corresponding terms.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_002_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The perfect square expression a squared plus 2 a b plus b squared is shown above the expression x squared plus 2 x b + b squared. Note that x has been substituted for a in the second equation and compare corresponding terms.\"><\/span><p id=\"fs-id1167836408542\">What number, <em data-effect=\"italics\">b,<\/em> when multiplied with 2<em data-effect=\"italics\">x<\/em> gives 6<em data-effect=\"italics\">x<\/em>? It would have to be 3, which is \\(\\frac{1}{2}\\left(6\\right).\\) So <em data-effect=\"italics\">b<\/em> = 3.<\/p><span data-type=\"media\" id=\"fs-id1167825703475\" data-alt=\"The perfect square expression a squared plus 2 a b plus b squared is shown above the expression x squared plus 2 times 3 times x plus an unknown value to help compare terms.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_003_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The perfect square expression a squared plus 2 a b plus b squared is shown above the expression x squared plus 2 times 3 times x plus an unknown value to help compare terms.\"><\/span><p id=\"fs-id1167824767298\">Now to complete the perfect square trinomial, we will find the last term by squaring <em data-effect=\"italics\">b<\/em>, which is 3<sup>2<\/sup> = 9.<\/p><span data-type=\"media\" id=\"fs-id1167833227142\" data-alt=\"The perfect square expression a squared plus 2 a b plus b squared is shown above the expression x squared plus 6 x plus 9.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_004a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The perfect square expression a squared plus 2 a b plus b squared is shown above the expression x squared plus 6 x plus 9.\"><\/span><p id=\"fs-id1171791481186\">We can now factor.<\/p><span data-type=\"media\" id=\"fs-id1167836698501\" data-alt=\"The factored expression, the square of a plus b, is shown over the square of the expression x + 3.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_004b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The factored expression, the square of a plus b, is shown over the square of the expression x + 3.\"><\/span><p id=\"fs-id1167836673411\">So we found that adding 9 to <em data-effect=\"italics\">x<\/em><sup>2<\/sup> + 6<em data-effect=\"italics\">x<\/em> \u2018completes the square\u2019, and we write it as (<em data-effect=\"italics\">x<\/em> + 3)<sup>2<\/sup>.<\/p><div data-type=\"note\" id=\"fs-id1167836299468\" class=\"howto\"><div data-type=\"title\">Complete a square of \\({x}^{2}+bx.\\)<\/div><ol id=\"fs-id1167826162778\" type=\"1\" class=\"stepwise\"><li>Identify <em data-effect=\"italics\">b<\/em>, the coefficient of <em data-effect=\"italics\">x<\/em>.<\/li><li>Find \\({\\left(\\frac{1}{2}b\\right)}^{2},\\) the number to complete the square.<\/li><li>Add the \\({\\left(\\frac{1}{2}b\\right)}^{2}\\) to <em data-effect=\"italics\">x<\/em><sup>2<\/sup> + <em data-effect=\"italics\">bx<\/em>.<\/li><li>Factor the perfect square trinomial, writing it as a binomial squared.<\/li><\/ol><\/div><div data-type=\"example\" id=\"fs-id1167836299751\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167833007358\"><div data-type=\"problem\" id=\"fs-id1167825836369\"><p id=\"fs-id1167836620013\">Complete the square to make a perfect square trinomial. Then write the result as a binomial squared.<\/p><p id=\"fs-id1167836485755\"><span class=\"token\">\u24d0<\/span>\\({x}^{2}-26x\\)<span class=\"token\">\u24d1<\/span>\\({y}^{2}-9y\\)<span class=\"token\">\u24d2<\/span>\\({n}^{2}+\\frac{1}{2}n\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829720861\"><p id=\"fs-id1167836713860\"><span class=\"token\">\u24d0<\/span><\/p><div data-type=\"newline\"><br><\/div><table id=\"fs-id1167836664279\" class=\"unnumbered unstyled\" summary=\"The expression x squared minus b x is shown above the expression x squared minus 26 x. Note that the coefficient of x is negative 26. To complete the square, find the square of one half times b. Substitute negative 26 for b, rewriting the expression as the square of one half times negative 26. Simplifying the product gives the square of negative 13, and evaluating the square gives 169. Add 169 to the binomial to complete the square so that the expression becomes x squared minus 26 x plus 169. Factor the resulting perfect square trinomial, writing it as the square of x minus 26, a binomial squared.\" data-label=\"\"><tbody><tr><td><\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167829930011\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_005b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">The coefficient of \\(x\\) is \u221226.<\/td><td><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">\\(\\begin{array}{}\\\\ \\\\ \\\\ \\hfill \\text{Find}\\phantom{\\rule{0.2em}{0ex}}{\\left(\\frac{1}{2}b\\right)}^{2}.\\hfill \\\\ \\hfill {\\left(\\frac{1}{2}\u00b7\\left(-26\\right)\\right)}^{2}\\hfill \\\\ \\hfill {\\left(13\\right)}^{2}\\hfill \\\\ \\hfill 169\\hfill \\end{array}\\)<\/td><td><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Add 169 to the binomial to complete the square.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167833326353\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_005c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Factor the perfect square trinomial, writing it as<div data-type=\"newline\"><br><\/div>a binomial squared.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167829743340\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_005d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><p id=\"fs-id1167829742072\"><span class=\"token\">\u24d1<\/span><\/p><div data-type=\"newline\"><br><\/div><table id=\"fs-id1167829695497\" class=\"unnumbered unstyled\" summary=\"The expression x squared minus b x is shown above the expression y squared minus 9 y. Note that the coefficient of y is negative 9. To complete the square, find the square of one half times b. Substitute negative 9 for b, rewriting the expression as the square of one half times negative 9. Simplifying the product gives the square of negative nine halves, and evaluating the square gives eighty-one fourths. Add eighty-one fourths to the binomial to complete the square so that the expression becomes y squared minus 9 y plus eighty-one fourths. Factor the resulting perfect square trinomial, writing it as the square of y minus nine halves, a binomial squared.\" data-label=\"\"><tbody><tr><td><\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167829715936\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_006b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">The coefficient of \\(y\\) is \\(-9\\).<\/td><td><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">\\(\\begin{array}{}\\\\ \\\\ \\\\ \\hfill \\text{Find}\\phantom{\\rule{0.2em}{0ex}}{\\left(\\frac{1}{2}b\\right)}^{2}.\\hfill \\\\ \\hfill {\\left(\\frac{1}{2}\u00b7\\left(-9\\right)\\right)}^{2}\\hfill \\\\ \\hfill {\\left(\\text{\u2212}\\frac{9}{2}\\right)}^{2}\\hfill \\\\ \\hfill \\frac{81}{4}\\hfill \\end{array}\\)<\/td><td><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Add \\(\\frac{81}{4}\\) to the binomial to complete the square.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_006c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Factor the perfect square trinomial, writing it as<div data-type=\"newline\"><br><\/div>a binomial squared.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836417244\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_006d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><p><span class=\"token\">\u24d2<\/span><\/p><div data-type=\"newline\"><br><\/div><table id=\"fs-id1167836621388\" class=\"unnumbered unstyled can-break\" summary=\"The expression x squared plus b x is shown above the expression n squared plus one-half n. The coefficient of n is one half. To complete the square, find the square of one half times b. Substitute the coefficient of n, b equals one half to get the square of one half times one half. Simplify the product, one fourth squared. Find the square to get the constant value one sixteenth. This number completes the square, yielding the expression n squared plus one-half n plus one sixteenth. Rewrite as a binomial square, the square of n plus one fourth.\" data-label=\"\"><tbody><tr><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836595584\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_007b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">The coefficient of \\(n\\) is \\(\\frac{1}{2}.\\)<\/td><td><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">\\(\\begin{array}{}\\\\ \\\\ \\\\ \\hfill \\text{Find}\\phantom{\\rule{0.2em}{0ex}}{\\left(\\frac{1}{2}b\\right)}^{2}.\\hfill \\\\ \\hfill {\\left(\\frac{1}{2}\u00b7\\frac{1}{2}\\right)}^{2}\\hfill \\\\ \\hfill {\\left(\\frac{1}{4}\\right)}^{2}\\hfill \\\\ \\hfill \\frac{1}{16}\\hfill \\end{array}\\)<\/td><td><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Add \\(\\frac{1}{16}\\) to the binomial to complete the square.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873803093\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_007c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Rewrite as a binomial square.<\/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_02_007d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836449723\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167829579073\"><div data-type=\"problem\" id=\"fs-id1167829751661\"><p id=\"fs-id1167833060830\">Complete the square to make a perfect square trinomial. Then write the result as a binomial squared.<\/p><p id=\"fs-id1167829826723\"><span class=\"token\">\u24d0<\/span>\\({a}^{2}-20a\\)<span class=\"token\">\u24d1<\/span>\\({m}^{2}-5m\\)<span class=\"token\">\u24d2<\/span>\\({p}^{2}+\\frac{1}{4}p\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833350873\"><p id=\"fs-id1167833239139\"><span class=\"token\">\u24d0<\/span>\\({\\left(a-10\\right)}^{2}\\)<span class=\"token\">\u24d1<\/span>\\({\\left(b-\\frac{5}{2}\\right)}^{2}\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\({\\left(p+\\frac{1}{8}\\right)}^{2}\\)<\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836552490\" class=\"try\"><div data-type=\"exercise\"><div data-type=\"problem\"><p id=\"fs-id1167836510598\">Complete the square to make a perfect square trinomial. Then write the result as a binomial squared.<\/p><p><span class=\"token\">\u24d0<\/span>\\({b}^{2}-4b\\)<span class=\"token\">\u24d1<\/span>\\({n}^{2}+13n\\)<span class=\"token\">\u24d2<\/span>\\({q}^{2}-\\frac{2}{3}q\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836334559\"><p id=\"fs-id1167833365706\"><span class=\"token\">\u24d0<\/span>\\({\\left(b-2\\right)}^{2}\\)<span class=\"token\">\u24d1<\/span>\\({\\left(n+\\frac{13}{2}\\right)}^{2}\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\({\\left(q-\\frac{1}{3}\\right)}^{2}\\)<\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167836800584\"><h3 data-type=\"title\">Solve Quadratic Equations of the Form <em data-effect=\"italics\">x<\/em><sup>2<\/sup> + <em data-effect=\"italics\">bx<\/em> + <em data-effect=\"italics\">c<\/em> = 0 by Completing the Square<\/h3><p>In solving equations, we must always do the same thing to both sides of the equation. This is true, of course, when we solve a <span data-type=\"term\" class=\"no-emphasis\">quadratic equation<\/span> by <span data-type=\"term\" class=\"no-emphasis\">completing the square<\/span> too. When we add a term to one side of the equation to make a perfect square trinomial, we must also add the same term to the other side of the equation.<\/p><p id=\"fs-id1167836509121\">For example, if we start with the equation <em data-effect=\"italics\">x<\/em><sup>2<\/sup> + 6<em data-effect=\"italics\">x<\/em> = 40, and we want to complete the square on the left, we will add 9 to both sides of the equation.<\/p><table id=\"fs-id1167836507749\" class=\"unnumbered unstyled\" summary=\"Start with the equation x squared plus 6 x equals 40. In the next step, insert spaces to use when completed the square. Write the equation as x squared plus 6 x plus space equals 40 plus space. Complete the square, inserting 9 into each space so that the equation becomes x squared plus 6 x plus 9 equals 40 plus 9. Factor the expression on the left side of the equation and simplify on the right to yield the square of x plus 3 equals 49.\" data-label=\"\"><tbody><tr><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832981250\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_008a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836518758\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_008b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\"><\/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_02_008c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Add 9 to both sides to complete the square.<\/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_02_008d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><p id=\"fs-id1167829749817\">Now the equation is in the form to solve using the <span data-type=\"term\" class=\"no-emphasis\">Square Root Property<\/span>! Completing the square is a way to transform an equation into the form we need to be able to use the Square Root Property.<\/p><div data-type=\"example\" id=\"fs-id1167829894368\" class=\"textbox textbox--examples\"><div data-type=\"title\">How to Solve a Quadratic Equation of the Form \\({x}^{2}+bx+c=0\\) by Completing the Square<\/div><div data-type=\"exercise\" id=\"fs-id1167829628058\"><div data-type=\"problem\"><p>Solve by completing the square: \\({x}^{2}+8x=48.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836542585\"><span data-type=\"media\" id=\"fs-id1167836440966\" data-alt=\"Step 1 is to isolate the variable terms on one side and the constant terms on the other. This equation, x squared plus 8 x equals 48 already has all variable terms on the left. Note that the leading coefficient is 1, so b equals 8.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_009a_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 1 is to isolate the variable terms on one side and the constant terms on the other. This equation, x squared plus 8 x equals 48 already has all variable terms on the left. Note that the leading coefficient is 1, so b equals 8.\"><\/span><span data-type=\"media\" id=\"fs-id1167833053781\" data-alt=\"In step 2, find the expression one half times b, squared, the number needed to complete the square. Add this value to both sides of the equation. Take half of 8 and square it. The square of one half times 8 equals 16, so add 16 to BOTH sides of the equation. The equation becomes x squared plus 8 x plus 16 equals 48 plus 16.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_009b_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"In step 2, find the expression one half times b, squared, the number needed to complete the square. Add this value to both sides of the equation. Take half of 8 and square it. The square of one half times 8 equals 16, so add 16 to BOTH sides of the equation. The equation becomes x squared plus 8 x plus 16 equals 48 plus 16.\"><\/span><span data-type=\"media\" id=\"fs-id1167836286060\" data-alt=\"In step 3, factor the perfect square trinomial, writing it as a binomial squared on the left and simplify by adding the terms on the right. Factor x squared plus 8 x plus 16 on the left side. Add 48+16 on the right side. The equation becomes the square of x plus 4 equals 64.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_009c_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"In step 3, factor the perfect square trinomial, writing it as a binomial squared on the left and simplify by adding the terms on the right. Factor x squared plus 8 x plus 16 on the left side. Add 48+16 on the right side. The equation becomes the square of x plus 4 equals 64.\"><\/span><span data-type=\"media\" data-alt=\"Step 4 is to use the Square Root Property. Take the square root of both sides of the equation to yield x plus 4 equals the positive or negative square root of 64.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_009d_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 4 is to use the Square Root Property. Take the square root of both sides of the equation to yield x plus 4 equals the positive or negative square root of 64.\"><\/span><span data-type=\"media\" id=\"fs-id1167836743243\" data-alt=\"In step 5, simplify the radical and then solve the two resulting equations. X plus 4 equals positive 8 or negative 8. If x plus 4 equals 8, then x equals 4. If x plus 4 equals negative 8, then x equals negative 12.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_009e_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"In step 5, simplify the radical and then solve the two resulting equations. X plus 4 equals positive 8 or negative 8. If x plus 4 equals 8, then x equals 4. If x plus 4 equals negative 8, then x equals negative 12.\"><\/span><span data-type=\"media\" id=\"fs-id1167832994384\" data-alt=\"Finally, step 6, check the solutions. Put each answer in the original equation to check. First substitute x equals 4. We need to show that 4 squared plus 8 times 4 equals 48. Simplify. The expression 4 squared plus 8 times 4 is equivalent to 16 plus 32, or 48. X equals 4 is a solution. Next substitute x equals negative 12 into the original equation, x squared plus 8 x equals 48. The square of negative 12 plus 8 times negative 12 equals 144 minus 96, or 48. X equals negative 12 is also a solution.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_009f_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Finally, step 6, check the solutions. Put each answer in the original equation to check. First substitute x equals 4. We need to show that 4 squared plus 8 times 4 equals 48. Simplify. The expression 4 squared plus 8 times 4 is equivalent to 16 plus 32, or 48. X equals 4 is a solution. Next substitute x equals negative 12 into the original equation, x squared plus 8 x equals 48. The square of negative 12 plus 8 times negative 12 equals 144 minus 96, or 48. X equals negative 12 is also a solution.\"><\/span><\/div><\/div><\/div><div data-type=\"note\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836625042\"><div data-type=\"problem\" id=\"fs-id1167829586206\"><p>Solve by completing the square: \\({x}^{2}+4x=5.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836625698\"><p id=\"fs-id1167836570280\">\\(x=-5,x=-1\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167829755835\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167833058751\"><div data-type=\"problem\" id=\"fs-id1167836484698\"><p id=\"fs-id1167829740784\">Solve by completing the square: \\({y}^{2}-10y=-9.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833139723\"><p>\\(y=1,y=9\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1167826211837\">The steps to solve a quadratic equation by completing the square are listed here.<\/p><div data-type=\"note\" id=\"fs-id1167833381337\" class=\"howto\"><div data-type=\"title\">Solve a quadratic equation of the form \\({x}^{2}+bx+c=0\\) by completing the square.<\/div><ol id=\"fs-id1167829791692\" type=\"1\" class=\"stepwise\"><li>Isolate the variable terms on one side and the constant terms on the other.<\/li><li>Find \\({\\left(\\frac{1}{2}\\phantom{\\rule{0.2em}{0ex}}\u00b7\\phantom{\\rule{0.2em}{0ex}}b\\right)}^{2},\\) the number needed to complete the square. Add it to both sides of the equation.<\/li><li>Factor the perfect square trinomial, writing it as a binomial squared on the left and simplify by adding the terms on the right<\/li><li>Use the Square Root Property.<\/li><li>Simplify the radical and then solve the two resulting equations.<\/li><li>Check the solutions.<\/li><\/ol><\/div><p id=\"fs-id1167829689257\">When we solve an equation by completing the square, the answers will not always be integers.<\/p><div data-type=\"example\" class=\"textbox textbox--examples\"><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167836538119\"><p>Solve by completing the square: \\({x}^{2}+4x=-21.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836417144\"><table id=\"fs-id1167829711983\" class=\"unnumbered unstyled\" summary=\"In the equation x squared plus 4 x equals negative 21, the variable terms are on the left side of the equation and the constant term is on the right. To complete the square, take half of the coefficient of x, 4, and square it. The square of the product one half times 4 is 4, so add 4 to both sides. The equation becomes x squared plus 4 x plus 4 equals negative 21 plus 4. Factor the perfect square trinomial on the left and add the values on the right to yield the square of the sum x plus 2 equals negative 17. Use the Square Root Property. The equation becomes x plus 2 equals the positive or negative square root of negative 17. Simplify using complex numbers. X plus 2 equals positive or negative square root 17 times I. Subtract 2 from each side of the equation. X equals negative 2 plus or minus square root 17 I. Rewrite to show two solutions. x equals negative 2 plus square root 17 I and x equals negative 2 minus square root 17 I. We leave the check to you.\" data-label=\"\"><tbody><tr><td><\/td><td><\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167833050840\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_010b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">The variable terms are on the left side.<div data-type=\"newline\"><br><\/div> Take half of 4 and square it.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836408948\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_010c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">\\({\\left(\\frac{1}{2}\\left(4\\right)\\right)}^{2}=4\\)<\/td><td><\/td><\/tr><tr><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Add 4 to both sides.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_010d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Factor the perfect square trinomial,<div data-type=\"newline\"><br><\/div>writing it as a binomial squared.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836520552\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_010e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Use the Square Root Property.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_010f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Simplify using complex numbers.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836313583\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_010g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Subtract 2 from each side.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167833030947\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_010h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Rewrite to show two solutions.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836692571\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_010i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">We leave the check to you.<\/td><td><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167833369584\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167833379532\"><div data-type=\"problem\"><p>Solve by completing the square: \\({y}^{2}-10y=-35.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836424149\"><p id=\"fs-id1167833142274\">\\(y=5+\\sqrt{15}i,\\phantom{\\rule{0.2em}{0ex}}\\text{}\\phantom{\\rule{0.2em}{0ex}}y=5-\\sqrt{15}i\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836556313\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167824754841\"><div data-type=\"problem\" id=\"fs-id1167829843220\"><p id=\"fs-id1167829691124\">Solve by completing the square: \\({z}^{2}+8z=-19.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829718166\"><p id=\"fs-id1167826188105\">\\(z=-4+\\sqrt{3}i,\\phantom{\\rule{0.2em}{0ex}}\\text{z}=-4-\\sqrt{3}i\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1167829595092\">In the previous example, our solutions were complex numbers. In the next example, the solutions will be irrational numbers.<\/p><div data-type=\"example\" id=\"fs-id1167829687996\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167829692780\"><div data-type=\"problem\"><p id=\"fs-id1167829907184\">Solve by completing the square: \\({y}^{2}-18y=-6.\\)<\/p><\/div><div data-type=\"solution\"><table id=\"fs-id1167836548249\" class=\"unnumbered unstyled can-break\" summary=\"In the equation y squared minus 18 y equals negative 6, the variable terms are on the left side of the equation and the constant term is on the right. To complete the square, take half of the coefficient of y, \u221218, and square it. The square of the product one half times negative 18 is 81, so add 81 to both sides. The equation becomes y squared minus 18 y plus 81 equals negative 6 plus 81. Factor the perfect square trinomial on the left and add the values on the right to yield the square of the difference y minus 9 equals 75. Use the Square Root Property. The equation becomes y minus 9 equals the positive or negative square root of 75. Simplify the radical. Y minus 9 equals positive or negative 5 square root 3. Add 9 to each side of the equation. y equals 9 plus or minus 5 square root 3. Check. Substitute 9 plus 5 square root 3 into the original equation, y squared minus 18 y equals negative 6. The expression on the left becomes the square of 9 plus 5 square root 3 minus 18 times the sum 9 plus 5 square root 3. We need to show that this expression equals negative 6. Expanding the square and multiplying yields 81 plus 90 square root 3 plus 75 minus 162 minus 90 square root 3, which equals negative 6. Next substitute 9 minus 5 square root 3 into the original equation, y squared minus 18 y equals negative 6. The expression on the left becomes the square of 9 minus 5 square root 3 minus 18 times the sum 9 minus 5 square root 3. We need to show that this expression equals negative 6. Expanding the square and multiplying yields 81 minus 90 square root 3 plus 75 minus 162 plus 90 square root 3, which equals negative 6.\" data-label=\"\"><tbody><tr><td><\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_011c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">The variable terms are on the left side.<div data-type=\"newline\"><br><\/div> Take half of \\(-18\\) and square it.<\/td><td><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">\\({\\left(\\frac{1}{2}\\left(-18\\right)\\right)}^{2}=81\\)<\/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_02_011d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Add 81 to both sides.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836614906\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_011e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Factor the perfect square trinomial,<div data-type=\"newline\"><br><\/div>writing it as a binomial squared.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167829753570\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_011f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Use the Square Root Property.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836424051\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_011g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Simplify the radical.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_011h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Solve for \\(y\\).<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836407199\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_011i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Check.<div data-type=\"newline\"><br><\/div> <span data-type=\"media\" id=\"fs-id1167833059590\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_011b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span> <\/td><\/tr><\/tbody><\/table><p id=\"fs-id1167836378576\">Another way to check this would be to use a calculator. Evaluate \\({y}^{2}-18y\\)for both of the solutions. The answer should be \\(-6.\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836731428\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167829832718\"><div data-type=\"problem\" id=\"fs-id1167836526026\"><p id=\"fs-id1167833379137\">Solve by completing the square: \\({x}^{2}-16x=-16.\\)<\/p><\/div><div data-type=\"solution\"><p id=\"fs-id1167836513723\">\\(x=8+4\\sqrt{3},\\phantom{\\rule{0.2em}{0ex}}\\text{}\\phantom{\\rule{0.2em}{0ex}}x=8-4\\sqrt{3}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836527820\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167824720413\"><div data-type=\"problem\" id=\"fs-id1167833369838\"><p id=\"fs-id1167836392691\">Solve by completing the square: \\({y}^{2}+8y=11.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836662923\"><p id=\"fs-id1167833137978\">\\(y=-4+3\\sqrt{3},\\phantom{\\rule{0.2em}{0ex}}\\text{y}=-4-3\\sqrt{3}\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1167836627782\">We will start the next example by isolating the variable terms on the left side of the equation.<\/p><div data-type=\"example\" id=\"fs-id1167836507963\" class=\"textbox textbox--examples\"><div data-type=\"exercise\"><div data-type=\"problem\"><p id=\"fs-id1167824767011\">Solve by completing the square: \\({x}^{2}+10x+4=15.\\)<\/p><\/div><div data-type=\"solution\"><table id=\"fs-id1167836549726\" class=\"unnumbered unstyled can-break\" summary=\"Rewrite the original equation, x squared plus 10 x plus 4 equals 15 to isolate the variables on the left side. Subtract 4 from each side of the equation. X square plus 10 x equals 11. To complete the square, take half of the coefficient of x, 10, and square it. The square of the product one half times 10 is 25, so add 25 to both sides. The equation becomes x squared plus 10 x plus 25 equals 11 plus 25. Factor the perfect square trinomial on the left and add the values on the right to yield the square of the sum x plus 5 equals 36. Use the Square Root Property. The equation becomes x plus 5 equals the positive or negative square root of 36. Simplify the radical. X plus 5 equals positive or negative 6. Subtract 5 from each side of the equation. X equals negative 5 plus or minus 6. Rewrite to show 2 solutions, x equals negative 5 plus 6, or 1 and x equals negative 5 minus 6 or negative 11. Check. Substitute 1 into the original equation, x squared plus 10 x plus 4 equals 15. The expression on the left becomes 1 squared plus 10 times 1 plus 4. We need to show that this expression equals 15. Simplifying gives 1 plus 10 plus 4, or 15. Next substitute negative 11 into the original equation x squared plus 10 x plus 4 equals 15. The expression on the left becomes negative 11 squared plus 10 times negative 11 plus 4. We need to show that this expression equals 15. Simplifying gives 121 minus 110 plus 4, or 15.\" 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_02_012c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Isolate the variable terms on the left side.<div data-type=\"newline\"><br><\/div>Subtract 4 to get the constant terms on the right side.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836356824\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_012d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Take half of 10 and square it.<\/td><td><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">\\({\\left(\\frac{1}{2}\\left(10\\right)\\right)}^{2}=25\\)<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829833401\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_012e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Add 25 to both sides.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829878589\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_012f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Factor the perfect square trinomial, writing it as<div data-type=\"newline\"><br><\/div>a binomial squared.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833345938\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_012g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Use the Square Root Property.<\/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_02_012h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Simplify the radical.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833048657\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_012i_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>.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836646306\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_012j_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Rewrite to show two solutions.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167824590480\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_012k_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Solve the equations.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833055123\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_012l_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-id1167829627533\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_012b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span> <\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167833086970\" class=\"try\"><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167829713249\"><p id=\"fs-id1167836697056\">Solve by completing the square: \\({a}^{2}+4a+9=30.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836513135\"><p>\\(a=-7,a=3\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167833055847\"><div data-type=\"problem\" id=\"fs-id1167836554813\"><p id=\"fs-id1167836620313\">Solve by completing the square: \\({b}^{2}+8b-4=16.\\)<\/p><\/div><div data-type=\"solution\"><p id=\"fs-id1167833408069\">\\(b=-10,b=2\\)<\/p><\/div><\/div><\/div><p>To solve the next equation, we must first collect all the variable terms on the left side of the equation. Then we proceed as we did in the previous examples.<\/p><div data-type=\"example\" id=\"fs-id1167836323334\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167836537598\"><div data-type=\"problem\"><p id=\"fs-id1167836691587\">Solve by completing the square: \\({n}^{2}=3n+11.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167826025325\"><table id=\"fs-id1167836792924\" class=\"unnumbered unstyled can-break\" summary=\"Rewrite the equation n squared equals 3 n plus 11 to isolate the variable terms on the left side of the equation. Subtract 3 n from both sides of the equation. N squared minus 3 n equals 11. To complete the square, take half of the coefficient of n, negative 3, and square it. The square of the product one half times negative 3 is 9 divided by 4, so add nine fourths to both sides. The equation becomes n squared minus 3 n plus 9 fourths equals 11 plus 9 fourths. Factor the perfect square trinomial on the left, writing it as a binomial squared. On the right, express 11 as a fraction with denominator 4. The square of the difference n minus 3 halves equals 44 fourths plus 9 fourths. Add the fractions on the right side. The square of the difference n minus 3 halves equals 53 fourths Use the Square Root Property. The equation becomes n minus 3 halves equals the positive or negative square root of 53 fourths. Simplify the radical. N minus 3 halves equals positive or negative square root 35 divided by 2. Add 3 halves to both sides of the equation to solve for n. n equals 3 halves plus or minus square root 53 divided by 2. Rewrite to show two solutions. n equals three halves plus square root 53 divided by 2 and n equals three halves minus square root 53 divided by 2. We leave the check to you.\" data-label=\"\"><tbody><tr><td><\/td><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_02_013b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Subtract \\(3n\\) to get the variable terms on the left side.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833056846\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_013c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Take half of \\(-3\\) and square it.<\/td><td><\/td><\/tr><tr><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">\\({\\left(\\frac{1}{2}\\left(-3\\right)\\right)}^{2}=\\frac{9}{4}\\)<\/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_02_013d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Add \\(\\frac{9}{4}\\) to both sides.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836650053\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_013e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Factor the perfect square trinomial, writing it as<div data-type=\"newline\"><br><\/div>a binomial squared.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167824741373\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_013f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Add the fractions on the right side.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829598071\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_013g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Use the Square Root Property.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836619785\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_013h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Simplify the radical.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167826171765\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_013i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Solve for <em data-effect=\"italics\">n<\/em>.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836688616\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_013j_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Rewrite to show two solutions.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833086747\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_013k_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Check:<div data-type=\"newline\"><br><\/div> We leave the check for you!<\/td><td><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836407661\" class=\"try\"><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167836392033\"><p>Solve by completing the square: \\({p}^{2}=5p+9.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836753600\"><p id=\"fs-id1167829789466\">\\(p=\\frac{5}{2}+\\frac{\\sqrt{61}}{2},\\phantom{\\rule{0.2em}{0ex}}\\text{}\\phantom{\\rule{0.2em}{0ex}}p=\\frac{5}{2}-\\frac{\\sqrt{61}}{2}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836410437\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167833059138\"><div data-type=\"problem\" id=\"fs-id1167836509823\"><p id=\"fs-id1167836399192\">Solve by completing the square: \\({q}^{2}=7q-3.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836292523\"><p id=\"fs-id1167836688308\">\\(q=\\frac{7}{2}+\\frac{\\sqrt{37}}{2},\\phantom{\\rule{0.2em}{0ex}}\\text{q}=\\frac{7}{2}-\\frac{\\sqrt{37}}{2}\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1167829750235\">Notice that the left side of the next equation is in factored form. But the right side is not zero. So, we cannot use the <span data-type=\"term\" class=\"no-emphasis\">Zero Product Property<\/span> since it says \u201cIf \\(a\\phantom{\\rule{0.2em}{0ex}}\u00b7\\phantom{\\rule{0.2em}{0ex}}b=0,\\) then <em data-effect=\"italics\">a<\/em> = 0 or <em data-effect=\"italics\">b<\/em> = 0.\u201d Instead, we multiply the factors and then put the equation into standard form to solve by completing the square.<\/p><div data-type=\"example\" id=\"fs-id1167836357146\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167836492833\"><div data-type=\"problem\" id=\"fs-id1167836544202\"><p>Solve by completing the square: \\(\\left(x-3\\right)\\left(x+5\\right)=9.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836731606\"><table id=\"fs-id1167836407133\" class=\"unnumbered unstyled can-break\" summary=\"Write the equation. The product of x minus 3 and x plus 5 equals 9. Multiply the binomials on the left. X squared plus 2 x minus 15 equals 9. Add 15 to isolate the constant terms on the right. X squared plus 2 x equals 24. To complete the square, take half of the coefficient of x, 2, and square it. The square of the product one half times 2 is 1, so add 1 to both sides. X squared plus 2 x plus 1 equals 24 plus 1. Factor the perfect square trinomial to write it as a binomial squared. The square of x plus 1 equals 25. Use the Square Root Property. X plus 1 equals the positive or negative square root of 25. Solve for x. x equals negative 1 plus or minus 5. Rewrite to show 2 solutions. x equals negative 1 plus 5, or 4. X equals negative 1 minus 5, or negative 6. We leave the check of solutions for you!\" data-label=\"\"><tbody><tr><td><\/td><td><\/td><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167824658654\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_014b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">We multiply the binomials on the left.<\/td><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836667027\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_014c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Add 15 to isolate the constant terms on the right.<\/td><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836628619\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_014d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Take half of 2 and square it.<\/td><td><\/td><td><\/td><\/tr><tr><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">\\({\\left(\\frac{1}{2}\u00b7\\left(2\\right)\\right)}^{2}=1\\)<\/td><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167824754906\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_014e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Add 1 to both sides.<\/td><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829589062\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_014f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Factor the perfect square trinomial, writing it as<div data-type=\"newline\"><br><\/div>a binomial squared.<\/td><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836326096\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_014g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Use the Square Root Property.<\/td><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829719081\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_014h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Solve for <em data-effect=\"italics\">x<\/em>.<\/td><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836294554\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_014i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Rewrite to show two solutions.<\/td><td colspan=\"2\" data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167833025582\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_014j_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td colspan=\"2\" data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_014k_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Check:<div data-type=\"newline\"><br><\/div> We leave the check for you!<\/td><td><\/td><td><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167833245437\" class=\"try\"><div data-type=\"exercise\"><div data-type=\"problem\"><p id=\"fs-id1167836544930\">Solve by completing the square: \\(\\left(c-2\\right)\\left(c+8\\right)=11.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836621350\"><p id=\"fs-id1167836407485\">\\(c=-9,c=3\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167829715782\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167830077335\"><div data-type=\"problem\"><p id=\"fs-id1167832951207\">Solve by completing the square: \\(\\left(d-7\\right)\\left(d+3\\right)=56.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829693560\"><p>\\(d=11,d=-7\\)<\/p><\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167836342004\"><h3 data-type=\"title\">Solve Quadratic Equations 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 by Completing the Square<\/h3><p>The process of <span data-type=\"term\" class=\"no-emphasis\">completing the square<\/span> works best when the coefficient of <em data-effect=\"italics\">x<\/em><sup>2<\/sup> is 1, so the left side of the equation is of the form <em data-effect=\"italics\">x<\/em><sup>2<\/sup> + <em data-effect=\"italics\">bx<\/em> + <em data-effect=\"italics\">c<\/em>. If the <em data-effect=\"italics\">x<\/em><sup>2<\/sup> term has a coefficient other than 1, we take some preliminary steps to make the coefficient equal to 1.<\/p><p>Sometimes the coefficient can be factored from all three terms of the trinomial. This will be our strategy in the next example.<\/p><div data-type=\"example\" id=\"fs-id1167836667059\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167829714809\"><div data-type=\"problem\" id=\"fs-id1167836509636\"><p id=\"fs-id1167829716705\">Solve by completing the square: \\(3{x}^{2}-12x-15=0.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836484371\"><p id=\"fs-id1167829594643\">To complete the square, we need the coefficient of \\({x}^{2}\\) to be one. If we factor out the coefficient of \\({x}^{2}\\) as a common factor, we can continue with solving the equation by completing the square.<\/p><table id=\"fs-id1167836508427\" class=\"unnumbered unstyled can-break\" summary=\"Write the equation 3 x squared minus 12 x minus 15 equals 0. Factor out the greatest common factor. 3 times the expression x squared minus 4 x minus 5 equals 0. Divide both sides by 3 to isolate the trinomial. 3 times the expression x squared minus 4 x minus 5 divided by 3 equals 0 divided by 3. Simplify. x squared minus 4 x minus 5 equals 0. Add 5 to get the constant terms on the right side. X squared minus 4 x equals 5. To complete the square, take half of the coefficient of x, 4, and square it. The square of the product one half times 4 is 4, so add 4 to both sides. X squared minus 4 x plus 4 equals 5 plus 4. Factor the perfect square trinomial to write it as a binomial squared. The square of x minus 2 equals 9. Use the Square Root Property. X minus 2 equals the positive or negative square root of 9. Solve for x. x equals 2 plus or minus 3. Rewrite to show 2 solutions. x equals 2 plus 3, or 5. X equals 2 minus 3, or negative 1. Check the solutions by substituting each value into the original equation. Substitute x equals 5 into the equation 3 x squared minus 12 x minus 15 equals 0. The left side becomes 3 times the square of 5 minus 12 times 5 minus 15. We need to show that this equals 0. Simplifying yields 75 minus 60 minus 15 which is equal to 0. 5 is a solution. Substitute x equals negative 1 into the equation 3 x squared minus 12 x minus 15 equals 0. The left side becomes 3 times the square of negative 1 minus 12 times negative 1 minus 15. We need to show that this equals 0. Simplifying yields 3 plus 12 minus 15 which is equal to 0. Negative 1 is a solution.\" data-label=\"\"><tbody><tr><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167824733261\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_015c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Factor out the greatest common factor.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836497733\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_015d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Divide both sides by 3 to isolate the trinomial<div data-type=\"newline\"><br><\/div>with coefficient 1.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832951049\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_015e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833326573\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_015f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Add 5 to get the constant terms on the right side.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836626094\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_015g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Take half of 4 and square it.<\/td><td><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">\\({\\left(\\frac{1}{2}\\left(-4\\right)\\right)}^{2}=4\\)<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829692301\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_015h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Add 4 to both sides.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836493824\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_015i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Factor the perfect square trinomial, writing it<div data-type=\"newline\"><br><\/div>as a binomial squared.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833380251\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_015j_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Use the Square Root Property.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836732071\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_015k_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>.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836539657\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_015l_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Rewrite to show two solutions.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836524160\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_015m_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836300550\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_015n_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-id1167836550626\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_015b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><td><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836596811\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167833024381\"><div data-type=\"problem\"><p id=\"fs-id1167836787783\">Solve by completing the square: \\(2{m}^{2}+16m+14=0.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833407704\"><p id=\"fs-id1167836492528\">\\(m=-7,m=-1\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167830122896\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836282523\"><div data-type=\"problem\" id=\"fs-id1167833019986\"><p id=\"fs-id1167829833349\">Solve by completing the square: \\(4{n}^{2}-24n-56=8.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836450369\"><p id=\"fs-id1167836305741\">\\(n=-2,n=8\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1167836329650\">To complete the square, the coefficient of the <em data-effect=\"italics\">x<\/em><sup>2<\/sup> must be 1. When the <span data-type=\"term\" class=\"no-emphasis\">leading coefficient<\/span> is not a factor of all the terms, we will divide both sides of the equation by the leading coefficient! This will give us a fraction for the second coefficient. We have already seen how to complete the square with fractions in this section.<\/p><div data-type=\"example\" id=\"fs-id1167836406740\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167826172184\"><div data-type=\"problem\" id=\"fs-id1167829747125\"><p>Solve by completing the square: \\(2{x}^{2}-3x=20.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833356483\"><p id=\"fs-id1167836356365\">To complete the square we need the coefficient of \\({x}^{2}\\) to be one. We will divide both sides of the equation by the coefficient of <em data-effect=\"italics\">x<\/em><sup>2<\/sup>. Then we can continue with solving the equation by completing the square.<\/p><table id=\"fs-id1167836615944\" class=\"unnumbered unstyled\" summary=\"The equation 2 x squared minus 3 x equals 20 has the x terms isolated on the left side of the equation. Divide both sides by 2 to get the leading coefficient of x squared to be 1. The quotient 2 x squared minus 3 x divided by 2 equals 20 divided by 2. Simplify. x squared minus 3 halves x equals 10. To complete the square, take half of the coefficient of x, negative 3 halves, and square it. The square of the product one half times negative 3 halves is 9 sixteenths, so add 9 sixteenths to both sides. X squared minus 3 halves x plus 9 sixteenths equals 10 plus 9 sixteenths. Factor the perfect square trinomial to write it as a binomial squared and express the terms on the right as fractions with a common denominator. The square of x minus 3 fourths equals 160 divided by 16 plus 9 divided by 16. Add the fractions on the right. The square of x minus 3 fourths equals 169 divided by 16. Use the Square Root Property. X minus 3 fourths equals the positive or negative square root of 169 sixteenths. Simplify the radical. X minus 3 fourths equals positive or negative 13 fourths. Solve for x. x equals 3 fourths plus or minus 13 fourths. Rewrite to show 2 solutions. x equals 3 plus 13 fourths, or 4. x equals 3 minus 13 fourths, or negative 5 halves. We leave the check of the solutions for you!\" data-label=\"\"><tbody><tr><td><\/td><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_02_016b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Divide both sides by 2 to get the<div data-type=\"newline\"><br><\/div>coefficient of \\({x}^{2}\\) to be 1.<\/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_02_016c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836544118\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_016d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Take half of \\(-\\frac{3}{2}\\) and square it.<\/td><td><\/td><\/tr><tr><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">\\({\\left(\\frac{1}{2}\\left(-\\frac{3}{2}\\right)\\right)}^{2}=\\frac{9}{16}\\)<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836477509\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_016e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Add \\(\\frac{9}{16}\\) to both sides.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829712714\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_016f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Factor the perfect square trinomial,<div data-type=\"newline\"><br><\/div>writing it as a binomial squared.<\/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_02_016g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Add the fractions on the right side.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836362768\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_016h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Use the Square Root Property.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836433674\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_016i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Simplify the radical.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836543341\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_016j_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Solve for <em data-effect=\"italics\">x<\/em>.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833186420\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_016k_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Rewrite to show two solutions.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833270316\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_016l_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833021813\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_016m_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Check:<div data-type=\"newline\"><br><\/div> We leave the check for you!<\/td><td><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836571353\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836410349\"><div data-type=\"problem\" id=\"fs-id1167836387744\"><p id=\"fs-id1167836481605\">Solve by completing the square: \\(3{r}^{2}-2r=21.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836514219\"><p id=\"fs-id1167833018598\">\\(r=-\\frac{7}{3},r=3\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167829714482\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167829807666\"><div data-type=\"problem\" id=\"fs-id1167836533308\"><p id=\"fs-id1167829930812\">Solve by completing the square: \\(4{t}^{2}+2t=20.\\)<\/p><\/div><div data-type=\"solution\"><p id=\"fs-id1167836433690\">\\(t=-\\frac{5}{2},t=2\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1167836299186\">Now that we have seen that the coefficient of <em data-effect=\"italics\">x<\/em><sup>2<\/sup> must be 1 for us to complete the square, we update our procedure for solving a <span data-type=\"term\" class=\"no-emphasis\">quadratic equation<\/span> by completing the square to include equations 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.<\/p><div data-type=\"note\" id=\"fs-id1167833051633\" class=\"howto\"><div data-type=\"title\">Solve a quadratic equation of the form \\(a{x}^{2}+bx+c=0\\) by completing the square.<\/div><ol id=\"fs-id1167829693205\" type=\"1\" class=\"stepwise\"><li>Divide by \\(a\\) to make the coefficient of <em data-effect=\"italics\">x<\/em><sup>2<\/sup> term 1.<\/li><li>Isolate the variable terms on one side and the constant terms on the other.<\/li><li>Find \\({\\left(\\frac{1}{2}\\phantom{\\rule{0.2em}{0ex}}\u00b7\\phantom{\\rule{0.2em}{0ex}}b\\right)}^{2},\\) the number needed to complete the square. Add it to both sides of the equation.<\/li><li>Factor the perfect square trinomial, writing it as a binomial squared on the left and simplify by adding the terms on the right<\/li><li>Use the Square Root Property.<\/li><li>Simplify the radical and then solve the two resulting equations.<\/li><li>Check the solutions.<\/li><\/ol><\/div><div data-type=\"example\" id=\"fs-id1167836299458\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167836501815\"><div data-type=\"problem\" id=\"fs-id1167836627532\"><p id=\"fs-id1167829746209\">Solve by completing the square: \\(3{x}^{2}+2x=4.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833022209\"><p id=\"fs-id1167836536066\">Again, our first step will be to make the coefficient of <em data-effect=\"italics\">x<\/em><sup>2<\/sup> one. By dividing both sides of the equation by the coefficient of <em data-effect=\"italics\">x<\/em><sup>2<\/sup>, we can then continue with solving the equation by completing the square.<\/p><table id=\"fs-id1167829909536\" class=\"unnumbered unstyled can-break\" summary=\"The equation 3 x squared plus 2 x equals 4 has the x terms isolated on the left side of the equation. Divide both sides by 3 to get the leading coefficient of x squared to be 1. The quotient 3 x squared plus 2 x divided by 3 equals 4 divided by 3. Simplify. x squared plus 2 thirds x equals 4 thirds. To complete the square, take half of the coefficient of x, 2 thirds, and square it. The square of the product one half times 2 thirds is 1 ninth, so add 1 ninth to both sides. X squared plus 2 thirds x plus 1 ninth equals 4 thirds plus 1 ninth. Factor the perfect square trinomial to write it as a binomial squared and express the terms on the right as fractions with a common denominator. The square of x plus 1 thirds equals 12 divided by 9 plus 1 divided by 9. Add the fractions on the right and use the Square Root Property. X plus 1 third equals the positive or negative square root of 13 ninths. sixteenths. Simplify the radical. X plus 1 third equals positive or negative square root 13 divided by 3. Solve for x. x equals negative 1 thirds plus or minus square root 13 thirds. Rewrite to show 2 solutions. x equals negative 1 thirds plus square root 13 thirds, x equals negative 1 thirds minus square root 13 thirds. We leave the check of the solutions for you!\" data-label=\"\"><tbody><tr><td><\/td><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829785112\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_017b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Divide both sides by 3 to make the<div data-type=\"newline\"><br><\/div>coefficient of \\({x}^{2}\\) equal 1.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836687333\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_017c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Simplify.<\/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_02_017d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Take half of \\(\\frac{2}{3}\\) and square it.<\/td><td><\/td><\/tr><tr><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">\\({\\left(\\frac{1}{2}\u00b7\\frac{2}{3}\\right)}^{2}=\\frac{1}{9}\\)<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836613649\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_017e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Add \\(\\frac{1}{9}\\) to both sides.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833310353\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_017f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Factor the perfect square trinomial, writing it as<div data-type=\"newline\"><br><\/div>a binomial squared.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836375385\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_017g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Use the Square Root Property.<\/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_02_017h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Simplify the radical.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836408865\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_017i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Solve for <em data-effect=\"italics\">x<\/em> .<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836728833\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_017j_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Rewrite to show two solutions.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833059280\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_017k_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Check:<div data-type=\"newline\"><br><\/div> We leave the check for you!<\/td><td><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167829715966\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167833309872\"><div data-type=\"problem\" id=\"fs-id1167833309874\"><p id=\"fs-id1167829712104\">Solve by completing the square: \\(4{x}^{2}+3x=2.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167826204778\"><p id=\"fs-id1167826204780\">\\(x=-\\frac{3}{8}+\\frac{\\sqrt{41}}{8},\\phantom{\\rule{0.2em}{0ex}}\\text{}\\phantom{\\rule{0.2em}{0ex}}x=-\\frac{3}{8}-\\frac{\\sqrt{41}}{8}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836287199\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836287203\"><div data-type=\"problem\" id=\"fs-id1167829851940\"><p id=\"fs-id1167829851942\">Solve by completing the square: \\(3{y}^{2}-10y=-5.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829753405\"><p id=\"fs-id1167829753408\">\\(y=\\frac{5}{3}+\\frac{\\sqrt{10}}{3},\\phantom{\\rule{0.2em}{0ex}}\\text{}\\phantom{\\rule{0.2em}{0ex}}y=\\frac{5}{3}-\\frac{\\sqrt{10}}{3}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167829861822\" class=\"media-2\"><p>Access these online resources for additional instruction and practice with completing the square.<\/p><ul id=\"fs-id1167829744240\" data-display=\"block\"><li><a href=\"https:\/\/openstax.org\/l\/37CompTheSq1\">Completing Perfect Square Trinomials<\/a><\/li><li><a href=\"https:\/\/openstax.org\/l\/37CompTheSq2\">Completing the Square 1<\/a><\/li><li><a href=\"https:\/\/openstax.org\/l\/37CompTheSq3\">Completing the Square to Solve Quadratic Equations<\/a><\/li><li><a href=\"https:\/\/openstax.org\/l\/37CompTheSq4\">Completing the Square to Solve Quadratic Equations: More Examples<\/a><\/li><li><a href=\"https:\/\/openstax.org\/l\/37CompTheSq5\">Completing the Square 4<\/a><\/li><\/ul><\/div><\/div><div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167836544017\"><h3 data-type=\"title\">Key Concepts<\/h3><ul id=\"fs-id1167833339296\" data-bullet-style=\"bullet\"><li>Binomial Squares Pattern<div data-type=\"newline\"><br><\/div> If <em data-effect=\"italics\">a<\/em> and <em data-effect=\"italics\">b<\/em> are real numbers,<div data-type=\"newline\"><br><\/div> <span data-type=\"media\" data-alt=\"Quantity a plus b squared equals a squared plus 2 a b plus b2 where the binomial squared equals the first term squared plus 2 times the product of terms plus the second term squared. Quantity a minus b squared equals a squared minus 2 a b plus b2 where the binomial squared equals the first term squared minus 2 times the product of terms plus the second term squared.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_018_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Quantity a plus b squared equals a squared plus 2 a b plus b2 where the binomial squared equals the first term squared plus 2 times the product of terms plus the second term squared. Quantity a minus b squared equals a squared minus 2 a b plus b2 where the binomial squared equals the first term squared minus 2 times the product of terms plus the second term squared.\"><\/span><\/li><li>How to Complete a Square <ol id=\"fs-id1167829783787\" type=\"1\" class=\"stepwise\"><li>Identify <em data-effect=\"italics\">b<\/em>, the coefficient of <em data-effect=\"italics\">x<\/em>.<\/li><li>Find \\({\\left(\\frac{1}{2}b\\right)}^{2},\\) the number to complete the square.<\/li><li>Add the \\({\\left(\\frac{1}{2}b\\right)}^{2}\\) to <em data-effect=\"italics\">x<\/em><sup>2<\/sup> + <em data-effect=\"italics\">bx<\/em><\/li><li>Rewrite the trinomial as a binomial square<\/li><\/ol><\/li><li>How to solve a quadratic 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 by completing the square. <ol id=\"fs-id1167836529877\" type=\"1\" class=\"stepwise\"><li>Divide by <em data-effect=\"italics\">a<\/em> to make the coefficient of <em data-effect=\"italics\">x<\/em><sup>2<\/sup> term 1.<\/li><li>Isolate the variable terms on one side and the constant terms on the other.<\/li><li>Find \\({\\left(\\frac{1}{2}\\phantom{\\rule{0.2em}{0ex}}\u00b7\\phantom{\\rule{0.2em}{0ex}}b\\right)}^{2},\\) the number needed to complete the square. Add it to both sides of the equation.<\/li><li>Factor the perfect square trinomial, writing it as a binomial squared on the left and simplify by adding the terms on the right.<\/li><li>Use the Square Root Property.<\/li><li>Simplify the radical and then solve the two resulting equations.<\/li><li>Check the solutions.<\/li><\/ol><\/li><\/ul><\/div><div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167836334125\"><div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1167836518584\"><h4 data-type=\"title\">Practice Makes Perfect<\/h4><p><strong data-effect=\"bold\">Complete the Square of a Binomial Expression<\/strong><\/p><p id=\"fs-id1167836507896\">In the following exercises, complete the square to make a perfect square trinomial. Then write the result as a binomial squared.<\/p><div data-type=\"exercise\" id=\"fs-id1167836390279\"><div data-type=\"problem\" id=\"fs-id1167836390281\"><p id=\"fs-id1167836390284\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\({m}^{2}-24m\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\({x}^{2}-11x\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\({p}^{2}-\\frac{1}{3}p\\)<\/div><div data-type=\"solution\" id=\"fs-id1167836399357\"><p id=\"fs-id1167836408168\"><span class=\"token\">\u24d0<\/span>\\({\\left(m-12\\right)}^{2}\\)<span class=\"token\">\u24d1<\/span>\\({\\left(x-\\frac{11}{2}\\right)}^{2}\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\({\\left(p-\\frac{1}{6}\\right)}^{2}\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829746316\"><div data-type=\"problem\" id=\"fs-id1167836613711\"><p id=\"fs-id1167836613713\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\({n}^{2}-16n\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\({y}^{2}+15y\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\({q}^{2}+\\frac{3}{4}q\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836323142\"><div data-type=\"problem\" id=\"fs-id1167833245761\"><p id=\"fs-id1167833245763\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\({p}^{2}-22p\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\({y}^{2}+5y\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\({m}^{2}+\\frac{2}{5}m\\)<\/div><div data-type=\"solution\" id=\"fs-id1167836549924\"><p id=\"fs-id1167836549926\"><span class=\"token\">\u24d0<\/span>\\({\\left(p-11\\right)}^{2}\\)<span class=\"token\">\u24d1<\/span>\\({\\left(y+\\frac{5}{2}\\right)}^{2}\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\({\\left(m+\\frac{1}{5}\\right)}^{2}\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167830077402\"><div data-type=\"problem\" id=\"fs-id1167830077404\"><p id=\"fs-id1167830077406\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\({q}^{2}-6q\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\({x}^{2}-7x\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\({n}^{2}-\\frac{2}{3}n\\)<\/div><\/div><p id=\"fs-id1167833061042\"><strong data-effect=\"bold\">Solve Quadratic Equations of the form <em data-effect=\"italics\">x<\/em><sup>2<\/sup> + <em data-effect=\"italics\">bx<\/em> + <em data-effect=\"italics\">c<\/em> = 0 by Completing the Square<\/strong><\/p><p id=\"fs-id1167836611918\">In the following exercises, solve by completing the square.<\/p><div data-type=\"exercise\" id=\"fs-id1167825702838\"><div data-type=\"problem\" id=\"fs-id1167825702840\"><p id=\"fs-id1167825702842\">\\({u}^{2}+2u=3\\)<\/p><\/div><div data-type=\"solution\"><p>\\(u=-3,u=1\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836349336\"><div data-type=\"problem\" id=\"fs-id1167836288654\"><p id=\"fs-id1167836288656\">\\({z}^{2}+12z=-11\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832982349\"><div data-type=\"problem\" id=\"fs-id1167832982351\"><p id=\"fs-id1167836704159\">\\({x}^{2}-20x=21\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836320064\"><p id=\"fs-id1167836320066\">\\(x=-1,x=21\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836685027\"><div data-type=\"problem\" id=\"fs-id1167836685029\"><p id=\"fs-id1167836685032\">\\({y}^{2}-2y=8\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836536608\"><div data-type=\"problem\" id=\"fs-id1167824648900\"><p id=\"fs-id1167824648902\">\\({m}^{2}+4m=-44\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836511692\"><p id=\"fs-id1167833081892\">\\(m=-2\u00b12\\sqrt{10}i\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836322851\"><div data-type=\"problem\" id=\"fs-id1167836624579\"><p id=\"fs-id1167836624581\">\\({n}^{2}-2n=-3\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833021153\"><div data-type=\"problem\" id=\"fs-id1167833021155\"><p id=\"fs-id1167826171748\">\\({r}^{2}+6r=-11\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833274539\"><p id=\"fs-id1167833274542\">\\(r=-3\u00b1\\sqrt{2}i\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833024011\"><div data-type=\"problem\" id=\"fs-id1167833381494\"><p id=\"fs-id1167833381496\">\\({t}^{2}-14t=-50\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826077597\"><div data-type=\"problem\" id=\"fs-id1167833334791\"><p id=\"fs-id1167833334794\">\\({a}^{2}-10a=-5\\)<\/p><\/div><div data-type=\"solution\"><p id=\"fs-id1167836579068\">\\(a=5\u00b12\\sqrt{5}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836717039\"><div data-type=\"problem\" id=\"fs-id1167836717041\"><p id=\"fs-id1167836693277\">\\({b}^{2}+6b=41\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836295347\"><div data-type=\"problem\" id=\"fs-id1167829597930\"><p id=\"fs-id1167829597932\">\\({x}^{2}+5x=2\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829717765\"><p id=\"fs-id1167836520796\">\\(x=-\\frac{5}{2}\u00b1\\frac{\\sqrt{33}}{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829752498\"><div data-type=\"problem\" id=\"fs-id1167833377155\"><p id=\"fs-id1167833377157\">\\({y}^{2}-3y=2\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833338934\"><div data-type=\"problem\" id=\"fs-id1167833338936\"><p id=\"fs-id1167833338938\">\\({u}^{2}-14u+12=-1\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833378999\"><p id=\"fs-id1167836516560\">\\(u=1,u=13\\)<\/p><\/div><\/div><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167829739306\"><p id=\"fs-id1167829930891\">\\({z}^{2}+2z-5=2\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833060172\"><div data-type=\"problem\" id=\"fs-id1167829861776\"><p id=\"fs-id1167829861778\">\\({r}^{2}-4r-3=9\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836534005\"><p id=\"fs-id1167836534007\">\\(r=-2,r=6\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836520251\"><div data-type=\"problem\" id=\"fs-id1167836520253\"><p id=\"fs-id1167836520255\">\\({t}^{2}-10t-6=5\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836528272\"><div data-type=\"problem\" id=\"fs-id1167836528275\"><p id=\"fs-id1167829593604\">\\({v}^{2}=9v+2\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829747213\"><p id=\"fs-id1167829747215\">\\(v=\\frac{9}{2}\u00b1\\frac{\\sqrt{89}}{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829809842\"><div data-type=\"problem\" id=\"fs-id1167836788437\"><p id=\"fs-id1167836788439\">\\({w}^{2}=5w-1\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833025559\"><div data-type=\"problem\" id=\"fs-id1167833025561\"><p>\\({x}^{2}-5=10x\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829742112\"><p id=\"fs-id1167829742114\">\\(x=5\u00b1\\sqrt{30}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833396393\"><div data-type=\"problem\" id=\"fs-id1167833263810\"><p id=\"fs-id1167833263812\">\\({y}^{2}-14=6y\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833085342\"><div data-type=\"problem\" id=\"fs-id1167833085344\"><p id=\"fs-id1167833085346\">\\(\\left(x+6\\right)\\left(x-2\\right)=9\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836502743\"><p id=\"fs-id1167836502745\">\\(x=-7,x=3\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833051595\"><div data-type=\"problem\" id=\"fs-id1167836774990\"><p id=\"fs-id1167836774992\">\\(\\left(y+9\\right)\\left(y+7\\right)=80\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836706352\"><div data-type=\"problem\" id=\"fs-id1167836706354\"><p id=\"fs-id1167836706356\">\\(\\left(x+2\\right)\\left(x+4\\right)=3\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836685173\"><p id=\"fs-id1167836685175\">\\(x=-5,x=-1\\)<\/p><\/div><\/div><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167836349281\"><p id=\"fs-id1167829686605\">\\(\\left(x-2\\right)\\left(x-6\\right)=5\\)<\/p><\/div><\/div><p><strong data-effect=\"bold\">Solve Quadratic Equations 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 by Completing the Square<\/strong><\/p><p id=\"fs-id1167836704842\">In the following exercises, solve by completing the square.<\/p><div data-type=\"exercise\" id=\"fs-id1167836554015\"><div data-type=\"problem\" id=\"fs-id1167836554017\"><p id=\"fs-id1167836554019\">\\(3{m}^{2}+30m-27=6\\)<\/p><\/div><div data-type=\"solution\"><p id=\"fs-id1167836319949\">\\(m=-11,m=1\\)<\/p><\/div><\/div><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167833224435\"><p id=\"fs-id1167833054082\">\\(2{x}^{2}-14x+12=0\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836558013\"><div data-type=\"problem\" id=\"fs-id1167836558015\"><p id=\"fs-id1167836756892\">\\(2{n}^{2}+4n=26\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167830122893\"><p id=\"fs-id1167832927598\">\\(n=1\u00b1\\sqrt{14}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833334776\"><div data-type=\"problem\" id=\"fs-id1167833334778\"><p id=\"fs-id1167833334780\">\\(5{x}^{2}+20x=15\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836599358\"><div data-type=\"problem\" id=\"fs-id1167836599360\"><p id=\"fs-id1167836599363\">\\(2{c}^{2}+c=6\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836601206\"><p id=\"fs-id1167836601208\">\\(c=-2,c=\\frac{3}{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829714659\"><div data-type=\"problem\" id=\"fs-id1167829714662\"><p id=\"fs-id1167829714664\">\\(3{d}^{2}-4d=15\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833138092\"><div data-type=\"problem\" id=\"fs-id1167836536049\"><p id=\"fs-id1167836536051\">\\(2{x}^{2}+7x-15=0\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836523515\"><p id=\"fs-id1167836523518\">\\(x=-5,x=\\frac{3}{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836542019\"><div data-type=\"problem\" id=\"fs-id1167836542021\"><p id=\"fs-id1167829833202\">\\(3{x}^{2}-14x+8=0\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836524254\"><div data-type=\"problem\" id=\"fs-id1167836524256\"><p id=\"fs-id1167833227114\">\\(2{p}^{2}+7p=14\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829719417\"><p id=\"fs-id1167829952795\">\\(p=-\\frac{7}{4}\u00b1\\frac{\\sqrt{161}}{4}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836543379\"><div data-type=\"problem\" id=\"fs-id1167836543381\"><p id=\"fs-id1167836543383\">\\(3{q}^{2}-5q=9\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836731358\"><div data-type=\"problem\" id=\"fs-id1167836662626\"><p id=\"fs-id1167836662628\">\\(5{x}^{2}-3x=-10\\)<\/p><\/div><div data-type=\"solution\"><p id=\"fs-id1167829783759\">\\(x=\\frac{3}{10}\u00b1\\frac{\\sqrt{191}}{10}i\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167824578910\"><div data-type=\"problem\" id=\"fs-id1167824578912\"><p id=\"fs-id1167824578914\">\\(7{x}^{2}+4x=-3\\)<\/p><\/div><\/div><\/div><div class=\"writing\" data-depth=\"2\" id=\"fs-id1167836619200\"><h4 data-type=\"title\">Writing Exercises<\/h4><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167833274696\"><p>Solve the equation \\({x}^{2}+10x=-25\\)<\/p><p id=\"fs-id1167836665108\"><span class=\"token\">\u24d0<\/span> by using the Square Root Property<\/p><p id=\"fs-id1167829753427\"><span class=\"token\">\u24d1<\/span> by Completing the Square<\/p><p id=\"fs-id1167829753433\"><span class=\"token\">\u24d2<\/span> Which method do you prefer? Why?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836310437\"><p id=\"fs-id1167836310440\">Answers will vary.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836730436\"><div data-type=\"problem\" id=\"fs-id1167836730438\"><p id=\"fs-id1167836730440\">Solve the equation \\({y}^{2}+8y=48\\) by completing the square and explain all your steps.<\/p><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167832945882\"><h4 data-type=\"title\">Self Check<\/h4><p id=\"fs-id1167829827835\"><span class=\"token\">\u24d0<\/span> After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.<\/p><span data-type=\"media\" id=\"fs-id1167833290114\" 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 complete the square of a binomial expression.\u201d \u201cConfidently,\u201d \u201cwith some help,\u201d or \u201cNo, I don\u2019t get it.\u201d Choose how would you respond to the statement \u201cI can solve quadratic equations of the form x squared plus b times x plus c equals 0 by completing the square.\u201d \u201cConfidently,\u201d \u201cwith some help,\u201d or \u201cNo, I don\u2019t get it.\u201d Choose how would you respond to the statement \u201cI can solve quadratic equations of the form a times x squared plus b times x plus c equals 0 by completing the square.\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_02_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 complete the square of a binomial expression.\u201d \u201cConfidently,\u201d \u201cwith some help,\u201d or \u201cNo, I don\u2019t get it.\u201d Choose how would you respond to the statement \u201cI can solve quadratic equations of the form x squared plus b times x plus c equals 0 by completing the square.\u201d \u201cConfidently,\u201d \u201cwith some help,\u201d or \u201cNo, I don\u2019t get it.\u201d Choose how would you respond to the statement \u201cI can solve quadratic equations of the form a times x squared plus b times x plus c equals 0 by completing the square.\u201d \u201cConfidently,\u201d \u201cwith some help,\u201d or \u201cNo, I don\u2019t get it.\u201d\"><\/span><p id=\"fs-id1167833290123\"><span class=\"token\">\u24d1<\/span> After reviewing this checklist, what will you do to become confident for all objectives?<\/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>Complete the square of a binomial expression<\/li>\n<li>Solve quadratic equations of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-265159aa9340b296f88aeb962d76639a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#120;&#43;&#99;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"119\" style=\"vertical-align: -2px;\" \/> by completing the square<\/li>\n<li>Solve quadratic equations of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-989d634a4501459153f133ca0d00e2db_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#120;&#43;&#99;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"129\" style=\"vertical-align: -2px;\" \/> by completing the square<\/li>\n<\/ul>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167829746843\" class=\"be-prepared\">\n<p id=\"fs-id1167824737670\">Before you get started, take this readiness quiz.<\/p>\n<ol id=\"fs-id1167836575945\" type=\"1\">\n<li>Expand: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-31d2e04a47a4b5156ef2d66f2d1d7301_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"65\" style=\"vertical-align: -4px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p> If you missed this problem, review <a href=\"\/contents\/0b9be1db-21c4-4bd0-8f8e-d809f6ff7c8c#fs-id1167836392219\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<li>Factor <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bb05ef9baba18724fede8d1bfee811fc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#52;&#121;&#43;&#52;&#57;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"109\" 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-id1167836732680\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<li>Factor <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7d22de75296d6977d8de4ee41f4d6a7d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#48;&#110;&#43;&#56;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"121\" style=\"vertical-align: -2px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p> If you missed this problem, review <a href=\"\/contents\/edd9c403-9825-4cf2-b237-2e05552ea3ec#fs-id1167836415146\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<\/ol>\n<\/div>\n<p id=\"fs-id1167833025786\">So far we have solved quadratic equations by factoring and using the Square Root Property. In this section, we will solve quadratic equations by a process called <span data-type=\"term\">completing the square<\/span>, which is important for our work on conics later.<\/p>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167829811989\">\n<h3 data-type=\"title\">Complete the Square of a Binomial Expression<\/h3>\n<p id=\"fs-id1167829714915\">In the last section, we were able to use the <span data-type=\"term\" class=\"no-emphasis\">Square Root Property<\/span> to solve the equation (<em data-effect=\"italics\">y<\/em> \u2212 7)<sup>2<\/sup> = 12 because the left side was a perfect square.<\/p>\n<div data-type=\"equation\" id=\"fs-id1167829908200\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9b2a28e4c9ecfb49834fe539ebaa0a13_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#49;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#121;&#45;&#55;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&plusmn;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#121;&#45;&#55;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&plusmn;&#50;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#55;&plusmn;&#50;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"90\" width=\"148\" style=\"vertical-align: -39px;\" \/><\/div>\n<p id=\"fs-id1167833311254\">We also solved an equation in which the left side was a perfect square trinomial, but we had to rewrite it the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-65dc4ce98e440a287e019c345645ad22_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#107;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"61\" style=\"vertical-align: -4px;\" \/> in order to use the Square Root Property.<\/p>\n<div data-type=\"equation\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cb5024f2c2d1e2aabd276df7a98401cc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#48;&#120;&#43;&#50;&#53;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#49;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#49;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"171\" style=\"vertical-align: -16px;\" \/><\/div>\n<p id=\"fs-id1167824754928\">What happens if the variable is not part of a perfect square? Can we use algebra to make a perfect square?<\/p>\n<p id=\"fs-id1167832925970\">Let\u2019s look at two examples to help us recognize the patterns.<\/p>\n<div data-type=\"equation\" id=\"fs-id1167836579092\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e2dccec71a611902caefd57f4cd9ab07_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#57;&#120;&#43;&#57;&#120;&#43;&#56;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#55;&#121;&#45;&#55;&#121;&#43;&#52;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#56;&#120;&#43;&#56;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#52;&#121;&#43;&#52;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"88\" width=\"377\" style=\"vertical-align: -38px;\" \/><\/div>\n<p id=\"fs-id1167829688618\">We restate the patterns here for reference.<\/p>\n<div data-type=\"note\" id=\"fs-id1167836363097\">\n<div data-type=\"title\">Binomial Squares Pattern<\/div>\n<p id=\"fs-id1167826132392\">If <em data-effect=\"italics\">a<\/em> and <em data-effect=\"italics\">b<\/em> are real numbers,<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167829739760\" data-alt=\"Quantity a plus b squared equals a squared plus 2 a b plus b2 where the binomial squared equals the first term squared plus 2 times the product of terms plus the second term squared. Quantity a minus b squared equals a squared minus 2 a b plus b2 where the binomial squared equals the first term squared minus 2 times the product of terms plus the second term squared.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_019_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Quantity a plus b squared equals a squared plus 2 a b plus b2 where the binomial squared equals the first term squared plus 2 times the product of terms plus the second term squared. Quantity a minus b squared equals a squared minus 2 a b plus b2 where the binomial squared equals the first term squared minus 2 times the product of terms plus the second term squared.\" \/><\/span><\/div>\n<p id=\"fs-id1167836526163\">We can use this pattern to \u201cmake\u201d a perfect square.<\/p>\n<p id=\"fs-id1167836507775\">We will start with the expression <em data-effect=\"italics\">x<\/em><sup>2<\/sup> + 6<em data-effect=\"italics\">x<\/em>. Since there is a plus sign between the two terms, we will use the (<em data-effect=\"italics\">a<\/em> + <em data-effect=\"italics\">b<\/em>)<sup>2<\/sup> pattern, <em data-effect=\"italics\">a<\/em><sup>2<\/sup> + 2<em data-effect=\"italics\">ab<\/em> + <em data-effect=\"italics\">b<\/em><sup>2<\/sup> = (<em data-effect=\"italics\">a<\/em> + <em data-effect=\"italics\">b<\/em>)<sup>2<\/sup>.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167833272243\" data-alt=\"The perfect square expression a squared plus 2 a b plus b squared is shown above the expression x squared plus 6x plus an unknown to allow a comparison of the corresponding terms of the expressions.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_001_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The perfect square expression a squared plus 2 a b plus b squared is shown above the expression x squared plus 6x plus an unknown to allow a comparison of the corresponding terms of the expressions.\" \/><\/span><\/p>\n<p>We ultimately need to find the last term of this trinomial that will make it a perfect square trinomial. To do that we will need to find <em data-effect=\"italics\">b<\/em>. But first we start with determining <em data-effect=\"italics\">a<\/em>. Notice that the first term of <em data-effect=\"italics\">x<\/em><sup>2<\/sup> + 6<em data-effect=\"italics\">x<\/em> is a square, <em data-effect=\"italics\">x<\/em><sup>2<\/sup>. This tells us that <em data-effect=\"italics\">a<\/em> = <em data-effect=\"italics\">x<\/em>.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167836448681\" data-alt=\"The perfect square expression a squared plus 2 a b plus b squared is shown above the expression x squared plus 2 x b + b squared. Note that x has been substituted for a in the second equation and compare corresponding terms.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_002_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The perfect square expression a squared plus 2 a b plus b squared is shown above the expression x squared plus 2 x b + b squared. Note that x has been substituted for a in the second equation and compare corresponding terms.\" \/><\/span><\/p>\n<p id=\"fs-id1167836408542\">What number, <em data-effect=\"italics\">b,<\/em> when multiplied with 2<em data-effect=\"italics\">x<\/em> gives 6<em data-effect=\"italics\">x<\/em>? It would have to be 3, which is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5bc61e507bd63e00f96ad0e29a14a2c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"41\" style=\"vertical-align: -6px;\" \/> So <em data-effect=\"italics\">b<\/em> = 3.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167825703475\" data-alt=\"The perfect square expression a squared plus 2 a b plus b squared is shown above the expression x squared plus 2 times 3 times x plus an unknown value to help compare terms.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_003_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The perfect square expression a squared plus 2 a b plus b squared is shown above the expression x squared plus 2 times 3 times x plus an unknown value to help compare terms.\" \/><\/span><\/p>\n<p id=\"fs-id1167824767298\">Now to complete the perfect square trinomial, we will find the last term by squaring <em data-effect=\"italics\">b<\/em>, which is 3<sup>2<\/sup> = 9.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167833227142\" data-alt=\"The perfect square expression a squared plus 2 a b plus b squared is shown above the expression x squared plus 6 x plus 9.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_004a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The perfect square expression a squared plus 2 a b plus b squared is shown above the expression x squared plus 6 x plus 9.\" \/><\/span><\/p>\n<p id=\"fs-id1171791481186\">We can now factor.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167836698501\" data-alt=\"The factored expression, the square of a plus b, is shown over the square of the expression x + 3.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_004b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The factored expression, the square of a plus b, is shown over the square of the expression x + 3.\" \/><\/span><\/p>\n<p id=\"fs-id1167836673411\">So we found that adding 9 to <em data-effect=\"italics\">x<\/em><sup>2<\/sup> + 6<em data-effect=\"italics\">x<\/em> \u2018completes the square\u2019, and we write it as (<em data-effect=\"italics\">x<\/em> + 3)<sup>2<\/sup>.<\/p>\n<div data-type=\"note\" id=\"fs-id1167836299468\" class=\"howto\">\n<div data-type=\"title\">Complete a square of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fa2c9948b8f5344d5932dd6fd3679d2f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#120;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"61\" style=\"vertical-align: -2px;\" \/><\/div>\n<ol id=\"fs-id1167826162778\" type=\"1\" class=\"stepwise\">\n<li>Identify <em data-effect=\"italics\">b<\/em>, the coefficient of <em data-effect=\"italics\">x<\/em>.<\/li>\n<li>Find <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e6e29d9e6be01c203d706b9ee5068348_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"44\" style=\"vertical-align: -7px;\" \/> the number to complete the square.<\/li>\n<li>Add the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c703af533a55744c01c07c29c5fe238a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"40\" style=\"vertical-align: -7px;\" \/> to <em data-effect=\"italics\">x<\/em><sup>2<\/sup> + <em data-effect=\"italics\">bx<\/em>.<\/li>\n<li>Factor the perfect square trinomial, writing it as a binomial squared.<\/li>\n<\/ol>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1167836299751\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167833007358\">\n<div data-type=\"problem\" id=\"fs-id1167825836369\">\n<p id=\"fs-id1167836620013\">Complete the square to make a perfect square trinomial. Then write the result as a binomial squared.<\/p>\n<p id=\"fs-id1167836485755\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b5bd1639a976c8c3d4e041ca0aba4210_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#54;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"67\" 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-c76fdbaa12b70ef48921196d1c84a442_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#57;&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"57\" style=\"vertical-align: -4px;\" \/><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c29a7bfecda767ebaaad75f8be95f3e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"62\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829720861\">\n<p id=\"fs-id1167836713860\"><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1167836664279\" class=\"unnumbered unstyled\" summary=\"The expression x squared minus b x is shown above the expression x squared minus 26 x. Note that the coefficient of x is negative 26. To complete the square, find the square of one half times b. Substitute negative 26 for b, rewriting the expression as the square of one half times negative 26. Simplifying the product gives the square of negative 13, and evaluating the square gives 169. Add 169 to the binomial to complete the square so that the expression becomes x squared minus 26 x plus 169. Factor the resulting perfect square trinomial, writing it as the square of x minus 26, a binomial squared.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167829930011\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_005b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">The coefficient of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/> is \u221226.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">\n<pre class=\"ql-errors\">*** QuickLaTeX cannot compile formula:\n&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#125;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#70;&#105;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&middot;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#54;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\n\n*** Error message:\n&#77;&#105;&#115;&#115;&#105;&#110;&#103;&#32;&#35;&#32;&#105;&#110;&#115;&#101;&#114;&#116;&#101;&#100;&#32;&#105;&#110;&#32;&#97;&#108;&#105;&#103;&#110;&#109;&#101;&#110;&#116;&#32;&#112;&#114;&#101;&#97;&#109;&#98;&#108;&#101;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#36;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#125;\r\n&#77;&#105;&#115;&#115;&#105;&#110;&#103;&#32;&#36;&#32;&#105;&#110;&#115;&#101;&#114;&#116;&#101;&#100;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#46;&#46;&#46;&#116;&#101;&#120;&#116;&#123;&#70;&#105;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#108;&#101;&#102;&#116;\r\n&#69;&#120;&#116;&#114;&#97;&#32;&#125;&#44;&#32;&#111;&#114;&#32;&#102;&#111;&#114;&#103;&#111;&#116;&#116;&#101;&#110;&#32;&#36;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#46;&#46;&#46;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;\r\n&#77;&#105;&#115;&#115;&#105;&#110;&#103;&#32;&#125;&#32;&#105;&#110;&#115;&#101;&#114;&#116;&#101;&#100;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#46;&#46;&#46;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;\r\n&#69;&#120;&#116;&#114;&#97;&#32;&#125;&#44;&#32;&#111;&#114;&#32;&#102;&#111;&#114;&#103;&#111;&#116;&#116;&#101;&#110;&#32;&#36;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#46;&#46;&#46;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;\r\n&#77;&#105;&#115;&#115;&#105;&#110;&#103;&#32;&#125;&#32;&#105;&#110;&#115;&#101;&#114;&#116;&#101;&#100;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#46;&#46;&#46;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;\r\n&#69;&#120;&#116;&#114;&#97;&#32;&#125;&#44;&#32;&#111;&#114;&#32;&#102;&#111;&#114;&#103;&#111;&#116;&#116;&#101;&#110;&#32;&#36;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#46;&#46;&#46;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;\r\n&#77;&#105;&#115;&#115;&#105;&#110;&#103;&#32;&#125;&#32;&#105;&#110;&#115;&#101;&#114;&#116;&#101;&#100;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#46;&#46;&#46;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;\r\n&#69;&#120;&#116;&#114;&#97;&#32;&#125;&#44;&#32;&#111;&#114;&#32;&#102;&#111;&#114;&#103;&#111;&#116;&#116;&#101;&#110;&#32;&#36;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#46;&#46;&#46;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;\r\n&#77;&#105;&#115;&#115;&#105;&#110;&#103;&#32;&#125;&#32;&#105;&#110;&#115;&#101;&#114;&#116;&#101;&#100;&#46;\r\n\n<\/pre>\n<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Add 169 to the binomial to complete the square.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167833326353\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_005c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Factor the perfect square trinomial, writing it as<\/p>\n<div data-type=\"newline\"><\/div>\n<p>a binomial squared.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167829743340\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_005d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167829742072\"><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1167829695497\" class=\"unnumbered unstyled\" summary=\"The expression x squared minus b x is shown above the expression y squared minus 9 y. Note that the coefficient of y is negative 9. To complete the square, find the square of one half times b. Substitute negative 9 for b, rewriting the expression as the square of one half times negative 9. Simplifying the product gives the square of negative nine halves, and evaluating the square gives eighty-one fourths. Add eighty-one fourths to the binomial to complete the square so that the expression becomes y squared minus 9 y plus eighty-one fourths. Factor the resulting perfect square trinomial, writing it as the square of y minus nine halves, a binomial squared.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167829715936\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_006b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">The coefficient of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/> is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9672181aec15f8334b80ada7de4e4fc0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/>.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">\n<pre class=\"ql-errors\">*** QuickLaTeX cannot compile formula:\n&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#125;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#70;&#105;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&middot;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#125;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#49;&#125;&#123;&#52;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\n\n*** Error message:\n&#77;&#105;&#115;&#115;&#105;&#110;&#103;&#32;&#35;&#32;&#105;&#110;&#115;&#101;&#114;&#116;&#101;&#100;&#32;&#105;&#110;&#32;&#97;&#108;&#105;&#103;&#110;&#109;&#101;&#110;&#116;&#32;&#112;&#114;&#101;&#97;&#109;&#98;&#108;&#101;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#36;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#125;\r\n&#77;&#105;&#115;&#115;&#105;&#110;&#103;&#32;&#36;&#32;&#105;&#110;&#115;&#101;&#114;&#116;&#101;&#100;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#46;&#46;&#46;&#116;&#101;&#120;&#116;&#123;&#70;&#105;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#108;&#101;&#102;&#116;\r\n&#69;&#120;&#116;&#114;&#97;&#32;&#125;&#44;&#32;&#111;&#114;&#32;&#102;&#111;&#114;&#103;&#111;&#116;&#116;&#101;&#110;&#32;&#36;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#46;&#46;&#46;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;\r\n&#77;&#105;&#115;&#115;&#105;&#110;&#103;&#32;&#125;&#32;&#105;&#110;&#115;&#101;&#114;&#116;&#101;&#100;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#46;&#46;&#46;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;\r\n&#69;&#120;&#116;&#114;&#97;&#32;&#125;&#44;&#32;&#111;&#114;&#32;&#102;&#111;&#114;&#103;&#111;&#116;&#116;&#101;&#110;&#32;&#36;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#46;&#46;&#46;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;\r\n&#77;&#105;&#115;&#115;&#105;&#110;&#103;&#32;&#125;&#32;&#105;&#110;&#115;&#101;&#114;&#116;&#101;&#100;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#46;&#46;&#46;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;\r\n&#69;&#120;&#116;&#114;&#97;&#32;&#125;&#44;&#32;&#111;&#114;&#32;&#102;&#111;&#114;&#103;&#111;&#116;&#116;&#101;&#110;&#32;&#36;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#46;&#46;&#46;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;\r\n&#77;&#105;&#115;&#115;&#105;&#110;&#103;&#32;&#125;&#32;&#105;&#110;&#115;&#101;&#114;&#116;&#101;&#100;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#46;&#46;&#46;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;\r\n&#69;&#120;&#116;&#114;&#97;&#32;&#125;&#44;&#32;&#111;&#114;&#32;&#102;&#111;&#114;&#103;&#111;&#116;&#116;&#101;&#110;&#32;&#36;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#46;&#46;&#46;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;\r\n&#77;&#105;&#115;&#115;&#105;&#110;&#103;&#32;&#125;&#32;&#105;&#110;&#115;&#101;&#114;&#116;&#101;&#100;&#46;\r\n\n<\/pre>\n<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Add <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5fb3724910cf0037df3dc7e233560c84_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#49;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"14\" style=\"vertical-align: -6px;\" \/> to the binomial to complete the square.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><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_02_006c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Factor the perfect square trinomial, writing it as<\/p>\n<div data-type=\"newline\"><\/div>\n<p>a binomial squared.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836417244\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_006d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><span class=\"token\">\u24d2<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1167836621388\" class=\"unnumbered unstyled can-break\" summary=\"The expression x squared plus b x is shown above the expression n squared plus one-half n. The coefficient of n is one half. To complete the square, find the square of one half times b. Substitute the coefficient of n, b equals one half to get the square of one half times one half. Simplify the product, one fourth squared. Find the square to get the constant value one sixteenth. This number completes the square, yielding the expression n squared plus one-half n plus one sixteenth. Rewrite as a binomial square, the square of n plus one fourth.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836595584\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_007b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">The coefficient of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b170995d512c659d8668b4e42e1fef6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/> is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1ee3166445a46094a2ba22d5b5c61802_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"13\" style=\"vertical-align: -6px;\" \/><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">\n<pre class=\"ql-errors\">*** QuickLaTeX cannot compile formula:\n&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#125;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#70;&#105;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&middot;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#54;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\n\n*** Error message:\n&#77;&#105;&#115;&#115;&#105;&#110;&#103;&#32;&#35;&#32;&#105;&#110;&#115;&#101;&#114;&#116;&#101;&#100;&#32;&#105;&#110;&#32;&#97;&#108;&#105;&#103;&#110;&#109;&#101;&#110;&#116;&#32;&#112;&#114;&#101;&#97;&#109;&#98;&#108;&#101;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#36;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#125;\r\n&#77;&#105;&#115;&#115;&#105;&#110;&#103;&#32;&#36;&#32;&#105;&#110;&#115;&#101;&#114;&#116;&#101;&#100;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#46;&#46;&#46;&#116;&#101;&#120;&#116;&#123;&#70;&#105;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#108;&#101;&#102;&#116;\r\n&#69;&#120;&#116;&#114;&#97;&#32;&#125;&#44;&#32;&#111;&#114;&#32;&#102;&#111;&#114;&#103;&#111;&#116;&#116;&#101;&#110;&#32;&#36;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#46;&#46;&#46;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;\r\n&#77;&#105;&#115;&#115;&#105;&#110;&#103;&#32;&#125;&#32;&#105;&#110;&#115;&#101;&#114;&#116;&#101;&#100;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#46;&#46;&#46;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;\r\n&#69;&#120;&#116;&#114;&#97;&#32;&#125;&#44;&#32;&#111;&#114;&#32;&#102;&#111;&#114;&#103;&#111;&#116;&#116;&#101;&#110;&#32;&#36;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#46;&#46;&#46;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;\r\n&#77;&#105;&#115;&#115;&#105;&#110;&#103;&#32;&#125;&#32;&#105;&#110;&#115;&#101;&#114;&#116;&#101;&#100;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#46;&#46;&#46;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;\r\n&#69;&#120;&#116;&#114;&#97;&#32;&#125;&#44;&#32;&#111;&#114;&#32;&#102;&#111;&#114;&#103;&#111;&#116;&#116;&#101;&#110;&#32;&#36;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#46;&#46;&#46;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;\r\n&#77;&#105;&#115;&#115;&#105;&#110;&#103;&#32;&#125;&#32;&#105;&#110;&#115;&#101;&#114;&#116;&#101;&#100;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#46;&#46;&#46;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;\r\n&#69;&#120;&#116;&#114;&#97;&#32;&#125;&#44;&#32;&#111;&#114;&#32;&#102;&#111;&#114;&#103;&#111;&#116;&#116;&#101;&#110;&#32;&#36;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#46;&#46;&#46;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;\r\n&#77;&#105;&#115;&#115;&#105;&#110;&#103;&#32;&#125;&#32;&#105;&#110;&#115;&#101;&#114;&#116;&#101;&#100;&#46;\r\n\n<\/pre>\n<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Add <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e9355787b51ad9db1922ec074556dd6d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"14\" style=\"vertical-align: -7px;\" \/> to the binomial to complete the square.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873803093\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_007c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Rewrite as a binomial square.<\/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_02_007d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836449723\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167829579073\">\n<div data-type=\"problem\" id=\"fs-id1167829751661\">\n<p id=\"fs-id1167833060830\">Complete the square to make a perfect square trinomial. Then write the result as a binomial squared.<\/p>\n<p id=\"fs-id1167829826723\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-71d03a6e5d6c248560f29a29ef9d6669_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#48;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"66\" 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-0806bdcfd40695a152c6a3f2bb4a103e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#109;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"69\" style=\"vertical-align: 0px;\" \/><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8afdc782900943466b31183490a5c817_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#112;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"59\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833350873\">\n<p id=\"fs-id1167833239139\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-45da95dd23fe09ceda268fd9e960a314_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#45;&#49;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"69\" style=\"vertical-align: -4px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e0140ed0066a0c7edf9d967361abb494_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#98;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"61\" style=\"vertical-align: -7px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2975b958c46cf0930ca19af6662aa64c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#112;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#56;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"63\" style=\"vertical-align: -7px;\" \/><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836552490\" class=\"try\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167836510598\">Complete the square to make a perfect square trinomial. Then write the result as a binomial squared.<\/p>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a7dd85b4469c282d52ef9d523fd9eb28_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"54\" style=\"vertical-align: -1px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-832ec946a38f868f9c9b59cf4462d267_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#51;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"69\" style=\"vertical-align: -2px;\" \/><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ce0e9cbe3055a8f5128a2837c0791029_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#113;&#125;&#94;&#123;&#50;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#113;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"57\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836334559\">\n<p id=\"fs-id1167833365706\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7782e20c8b504b8e4af38d54782fdaab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#98;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"58\" style=\"vertical-align: -4px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2a9399892ac66b8c89a712fee6d9b0f5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#110;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#51;&#125;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"71\" style=\"vertical-align: -7px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-87625424bfdb20e4d3017f972076cd19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#113;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"62\" style=\"vertical-align: -7px;\" \/><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167836800584\">\n<h3 data-type=\"title\">Solve Quadratic Equations of the Form <em data-effect=\"italics\">x<\/em><sup>2<\/sup> + <em data-effect=\"italics\">bx<\/em> + <em data-effect=\"italics\">c<\/em> = 0 by Completing the Square<\/h3>\n<p>In solving equations, we must always do the same thing to both sides of the equation. This is true, of course, when we solve a <span data-type=\"term\" class=\"no-emphasis\">quadratic equation<\/span> by <span data-type=\"term\" class=\"no-emphasis\">completing the square<\/span> too. When we add a term to one side of the equation to make a perfect square trinomial, we must also add the same term to the other side of the equation.<\/p>\n<p id=\"fs-id1167836509121\">For example, if we start with the equation <em data-effect=\"italics\">x<\/em><sup>2<\/sup> + 6<em data-effect=\"italics\">x<\/em> = 40, and we want to complete the square on the left, we will add 9 to both sides of the equation.<\/p>\n<table id=\"fs-id1167836507749\" class=\"unnumbered unstyled\" summary=\"Start with the equation x squared plus 6 x equals 40. In the next step, insert spaces to use when completed the square. Write the equation as x squared plus 6 x plus space equals 40 plus space. Complete the square, inserting 9 into each space so that the equation becomes x squared plus 6 x plus 9 equals 40 plus 9. Factor the expression on the left side of the equation and simplify on the right to yield the square of x plus 3 equals 49.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832981250\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_008a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836518758\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_008b_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 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_02_008c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Add 9 to both sides to complete the square.<\/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_02_008d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167829749817\">Now the equation is in the form to solve using the <span data-type=\"term\" class=\"no-emphasis\">Square Root Property<\/span>! Completing the square is a way to transform an equation into the form we need to be able to use the Square Root Property.<\/p>\n<div data-type=\"example\" id=\"fs-id1167829894368\" class=\"textbox textbox--examples\">\n<div data-type=\"title\">How to Solve a Quadratic Equation of the Form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-265159aa9340b296f88aeb962d76639a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#120;&#43;&#99;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"119\" style=\"vertical-align: -2px;\" \/> by Completing the Square<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829628058\">\n<div data-type=\"problem\">\n<p>Solve by completing the square: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0a03ba50c741cd1a65b628aaf503c373_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#56;&#120;&#61;&#52;&#56;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"104\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836542585\"><span data-type=\"media\" id=\"fs-id1167836440966\" data-alt=\"Step 1 is to isolate the variable terms on one side and the constant terms on the other. This equation, x squared plus 8 x equals 48 already has all variable terms on the left. Note that the leading coefficient is 1, so b equals 8.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_009a_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 1 is to isolate the variable terms on one side and the constant terms on the other. This equation, x squared plus 8 x equals 48 already has all variable terms on the left. Note that the leading coefficient is 1, so b equals 8.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167833053781\" data-alt=\"In step 2, find the expression one half times b, squared, the number needed to complete the square. Add this value to both sides of the equation. Take half of 8 and square it. The square of one half times 8 equals 16, so add 16 to BOTH sides of the equation. The equation becomes x squared plus 8 x plus 16 equals 48 plus 16.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_009b_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"In step 2, find the expression one half times b, squared, the number needed to complete the square. Add this value to both sides of the equation. Take half of 8 and square it. The square of one half times 8 equals 16, so add 16 to BOTH sides of the equation. The equation becomes x squared plus 8 x plus 16 equals 48 plus 16.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167836286060\" data-alt=\"In step 3, factor the perfect square trinomial, writing it as a binomial squared on the left and simplify by adding the terms on the right. Factor x squared plus 8 x plus 16 on the left side. Add 48+16 on the right side. The equation becomes the square of x plus 4 equals 64.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_009c_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"In step 3, factor the perfect square trinomial, writing it as a binomial squared on the left and simplify by adding the terms on the right. Factor x squared plus 8 x plus 16 on the left side. Add 48+16 on the right side. The equation becomes the square of x plus 4 equals 64.\" \/><\/span><span data-type=\"media\" data-alt=\"Step 4 is to use the Square Root Property. Take the square root of both sides of the equation to yield x plus 4 equals the positive or negative square root of 64.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_009d_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 4 is to use the Square Root Property. Take the square root of both sides of the equation to yield x plus 4 equals the positive or negative square root of 64.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167836743243\" data-alt=\"In step 5, simplify the radical and then solve the two resulting equations. X plus 4 equals positive 8 or negative 8. If x plus 4 equals 8, then x equals 4. If x plus 4 equals negative 8, then x equals negative 12.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_009e_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"In step 5, simplify the radical and then solve the two resulting equations. X plus 4 equals positive 8 or negative 8. If x plus 4 equals 8, then x equals 4. If x plus 4 equals negative 8, then x equals negative 12.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167832994384\" data-alt=\"Finally, step 6, check the solutions. Put each answer in the original equation to check. First substitute x equals 4. We need to show that 4 squared plus 8 times 4 equals 48. Simplify. The expression 4 squared plus 8 times 4 is equivalent to 16 plus 32, or 48. X equals 4 is a solution. Next substitute x equals negative 12 into the original equation, x squared plus 8 x equals 48. The square of negative 12 plus 8 times negative 12 equals 144 minus 96, or 48. X equals negative 12 is also a solution.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_009f_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Finally, step 6, check the solutions. Put each answer in the original equation to check. First substitute x equals 4. We need to show that 4 squared plus 8 times 4 equals 48. Simplify. The expression 4 squared plus 8 times 4 is equivalent to 16 plus 32, or 48. X equals 4 is a solution. Next substitute x equals negative 12 into the original equation, x squared plus 8 x equals 48. The square of negative 12 plus 8 times negative 12 equals 144 minus 96, or 48. X equals negative 12 is also a solution.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836625042\">\n<div data-type=\"problem\" id=\"fs-id1167829586206\">\n<p>Solve by completing the square: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-22a1f68586b25660f79f0a2fad1d56ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#120;&#61;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"95\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836625698\">\n<p id=\"fs-id1167836570280\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8d66adcf0855e757500df1008f0dc299_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#53;&#44;&#120;&#61;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"120\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167829755835\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167833058751\">\n<div data-type=\"problem\" id=\"fs-id1167836484698\">\n<p id=\"fs-id1167829740784\">Solve by completing the square: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-468500931df6605c37c6c1cb5bebbdb6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#48;&#121;&#61;&#45;&#57;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"116\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833139723\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-449a475395b23d243a40c402bd1d05a6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#49;&#44;&#121;&#61;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"92\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167826211837\">The steps to solve a quadratic equation by completing the square are listed here.<\/p>\n<div data-type=\"note\" id=\"fs-id1167833381337\" class=\"howto\">\n<div data-type=\"title\">Solve a quadratic equation of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-265159aa9340b296f88aeb962d76639a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#120;&#43;&#99;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"119\" style=\"vertical-align: -2px;\" \/> by completing the square.<\/div>\n<ol id=\"fs-id1167829791692\" type=\"1\" class=\"stepwise\">\n<li>Isolate the variable terms on one side and the constant terms on the other.<\/li>\n<li>Find <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-10e9630f3384a327d7d81dc8e4fa7bac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&middot;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"51\" style=\"vertical-align: -7px;\" \/> the number needed to complete the square. Add it to both sides of the equation.<\/li>\n<li>Factor the perfect square trinomial, writing it as a binomial squared on the left and simplify by adding the terms on the right<\/li>\n<li>Use the Square Root Property.<\/li>\n<li>Simplify the radical and then solve the two resulting equations.<\/li>\n<li>Check the solutions.<\/li>\n<\/ol>\n<\/div>\n<p id=\"fs-id1167829689257\">When we solve an equation by completing the square, the answers will not always be integers.<\/p>\n<div data-type=\"example\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167836538119\">\n<p>Solve by completing the square: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-df24d5b6393b65c8ca242bb0c756b514_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#120;&#61;&#45;&#50;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"118\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836417144\">\n<table id=\"fs-id1167829711983\" class=\"unnumbered unstyled\" summary=\"In the equation x squared plus 4 x equals negative 21, the variable terms are on the left side of the equation and the constant term is on the right. To complete the square, take half of the coefficient of x, 4, and square it. The square of the product one half times 4 is 4, so add 4 to both sides. The equation becomes x squared plus 4 x plus 4 equals negative 21 plus 4. Factor the perfect square trinomial on the left and add the values on the right to yield the square of the sum x plus 2 equals negative 17. Use the Square Root Property. The equation becomes x plus 2 equals the positive or negative square root of negative 17. Simplify using complex numbers. X plus 2 equals positive or negative square root 17 times I. Subtract 2 from each side of the equation. X equals negative 2 plus or minus square root 17 I. Rewrite to show two solutions. x equals negative 2 plus square root 17 I and x equals negative 2 minus square root 17 I. We leave the check to you.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167833050840\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_010b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">The variable terms are on the left side.<\/p>\n<div data-type=\"newline\"><\/div>\n<p> Take half of 4 and square it.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836408948\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_010c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9b82b0b7c5d14e21942eac33f50297ed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"91\" style=\"vertical-align: -7px;\" \/><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Add 4 to both sides.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><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_02_010d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Factor the perfect square trinomial,<\/p>\n<div data-type=\"newline\"><\/div>\n<p>writing it as a binomial squared.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836520552\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_010e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Use the Square Root Property.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><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_02_010f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Simplify using complex numbers.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836313583\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_010g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Subtract 2 from each side.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167833030947\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_010h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Rewrite to show two solutions.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836692571\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_010i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">We leave the check to you.<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167833369584\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167833379532\">\n<div data-type=\"problem\">\n<p>Solve by completing the square: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-80f6bd2f53199d64d3fa2a6b47471014_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#48;&#121;&#61;&#45;&#51;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"125\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836424149\">\n<p id=\"fs-id1167833142274\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a633b2d959ecca79eb83425cb6af3610_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#53;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#53;&#125;&#105;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#121;&#61;&#53;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#53;&#125;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"219\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836556313\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167824754841\">\n<div data-type=\"problem\" id=\"fs-id1167829843220\">\n<p id=\"fs-id1167829691124\">Solve by completing the square: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dc85681d7dac799433ab14fbd55a957b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#122;&#125;&#94;&#123;&#50;&#125;&#43;&#56;&#122;&#61;&#45;&#49;&#57;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"116\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829718166\">\n<p id=\"fs-id1167826188105\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e5380464045ff13dc300ca28e40e83e8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#122;&#61;&#45;&#52;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#125;&#105;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#122;&#125;&#61;&#45;&#52;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#125;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"224\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167829595092\">In the previous example, our solutions were complex numbers. In the next example, the solutions will be irrational numbers.<\/p>\n<div data-type=\"example\" id=\"fs-id1167829687996\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167829692780\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167829907184\">Solve by completing the square: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0820bc88c364d0e700c68172a6088faa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#56;&#121;&#61;&#45;&#54;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"116\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<table id=\"fs-id1167836548249\" class=\"unnumbered unstyled can-break\" summary=\"In the equation y squared minus 18 y equals negative 6, the variable terms are on the left side of the equation and the constant term is on the right. To complete the square, take half of the coefficient of y, \u221218, and square it. The square of the product one half times negative 18 is 81, so add 81 to both sides. The equation becomes y squared minus 18 y plus 81 equals negative 6 plus 81. Factor the perfect square trinomial on the left and add the values on the right to yield the square of the difference y minus 9 equals 75. Use the Square Root Property. The equation becomes y minus 9 equals the positive or negative square root of 75. Simplify the radical. Y minus 9 equals positive or negative 5 square root 3. Add 9 to each side of the equation. y equals 9 plus or minus 5 square root 3. Check. Substitute 9 plus 5 square root 3 into the original equation, y squared minus 18 y equals negative 6. The expression on the left becomes the square of 9 plus 5 square root 3 minus 18 times the sum 9 plus 5 square root 3. We need to show that this expression equals negative 6. Expanding the square and multiplying yields 81 plus 90 square root 3 plus 75 minus 162 minus 90 square root 3, which equals negative 6. Next substitute 9 minus 5 square root 3 into the original equation, y squared minus 18 y equals negative 6. The expression on the left becomes the square of 9 minus 5 square root 3 minus 18 times the sum 9 minus 5 square root 3. We need to show that this expression equals negative 6. Expanding the square and multiplying yields 81 minus 90 square root 3 plus 75 minus 162 plus 90 square root 3, which equals negative 6.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"center\"><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_02_011c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">The variable terms are on the left side.<\/p>\n<div data-type=\"newline\"><\/div>\n<p> Take half of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bb9e52ddecc045b16dbf509b1c89f11b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"31\" style=\"vertical-align: -1px;\" \/> and square it.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-80a9535ffc69e9d1b9d81db38780e469_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#56;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"122\" style=\"vertical-align: -7px;\" \/><\/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_02_011d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Add 81 to both sides.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836614906\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_011e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Factor the perfect square trinomial,<\/p>\n<div data-type=\"newline\"><\/div>\n<p>writing it as a binomial squared.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167829753570\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_011f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Use the Square Root Property.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836424051\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_011g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Simplify the radical.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><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_02_011h_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-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/>.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836407199\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_011i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Check.<\/p>\n<div data-type=\"newline\"><\/div>\n<p> <span data-type=\"media\" id=\"fs-id1167833059590\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_011b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span> <\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167836378576\">Another way to check this would be to use a calculator. Evaluate <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-45478b66ee880973104483a2943a961f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#56;&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"66\" style=\"vertical-align: -4px;\" \/>for both of the solutions. The answer should be <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e45e9e13a8901226d7601cbfa15df84a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#54;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"26\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836731428\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167829832718\">\n<div data-type=\"problem\" id=\"fs-id1167836526026\">\n<p id=\"fs-id1167833379137\">Solve by completing the square: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-18d30f9c8cf0eb3854fdfe990040a09f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#54;&#120;&#61;&#45;&#49;&#54;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"127\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167836513723\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-777e2e4b018b1c09a64690a2a2d85078_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#56;&#43;&#52;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#125;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#61;&#56;&#45;&#52;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"209\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836527820\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167824720413\">\n<div data-type=\"problem\" id=\"fs-id1167833369838\">\n<p id=\"fs-id1167836392691\">Solve by completing the square: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ac6bfecc86944b008e60116d0db6d9a3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#56;&#121;&#61;&#49;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"103\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836662923\">\n<p id=\"fs-id1167833137978\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-318d9829c918bd214595088d4632872e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#52;&#43;&#51;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#125;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#121;&#125;&#61;&#45;&#52;&#45;&#51;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"231\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836627782\">We will start the next example by isolating the variable terms on the left side of the equation.<\/p>\n<div data-type=\"example\" id=\"fs-id1167836507963\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167824767011\">Solve by completing the square: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a999a8690b4f8ea7b41581e89ab0a1c8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#48;&#120;&#43;&#52;&#61;&#49;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"144\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<table id=\"fs-id1167836549726\" class=\"unnumbered unstyled can-break\" summary=\"Rewrite the original equation, x squared plus 10 x plus 4 equals 15 to isolate the variables on the left side. Subtract 4 from each side of the equation. X square plus 10 x equals 11. To complete the square, take half of the coefficient of x, 10, and square it. The square of the product one half times 10 is 25, so add 25 to both sides. The equation becomes x squared plus 10 x plus 25 equals 11 plus 25. Factor the perfect square trinomial on the left and add the values on the right to yield the square of the sum x plus 5 equals 36. Use the Square Root Property. The equation becomes x plus 5 equals the positive or negative square root of 36. Simplify the radical. X plus 5 equals positive or negative 6. Subtract 5 from each side of the equation. X equals negative 5 plus or minus 6. Rewrite to show 2 solutions, x equals negative 5 plus 6, or 1 and x equals negative 5 minus 6 or negative 11. Check. Substitute 1 into the original equation, x squared plus 10 x plus 4 equals 15. The expression on the left becomes 1 squared plus 10 times 1 plus 4. We need to show that this expression equals 15. Simplifying gives 1 plus 10 plus 4, or 15. Next substitute negative 11 into the original equation x squared plus 10 x plus 4 equals 15. The expression on the left becomes negative 11 squared plus 10 times negative 11 plus 4. We need to show that this expression equals 15. Simplifying gives 121 minus 110 plus 4, or 15.\" 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_02_012c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Isolate the variable terms on the left side.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>Subtract 4 to get the constant terms on the right side.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836356824\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_012d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Take half of 10 and square it.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-08576aca813e19daaf7d1a35d8f3cded_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#50;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"108\" style=\"vertical-align: -7px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829833401\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_012e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Add 25 to both sides.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829878589\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_012f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Factor the perfect square trinomial, writing it as<\/p>\n<div data-type=\"newline\"><\/div>\n<p>a binomial squared.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833345938\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_012g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Use the Square Root Property.<\/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_02_012h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Simplify the radical.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833048657\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_012i_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>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836646306\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_012j_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Rewrite to show two solutions.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167824590480\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_012k_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Solve the equations.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833055123\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_012l_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-id1167829627533\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_012b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span> <\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167833086970\" class=\"try\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167829713249\">\n<p id=\"fs-id1167836697056\">Solve by completing the square: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-559a493ec1da60dd4fd0a770537750cc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#97;&#43;&#57;&#61;&#51;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"133\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836513135\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4040fdda380bf3a340f29404dfe3c0f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&#45;&#55;&#44;&#97;&#61;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"106\" 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-id1167833055847\">\n<div data-type=\"problem\" id=\"fs-id1167836554813\">\n<p id=\"fs-id1167836620313\">Solve by completing the square: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-eb5bd4d2985f0efbc2f0242868e8d064_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#43;&#56;&#98;&#45;&#52;&#61;&#49;&#54;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"130\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167833408069\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-896f418b9ca0b7f0ca02c5560993d354_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;&#61;&#45;&#49;&#48;&#44;&#98;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"110\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>To solve the next equation, we must first collect all the variable terms on the left side of the equation. Then we proceed as we did in the previous examples.<\/p>\n<div data-type=\"example\" id=\"fs-id1167836323334\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167836537598\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167836691587\">Solve by completing the square: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5eab084b30635ff242722aef2a91c833_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#61;&#51;&#110;&#43;&#49;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"105\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826025325\">\n<table id=\"fs-id1167836792924\" class=\"unnumbered unstyled can-break\" summary=\"Rewrite the equation n squared equals 3 n plus 11 to isolate the variable terms on the left side of the equation. Subtract 3 n from both sides of the equation. N squared minus 3 n equals 11. To complete the square, take half of the coefficient of n, negative 3, and square it. The square of the product one half times negative 3 is 9 divided by 4, so add nine fourths to both sides. The equation becomes n squared minus 3 n plus 9 fourths equals 11 plus 9 fourths. Factor the perfect square trinomial on the left, writing it as a binomial squared. On the right, express 11 as a fraction with denominator 4. The square of the difference n minus 3 halves equals 44 fourths plus 9 fourths. Add the fractions on the right side. The square of the difference n minus 3 halves equals 53 fourths Use the Square Root Property. The equation becomes n minus 3 halves equals the positive or negative square root of 53 fourths. Simplify the radical. N minus 3 halves equals positive or negative square root 35 divided by 2. Add 3 halves to both sides of the equation to solve for n. n equals 3 halves plus or minus square root 53 divided by 2. Rewrite to show two solutions. n equals three halves plus square root 53 divided by 2 and n equals three halves minus square root 53 divided by 2. We leave the check to you.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\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_02_013b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Subtract <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0bd6fa46a14c151a72b15ba27270c306_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"20\" style=\"vertical-align: 0px;\" \/> to get the variable terms on the left side.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833056846\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_013c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Take half of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-470cb162cf92c55d139f4f69216225e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/> and square it.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-eb752b9ed503b2f629ca81c7616ba9b1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"105\" style=\"vertical-align: -7px;\" \/><\/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_02_013d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Add <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7f0c69f48cefc6c9a19a640e5157771c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/> to both sides.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836650053\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_013e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Factor the perfect square trinomial, writing it as<\/p>\n<div data-type=\"newline\"><\/div>\n<p>a binomial squared.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167824741373\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_013f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Add the fractions on the right side.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829598071\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_013g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Use the Square Root Property.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836619785\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_013h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Simplify the radical.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167826171765\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_013i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Solve for <em data-effect=\"italics\">n<\/em>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836688616\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_013j_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Rewrite to show two solutions.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833086747\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_013k_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Check:<\/p>\n<div data-type=\"newline\"><\/div>\n<p> We leave the check for you!<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836407661\" class=\"try\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167836392033\">\n<p>Solve by completing the square: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-200820506d76a90a66b74686112713a1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#61;&#53;&#112;&#43;&#57;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"94\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836753600\">\n<p id=\"fs-id1167829789466\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-080889f69a8c75938de013d4c785407c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#112;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#50;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#54;&#49;&#125;&#125;&#123;&#50;&#125;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#112;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#50;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#54;&#49;&#125;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"203\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836410437\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167833059138\">\n<div data-type=\"problem\" id=\"fs-id1167836509823\">\n<p id=\"fs-id1167836399192\">Solve by completing the square: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cef8aa5495a9f7b5197169d6a4a8a0c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#113;&#125;&#94;&#123;&#50;&#125;&#61;&#55;&#113;&#45;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"92\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836292523\">\n<p id=\"fs-id1167836688308\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0449b4a829e0c52aa5197c68175609f9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#113;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#50;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#55;&#125;&#125;&#123;&#50;&#125;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#113;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#50;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#55;&#125;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"199\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167829750235\">Notice that the left side of the next equation is in factored form. But the right side is not zero. So, we cannot use the <span data-type=\"term\" class=\"no-emphasis\">Zero Product Property<\/span> since it says \u201cIf <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-899c72d33ed5ab5627f04572996771d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&middot;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#98;&#61;&#48;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"61\" style=\"vertical-align: -4px;\" \/> then <em data-effect=\"italics\">a<\/em> = 0 or <em data-effect=\"italics\">b<\/em> = 0.\u201d Instead, we multiply the factors and then put the equation into standard form to solve by completing the square.<\/p>\n<div data-type=\"example\" id=\"fs-id1167836357146\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167836492833\">\n<div data-type=\"problem\" id=\"fs-id1167836544202\">\n<p>Solve by completing the square: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-92b0bd770a3055aeb7c56ba4f2c82960_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#57;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"147\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836731606\">\n<table id=\"fs-id1167836407133\" class=\"unnumbered unstyled can-break\" summary=\"Write the equation. The product of x minus 3 and x plus 5 equals 9. Multiply the binomials on the left. X squared plus 2 x minus 15 equals 9. Add 15 to isolate the constant terms on the right. X squared plus 2 x equals 24. To complete the square, take half of the coefficient of x, 2, and square it. The square of the product one half times 2 is 1, so add 1 to both sides. X squared plus 2 x plus 1 equals 24 plus 1. Factor the perfect square trinomial to write it as a binomial squared. The square of x plus 1 equals 25. Use the Square Root Property. X plus 1 equals the positive or negative square root of 25. Solve for x. x equals negative 1 plus or minus 5. Rewrite to show 2 solutions. x equals negative 1 plus 5, or 4. X equals negative 1 minus 5, or negative 6. We leave the check of solutions for you!\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167824658654\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_014b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">We multiply the binomials on the left.<\/td>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836667027\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_014c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Add 15 to isolate the constant terms on the right.<\/td>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836628619\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_014d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Take half of 2 and square it.<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b52518716b95c307d40f627d5e764a6c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&middot;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"90\" style=\"vertical-align: -7px;\" \/><\/td>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167824754906\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_014e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Add 1 to both sides.<\/td>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829589062\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_014f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Factor the perfect square trinomial, writing it as<\/p>\n<div data-type=\"newline\"><\/div>\n<p>a binomial squared.<\/td>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836326096\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_014g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Use the Square Root Property.<\/td>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829719081\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_014h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Solve for <em data-effect=\"italics\">x<\/em>.<\/td>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836294554\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_014i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Rewrite to show two solutions.<\/td>\n<td colspan=\"2\" data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167833025582\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_014j_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td colspan=\"2\" data-valign=\"top\" data-align=\"center\"><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_02_014k_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Check:<\/p>\n<div data-type=\"newline\"><\/div>\n<p> We leave the check for you!<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167833245437\" class=\"try\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167836544930\">Solve by completing the square: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d73aeaaf021359a7ae939e956292a905_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#99;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#99;&#43;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#49;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"152\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836621350\">\n<p id=\"fs-id1167836407485\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-751b89356514d3215adf05d54530dfae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;&#61;&#45;&#57;&#44;&#99;&#61;&#51;\" 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-id1167829715782\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167830077335\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167832951207\">Solve by completing the square: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-69c51f9da618208124cb3b60987f5020_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#100;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#100;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#53;&#54;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"155\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829693560\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b237d61ee1da22dc4a8c4398b0766a3d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#100;&#61;&#49;&#49;&#44;&#100;&#61;&#45;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"114\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167836342004\">\n<h3 data-type=\"title\">Solve Quadratic Equations 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 by Completing the Square<\/h3>\n<p>The process of <span data-type=\"term\" class=\"no-emphasis\">completing the square<\/span> works best when the coefficient of <em data-effect=\"italics\">x<\/em><sup>2<\/sup> is 1, so the left side of the equation is of the form <em data-effect=\"italics\">x<\/em><sup>2<\/sup> + <em data-effect=\"italics\">bx<\/em> + <em data-effect=\"italics\">c<\/em>. If the <em data-effect=\"italics\">x<\/em><sup>2<\/sup> term has a coefficient other than 1, we take some preliminary steps to make the coefficient equal to 1.<\/p>\n<p>Sometimes the coefficient can be factored from all three terms of the trinomial. This will be our strategy in the next example.<\/p>\n<div data-type=\"example\" id=\"fs-id1167836667059\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167829714809\">\n<div data-type=\"problem\" id=\"fs-id1167836509636\">\n<p id=\"fs-id1167829716705\">Solve by completing the square: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-13cdad3a6ba00fb1078231491e7f36ec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#50;&#120;&#45;&#49;&#53;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"152\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836484371\">\n<p id=\"fs-id1167829594643\">To complete the square, we need the coefficient of <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;\" \/> to be one. If we factor out the coefficient of <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;\" \/> as a common factor, we can continue with solving the equation by completing the square.<\/p>\n<table id=\"fs-id1167836508427\" class=\"unnumbered unstyled can-break\" summary=\"Write the equation 3 x squared minus 12 x minus 15 equals 0. Factor out the greatest common factor. 3 times the expression x squared minus 4 x minus 5 equals 0. Divide both sides by 3 to isolate the trinomial. 3 times the expression x squared minus 4 x minus 5 divided by 3 equals 0 divided by 3. Simplify. x squared minus 4 x minus 5 equals 0. Add 5 to get the constant terms on the right side. X squared minus 4 x equals 5. To complete the square, take half of the coefficient of x, 4, and square it. The square of the product one half times 4 is 4, so add 4 to both sides. X squared minus 4 x plus 4 equals 5 plus 4. Factor the perfect square trinomial to write it as a binomial squared. The square of x minus 2 equals 9. Use the Square Root Property. X minus 2 equals the positive or negative square root of 9. Solve for x. x equals 2 plus or minus 3. Rewrite to show 2 solutions. x equals 2 plus 3, or 5. X equals 2 minus 3, or negative 1. Check the solutions by substituting each value into the original equation. Substitute x equals 5 into the equation 3 x squared minus 12 x minus 15 equals 0. The left side becomes 3 times the square of 5 minus 12 times 5 minus 15. We need to show that this equals 0. Simplifying yields 75 minus 60 minus 15 which is equal to 0. 5 is a solution. Substitute x equals negative 1 into the equation 3 x squared minus 12 x minus 15 equals 0. The left side becomes 3 times the square of negative 1 minus 12 times negative 1 minus 15. We need to show that this equals 0. Simplifying yields 3 plus 12 minus 15 which is equal to 0. Negative 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\" id=\"fs-id1167824733261\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_015c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Factor out the greatest common factor.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836497733\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_015d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Divide both sides by 3 to isolate the trinomial<\/p>\n<div data-type=\"newline\"><\/div>\n<p>with coefficient 1.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832951049\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_015e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833326573\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_015f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Add 5 to get the constant terms on the right side.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836626094\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_015g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Take half of 4 and square it.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3947d80039724150055968a53dca1ac1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"105\" style=\"vertical-align: -7px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829692301\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_015h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Add 4 to both sides.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836493824\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_015i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Factor the perfect square trinomial, writing it<\/p>\n<div data-type=\"newline\"><\/div>\n<p>as a binomial squared.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833380251\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_015j_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Use the Square Root Property.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836732071\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_015k_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>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836539657\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_015l_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Rewrite to show two solutions.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836524160\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_015m_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836300550\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_015n_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-id1167836550626\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_015b_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-id1167836596811\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167833024381\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167836787783\">Solve by completing the square: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-eccda6bd2dd5c93d780ba4f1fee8848f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#54;&#109;&#43;&#49;&#52;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"163\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833407704\">\n<p id=\"fs-id1167836492528\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a22ba05285ff5dd44b2d20f2de784743_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#45;&#55;&#44;&#109;&#61;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"131\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167830122896\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836282523\">\n<div data-type=\"problem\" id=\"fs-id1167833019986\">\n<p id=\"fs-id1167829833349\">Solve by completing the square: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0f210d8dfdbd5c770f238028449148d6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#52;&#110;&#45;&#53;&#54;&#61;&#56;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"153\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836450369\">\n<p id=\"fs-id1167836305741\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b769e450e36126676545f411e6ee108d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#61;&#45;&#50;&#44;&#110;&#61;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"108\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836329650\">To complete the square, the coefficient of the <em data-effect=\"italics\">x<\/em><sup>2<\/sup> must be 1. When the <span data-type=\"term\" class=\"no-emphasis\">leading coefficient<\/span> is not a factor of all the terms, we will divide both sides of the equation by the leading coefficient! This will give us a fraction for the second coefficient. We have already seen how to complete the square with fractions in this section.<\/p>\n<div data-type=\"example\" id=\"fs-id1167836406740\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167826172184\">\n<div data-type=\"problem\" id=\"fs-id1167829747125\">\n<p>Solve by completing the square: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-77cb1e0a8d2a8cf9818d8481c29392a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#120;&#61;&#50;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"113\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833356483\">\n<p id=\"fs-id1167836356365\">To complete the square we need the coefficient of <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;\" \/> to be one. We will divide both sides of the equation by the coefficient of <em data-effect=\"italics\">x<\/em><sup>2<\/sup>. Then we can continue with solving the equation by completing the square.<\/p>\n<table id=\"fs-id1167836615944\" class=\"unnumbered unstyled\" summary=\"The equation 2 x squared minus 3 x equals 20 has the x terms isolated on the left side of the equation. Divide both sides by 2 to get the leading coefficient of x squared to be 1. The quotient 2 x squared minus 3 x divided by 2 equals 20 divided by 2. Simplify. x squared minus 3 halves x equals 10. To complete the square, take half of the coefficient of x, negative 3 halves, and square it. The square of the product one half times negative 3 halves is 9 sixteenths, so add 9 sixteenths to both sides. X squared minus 3 halves x plus 9 sixteenths equals 10 plus 9 sixteenths. Factor the perfect square trinomial to write it as a binomial squared and express the terms on the right as fractions with a common denominator. The square of x minus 3 fourths equals 160 divided by 16 plus 9 divided by 16. Add the fractions on the right. The square of x minus 3 fourths equals 169 divided by 16. Use the Square Root Property. X minus 3 fourths equals the positive or negative square root of 169 sixteenths. Simplify the radical. X minus 3 fourths equals positive or negative 13 fourths. Solve for x. x equals 3 fourths plus or minus 13 fourths. Rewrite to show 2 solutions. x equals 3 plus 13 fourths, or 4. x equals 3 minus 13 fourths, or negative 5 halves. We leave the check of the solutions for you!\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\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_02_016b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Divide both sides by 2 to get the<\/p>\n<div data-type=\"newline\"><\/div>\n<p>coefficient of <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;\" \/> to be 1.<\/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_02_016c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836544118\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_016d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Take half of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8418e48d32604e120f3a71dc30c024a6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"22\" style=\"vertical-align: -6px;\" \/> and square it.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cbcd3dd9aab4c3ca8e966967f220072e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#125;&#123;&#49;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"116\" style=\"vertical-align: -7px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836477509\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_016e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Add <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0d0ae1ffc25ea6e1a4965d816832886e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#125;&#123;&#49;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"14\" style=\"vertical-align: -7px;\" \/> to both sides.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829712714\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_016f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Factor the perfect square trinomial,<\/p>\n<div data-type=\"newline\"><\/div>\n<p>writing it as a binomial squared.<\/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_02_016g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Add the fractions on the right side.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836362768\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_016h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Use the Square Root Property.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836433674\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_016i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Simplify the radical.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836543341\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_016j_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Solve for <em data-effect=\"italics\">x<\/em>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833186420\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_016k_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Rewrite to show two solutions.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833270316\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_016l_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833021813\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_016m_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Check:<\/p>\n<div data-type=\"newline\"><\/div>\n<p> We leave the check for you!<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836571353\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836410349\">\n<div data-type=\"problem\" id=\"fs-id1167836387744\">\n<p id=\"fs-id1167836481605\">Solve by completing the square: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-95393d243cdd450cf88237be36d051ac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#123;&#114;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#114;&#61;&#50;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"110\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836514219\">\n<p id=\"fs-id1167833018598\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-17f602ca77faaed65b7c29afaf012e49_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#114;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#51;&#125;&#44;&#114;&#61;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"106\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167829714482\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167829807666\">\n<div data-type=\"problem\" id=\"fs-id1167836533308\">\n<p id=\"fs-id1167829930812\">Solve by completing the square: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2de2f507734be4b313f86d87b533798a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#116;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#116;&#61;&#50;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"106\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167836433690\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c966a6e3a9874b25a9fe70aded72e5f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#50;&#125;&#44;&#116;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"101\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836299186\">Now that we have seen that the coefficient of <em data-effect=\"italics\">x<\/em><sup>2<\/sup> must be 1 for us to complete the square, we update our procedure for solving a <span data-type=\"term\" class=\"no-emphasis\">quadratic equation<\/span> by completing the square to include equations 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.<\/p>\n<div data-type=\"note\" id=\"fs-id1167833051633\" class=\"howto\">\n<div data-type=\"title\">Solve a quadratic equation of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-29c8b40df2f348455a0b8aec28190877_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#120;&#43;&#99;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"129\" style=\"vertical-align: -2px;\" \/> by completing the square.<\/div>\n<ol id=\"fs-id1167829693205\" type=\"1\" class=\"stepwise\">\n<li>Divide by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/> to make the coefficient of <em data-effect=\"italics\">x<\/em><sup>2<\/sup> term 1.<\/li>\n<li>Isolate the variable terms on one side and the constant terms on the other.<\/li>\n<li>Find <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-10e9630f3384a327d7d81dc8e4fa7bac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&middot;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"51\" style=\"vertical-align: -7px;\" \/> the number needed to complete the square. Add it to both sides of the equation.<\/li>\n<li>Factor the perfect square trinomial, writing it as a binomial squared on the left and simplify by adding the terms on the right<\/li>\n<li>Use the Square Root Property.<\/li>\n<li>Simplify the radical and then solve the two resulting equations.<\/li>\n<li>Check the solutions.<\/li>\n<\/ol>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1167836299458\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167836501815\">\n<div data-type=\"problem\" id=\"fs-id1167836627532\">\n<p id=\"fs-id1167829746209\">Solve by completing the square: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-af4568d5ac6fc91d04dacf90a205d17a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#120;&#61;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"104\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833022209\">\n<p id=\"fs-id1167836536066\">Again, our first step will be to make the coefficient of <em data-effect=\"italics\">x<\/em><sup>2<\/sup> one. By dividing both sides of the equation by the coefficient of <em data-effect=\"italics\">x<\/em><sup>2<\/sup>, we can then continue with solving the equation by completing the square.<\/p>\n<table id=\"fs-id1167829909536\" class=\"unnumbered unstyled can-break\" summary=\"The equation 3 x squared plus 2 x equals 4 has the x terms isolated on the left side of the equation. Divide both sides by 3 to get the leading coefficient of x squared to be 1. The quotient 3 x squared plus 2 x divided by 3 equals 4 divided by 3. Simplify. x squared plus 2 thirds x equals 4 thirds. To complete the square, take half of the coefficient of x, 2 thirds, and square it. The square of the product one half times 2 thirds is 1 ninth, so add 1 ninth to both sides. X squared plus 2 thirds x plus 1 ninth equals 4 thirds plus 1 ninth. Factor the perfect square trinomial to write it as a binomial squared and express the terms on the right as fractions with a common denominator. The square of x plus 1 thirds equals 12 divided by 9 plus 1 divided by 9. Add the fractions on the right and use the Square Root Property. X plus 1 third equals the positive or negative square root of 13 ninths. sixteenths. Simplify the radical. X plus 1 third equals positive or negative square root 13 divided by 3. Solve for x. x equals negative 1 thirds plus or minus square root 13 thirds. Rewrite to show 2 solutions. x equals negative 1 thirds plus square root 13 thirds, x equals negative 1 thirds minus square root 13 thirds. We leave the check of the solutions for you!\" 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-id1167829785112\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_017b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Divide both sides by 3 to make the<\/p>\n<div data-type=\"newline\"><\/div>\n<p>coefficient of <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;\" \/> equal 1.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836687333\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_017c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Simplify.<\/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_02_017d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Take half of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6e89c4f69688b0dd6ab75a55841059de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/> and square it.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d0774d904ec6fb45aa77efb2db61a64a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&middot;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"76\" style=\"vertical-align: -7px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836613649\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_017e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Add <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6cad8eec3d159e52efa483f5f56c5d60_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/> to both sides.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833310353\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_017f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Factor the perfect square trinomial, writing it as<\/p>\n<div data-type=\"newline\"><\/div>\n<p>a binomial squared.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836375385\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_017g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Use the Square Root Property.<\/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_02_017h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Simplify the radical.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836408865\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_017i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Solve for <em data-effect=\"italics\">x<\/em> .<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836728833\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_017j_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Rewrite to show two solutions.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833059280\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_017k_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\" data-valign=\"top\" data-align=\"left\">Check:<\/p>\n<div data-type=\"newline\"><\/div>\n<p> We leave the check for you!<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167829715966\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167833309872\">\n<div data-type=\"problem\" id=\"fs-id1167833309874\">\n<p id=\"fs-id1167829712104\">Solve by completing the square: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9dcb8916c9cc8347861207a77fe87a54_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#120;&#61;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"104\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826204778\">\n<p id=\"fs-id1167826204780\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9886458bafb7110cec59aa09c8659f23_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#56;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#52;&#49;&#125;&#125;&#123;&#56;&#125;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#56;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#52;&#49;&#125;&#125;&#123;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"232\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836287199\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836287203\">\n<div data-type=\"problem\" id=\"fs-id1167829851940\">\n<p id=\"fs-id1167829851942\">Solve by completing the square: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-561a365ff3b10c0373cb301842c0f8fa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#48;&#121;&#61;&#45;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"125\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829753405\">\n<p id=\"fs-id1167829753408\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a4abe49f58f99c55bde0e207c6d5d838_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#51;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#48;&#125;&#125;&#123;&#51;&#125;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#51;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#48;&#125;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"203\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167829861822\" class=\"media-2\">\n<p>Access these online resources for additional instruction and practice with completing the square.<\/p>\n<ul id=\"fs-id1167829744240\" data-display=\"block\">\n<li><a href=\"https:\/\/openstax.org\/l\/37CompTheSq1\">Completing Perfect Square Trinomials<\/a><\/li>\n<li><a href=\"https:\/\/openstax.org\/l\/37CompTheSq2\">Completing the Square 1<\/a><\/li>\n<li><a href=\"https:\/\/openstax.org\/l\/37CompTheSq3\">Completing the Square to Solve Quadratic Equations<\/a><\/li>\n<li><a href=\"https:\/\/openstax.org\/l\/37CompTheSq4\">Completing the Square to Solve Quadratic Equations: More Examples<\/a><\/li>\n<li><a href=\"https:\/\/openstax.org\/l\/37CompTheSq5\">Completing the Square 4<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167836544017\">\n<h3 data-type=\"title\">Key Concepts<\/h3>\n<ul id=\"fs-id1167833339296\" data-bullet-style=\"bullet\">\n<li>Binomial Squares Pattern\n<div data-type=\"newline\"><\/div>\n<p> If <em data-effect=\"italics\">a<\/em> and <em data-effect=\"italics\">b<\/em> are real numbers,<\/p>\n<div data-type=\"newline\"><\/div>\n<p> <span data-type=\"media\" data-alt=\"Quantity a plus b squared equals a squared plus 2 a b plus b2 where the binomial squared equals the first term squared plus 2 times the product of terms plus the second term squared. Quantity a minus b squared equals a squared minus 2 a b plus b2 where the binomial squared equals the first term squared minus 2 times the product of terms plus the second term squared.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_02_018_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Quantity a plus b squared equals a squared plus 2 a b plus b2 where the binomial squared equals the first term squared plus 2 times the product of terms plus the second term squared. Quantity a minus b squared equals a squared minus 2 a b plus b2 where the binomial squared equals the first term squared minus 2 times the product of terms plus the second term squared.\" \/><\/span><\/li>\n<li>How to Complete a Square\n<ol id=\"fs-id1167829783787\" type=\"1\" class=\"stepwise\">\n<li>Identify <em data-effect=\"italics\">b<\/em>, the coefficient of <em data-effect=\"italics\">x<\/em>.<\/li>\n<li>Find <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e6e29d9e6be01c203d706b9ee5068348_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"44\" style=\"vertical-align: -7px;\" \/> the number to complete the square.<\/li>\n<li>Add the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c703af533a55744c01c07c29c5fe238a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"40\" style=\"vertical-align: -7px;\" \/> to <em data-effect=\"italics\">x<\/em><sup>2<\/sup> + <em data-effect=\"italics\">bx<\/em><\/li>\n<li>Rewrite the trinomial as a binomial square<\/li>\n<\/ol>\n<\/li>\n<li>How to solve a quadratic 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 by completing the square.\n<ol id=\"fs-id1167836529877\" type=\"1\" class=\"stepwise\">\n<li>Divide by <em data-effect=\"italics\">a<\/em> to make the coefficient of <em data-effect=\"italics\">x<\/em><sup>2<\/sup> term 1.<\/li>\n<li>Isolate the variable terms on one side and the constant terms on the other.<\/li>\n<li>Find <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-10e9630f3384a327d7d81dc8e4fa7bac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&middot;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"51\" style=\"vertical-align: -7px;\" \/> the number needed to complete the square. Add it to both sides of the equation.<\/li>\n<li>Factor the perfect square trinomial, writing it as a binomial squared on the left and simplify by adding the terms on the right.<\/li>\n<li>Use the Square Root Property.<\/li>\n<li>Simplify the radical and then solve the two resulting equations.<\/li>\n<li>Check the solutions.<\/li>\n<\/ol>\n<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167836334125\">\n<div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1167836518584\">\n<h4 data-type=\"title\">Practice Makes Perfect<\/h4>\n<p><strong data-effect=\"bold\">Complete the Square of a Binomial Expression<\/strong><\/p>\n<p id=\"fs-id1167836507896\">In the following exercises, complete the square to make a perfect square trinomial. Then write the result as a binomial squared.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167836390279\">\n<div data-type=\"problem\" id=\"fs-id1167836390281\">\n<p id=\"fs-id1167836390284\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-98854f30145e95cc96561a5fd0550856_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#52;&#109;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"78\" style=\"vertical-align: -1px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c34286718b9921ec55727ac4f4ab75c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#49;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"67\" style=\"vertical-align: -1px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e9ae58005de55866663c69be6846377e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#112;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"59\" style=\"vertical-align: -6px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1167836399357\">\n<p id=\"fs-id1167836408168\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-eab4ebea0eb5b2de8d6eb9cda24e8258_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#109;&#45;&#49;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"75\" style=\"vertical-align: -4px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-191ba78f098d415cc3c90c683a8d30fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#49;&#125;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"71\" style=\"vertical-align: -7px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ae868a6c482658f40aeb7315fd9abc50_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#112;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#54;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"63\" style=\"vertical-align: -7px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829746316\">\n<div data-type=\"problem\" id=\"fs-id1167836613711\">\n<p id=\"fs-id1167836613713\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cd27629f8243148fd9071a6cf58ff8e0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#54;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"69\" style=\"vertical-align: -1px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a912c81c2c9126631cd93019135608b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#53;&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"66\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6c836d13f59384472671485a7041e1a8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#113;&#125;&#94;&#123;&#50;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#113;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"57\" style=\"vertical-align: -6px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836323142\">\n<div data-type=\"problem\" id=\"fs-id1167833245761\">\n<p id=\"fs-id1167833245763\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a9be1d501ba47befc6f24d692426ec50_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#50;&#112;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"66\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-08d4f76e5b771a50d14105eccaeb5148_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#53;&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"57\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2c819deee8169dfbc0721efea120f09c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#53;&#125;&#109;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"71\" style=\"vertical-align: -6px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1167836549924\">\n<p id=\"fs-id1167836549926\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3223787cefadca6a859db9cba12f7b53_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#112;&#45;&#49;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"68\" style=\"vertical-align: -4px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-991d707d47a4dafcb60ca36c43ece52d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"63\" style=\"vertical-align: -7px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-57dfe7d433af1e0b1f4b0a21b11b6328_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#109;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#53;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"69\" style=\"vertical-align: -7px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830077402\">\n<div data-type=\"problem\" id=\"fs-id1167830077404\">\n<p id=\"fs-id1167830077406\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ada5f9d39829c2cab7e1bd49f2a78d40_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#113;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#113;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"55\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f5555f0f008554627df5f92d3c40f1c1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#55;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"58\" style=\"vertical-align: 0px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f8f77b5ca4684f35bfbacc953680bda0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"62\" style=\"vertical-align: -6px;\" \/><\/div>\n<\/div>\n<p id=\"fs-id1167833061042\"><strong data-effect=\"bold\">Solve Quadratic Equations of the form <em data-effect=\"italics\">x<\/em><sup>2<\/sup> + <em data-effect=\"italics\">bx<\/em> + <em data-effect=\"italics\">c<\/em> = 0 by Completing the Square<\/strong><\/p>\n<p id=\"fs-id1167836611918\">In the following exercises, solve by completing the square.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167825702838\">\n<div data-type=\"problem\" id=\"fs-id1167825702840\">\n<p id=\"fs-id1167825702842\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f7159d8c47ff013df6a3f7421c3a5286_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#117;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#117;&#61;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"91\" 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-319baa9a7708d06c3e34c41ef773211a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#61;&#45;&#51;&#44;&#117;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"106\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836349336\">\n<div data-type=\"problem\" id=\"fs-id1167836288654\">\n<p id=\"fs-id1167836288656\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-37ea66ade52554891d62d8442dba5adc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#122;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#50;&#122;&#61;&#45;&#49;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"120\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832982349\">\n<div data-type=\"problem\" id=\"fs-id1167832982351\">\n<p id=\"fs-id1167836704159\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-def2950447563bb2f43c4dda926cef46_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#48;&#120;&#61;&#50;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"108\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836320064\">\n<p id=\"fs-id1167836320066\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-10237f6ccf323fdbc53d634caf57512c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#49;&#44;&#120;&#61;&#50;&#49;\" 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-id1167836685027\">\n<div data-type=\"problem\" id=\"fs-id1167836685029\">\n<p id=\"fs-id1167836685032\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a5aae032cd9f60f64da151718e23294a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#121;&#61;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"90\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836536608\">\n<div data-type=\"problem\" id=\"fs-id1167824648900\">\n<p id=\"fs-id1167824648902\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-63b21a410a2c9e78abbc98f1b10d8676_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#109;&#61;&#45;&#52;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"125\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836511692\">\n<p id=\"fs-id1167833081892\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cfaee721e928ae7f8e9c818117181eda_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#45;&#50;&plusmn;&#50;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#48;&#125;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"109\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836322851\">\n<div data-type=\"problem\" id=\"fs-id1167836624579\">\n<p id=\"fs-id1167836624581\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-93e49db54d2d8fdcafe8cef70f52b362_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#110;&#61;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"106\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833021153\">\n<div data-type=\"problem\" id=\"fs-id1167833021155\">\n<p id=\"fs-id1167826171748\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7b63568b85f72536d728e98afbda6879_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#114;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#114;&#61;&#45;&#49;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"110\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833274539\">\n<p id=\"fs-id1167833274542\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-162a64217b88c20c08087b689176f063_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#114;&#61;&#45;&#51;&plusmn;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#125;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"84\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833024011\">\n<div data-type=\"problem\" id=\"fs-id1167833381494\">\n<p id=\"fs-id1167833381496\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-248770372a8ade32939776182b830ef6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#116;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#52;&#116;&#61;&#45;&#53;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"116\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826077597\">\n<div data-type=\"problem\" id=\"fs-id1167833334791\">\n<p id=\"fs-id1167833334794\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c1a81a72e4c3d00964406b1332ece8d8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#48;&#97;&#61;&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"111\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167836579068\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8e64a3e2b94d2bcff5d680ec97267024_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&#53;&plusmn;&#50;&#92;&#115;&#113;&#114;&#116;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"75\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836717039\">\n<div data-type=\"problem\" id=\"fs-id1167836717041\">\n<p id=\"fs-id1167836693277\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-28855bc112b5ebad991c74777e9b4cb2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#98;&#61;&#52;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"94\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836295347\">\n<div data-type=\"problem\" id=\"fs-id1167829597930\">\n<p id=\"fs-id1167829597932\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c5c0a92c7de4da3b610153146bd2d49d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#53;&#120;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"90\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829717765\">\n<p id=\"fs-id1167836520796\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8ab06c770da1ad3e434b7ceee8d818bb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#50;&#125;&plusmn;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#51;&#125;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"86\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829752498\">\n<div data-type=\"problem\" id=\"fs-id1167833377155\">\n<p id=\"fs-id1167833377157\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-85c1151fd974e956e276cac87eecc19f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#121;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"89\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833338934\">\n<div data-type=\"problem\" id=\"fs-id1167833338936\">\n<p id=\"fs-id1167833338938\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e127bde9fa67d1770706b6c46b95bc2f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#117;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#52;&#117;&#43;&#49;&#50;&#61;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"152\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833378999\">\n<p id=\"fs-id1167836516560\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9ed0e64ddfd01e2187d40b1daa9fd033_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#61;&#49;&#44;&#117;&#61;&#49;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"102\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167829739306\">\n<p id=\"fs-id1167829930891\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4b8df1e6757b148ff472b5cd49398386_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#122;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#122;&#45;&#53;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"118\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833060172\">\n<div data-type=\"problem\" id=\"fs-id1167829861776\">\n<p id=\"fs-id1167829861778\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-db69e9bb43332ccf78a79abb9ff79b72_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#114;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#114;&#45;&#51;&#61;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"118\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836534005\">\n<p id=\"fs-id1167836534007\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-495000bf1c57c0962970a5cd97e32bb6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#114;&#61;&#45;&#50;&#44;&#114;&#61;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"104\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836520251\">\n<div data-type=\"problem\" id=\"fs-id1167836520253\">\n<p id=\"fs-id1167836520255\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c728f7284fad6246d2607f994c051c93_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#116;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#48;&#116;&#45;&#54;&#61;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"122\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836528272\">\n<div data-type=\"problem\" id=\"fs-id1167836528275\">\n<p id=\"fs-id1167829593604\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-44c438af58522ff7a9ac387062130cb2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#94;&#123;&#50;&#125;&#61;&#57;&#118;&#43;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"88\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829747213\">\n<p id=\"fs-id1167829747215\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4a342c5bf77d86beda7e7177caa7c978_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#125;&#123;&#50;&#125;&plusmn;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#56;&#57;&#125;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"72\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829809842\">\n<div data-type=\"problem\" id=\"fs-id1167836788437\">\n<p id=\"fs-id1167836788439\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1e66a13e754cb27e9ee019ce3a95fc86_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#119;&#125;&#94;&#123;&#50;&#125;&#61;&#53;&#119;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"96\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833025559\">\n<div data-type=\"problem\" id=\"fs-id1167833025561\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bda0cbeb1f2fa25fc52960683df5e361_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#61;&#49;&#48;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"100\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829742112\">\n<p id=\"fs-id1167829742114\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-68b2faefd3e40d0c24f2cde2e6625050_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#53;&plusmn;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"75\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833396393\">\n<div data-type=\"problem\" id=\"fs-id1167833263810\">\n<p id=\"fs-id1167833263812\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9c3660366321676bab080f82d63a2797_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#52;&#61;&#54;&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"98\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833085342\">\n<div data-type=\"problem\" id=\"fs-id1167833085344\">\n<p id=\"fs-id1167833085346\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-af1a013c76b8e55bc883911055cc4bec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"143\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836502743\">\n<p id=\"fs-id1167836502745\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ddc43a69586489087b496e2b49c294ae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#55;&#44;&#120;&#61;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"107\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833051595\">\n<div data-type=\"problem\" id=\"fs-id1167836774990\">\n<p id=\"fs-id1167836774992\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-67450135ea5e905356850bc0c37ca971_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#56;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"151\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836706352\">\n<div data-type=\"problem\" id=\"fs-id1167836706354\">\n<p id=\"fs-id1167836706356\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-de00bb87edf3adf182e9db06c8959c98_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"143\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836685173\">\n<p id=\"fs-id1167836685175\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8d66adcf0855e757500df1008f0dc299_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#53;&#44;&#120;&#61;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"120\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167836349281\">\n<p id=\"fs-id1167829686605\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-81cfd9751b95964baf4a946fd10a081a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"142\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p><strong data-effect=\"bold\">Solve Quadratic Equations 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 by Completing the Square<\/strong><\/p>\n<p id=\"fs-id1167836704842\">In the following exercises, solve by completing the square.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167836554015\">\n<div data-type=\"problem\" id=\"fs-id1167836554017\">\n<p id=\"fs-id1167836554019\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f197ed6d3b44010719730ab427256749_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#48;&#109;&#45;&#50;&#55;&#61;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"159\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167836319949\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6f4bd4c1b171c928f15f1d6d8171fba2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#45;&#49;&#49;&#44;&#109;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"126\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167833224435\">\n<p id=\"fs-id1167833054082\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-de686bc9703812c484a7e09d95b60c05_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#52;&#120;&#43;&#49;&#50;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"148\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836558013\">\n<div data-type=\"problem\" id=\"fs-id1167836558015\">\n<p id=\"fs-id1167836756892\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-175eea2bd7e40182295cb6420e1ee677_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#110;&#61;&#50;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"110\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167830122893\">\n<p id=\"fs-id1167832927598\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-060214c09cbec8e8adcf0f1d26267ee4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#61;&#49;&plusmn;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"76\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833334776\">\n<div data-type=\"problem\" id=\"fs-id1167833334778\">\n<p id=\"fs-id1167833334780\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-287bfb0358ddc1090a7e87977a960cb6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#48;&#120;&#61;&#49;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"117\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836599358\">\n<div data-type=\"problem\" id=\"fs-id1167836599360\">\n<p id=\"fs-id1167836599363\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-81b3db042d2bb84fa0f30ef18290d475_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#123;&#99;&#125;&#94;&#123;&#50;&#125;&#43;&#99;&#61;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"86\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836601206\">\n<p id=\"fs-id1167836601208\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-513bcb1ca414eccaaf06abd563e082f7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;&#61;&#45;&#50;&#44;&#99;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"102\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829714659\">\n<div data-type=\"problem\" id=\"fs-id1167829714662\">\n<p id=\"fs-id1167829714664\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2660920564f6d3aff15d50bcf804e6fc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#123;&#100;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#100;&#61;&#49;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"106\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833138092\">\n<div data-type=\"problem\" id=\"fs-id1167836536049\">\n<p id=\"fs-id1167836536051\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d4b361f16801b2d22dccf4f66d02929e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#55;&#120;&#45;&#49;&#53;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"139\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836523515\">\n<p id=\"fs-id1167836523518\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cd8c698e512dc8d2a7a70e4770957bb0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#53;&#44;&#120;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#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-id1167836542019\">\n<div data-type=\"problem\" id=\"fs-id1167836542021\">\n<p id=\"fs-id1167829833202\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6b3a64da69c36e4a45c8058ee61b6b50_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#52;&#120;&#43;&#56;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"139\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836524254\">\n<div data-type=\"problem\" id=\"fs-id1167836524256\">\n<p id=\"fs-id1167833227114\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d5e208ea959d2cac0f4977f3899aef25_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#43;&#55;&#112;&#61;&#49;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"107\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829719417\">\n<p id=\"fs-id1167829952795\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3727c6b9c6caa27a2e61d8509cbedcd7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#112;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#52;&#125;&plusmn;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#54;&#49;&#125;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"93\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836543379\">\n<div data-type=\"problem\" id=\"fs-id1167836543381\">\n<p id=\"fs-id1167836543383\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d6af53c3a95d7708249cee5f1dcd3f36_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#123;&#113;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#113;&#61;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"97\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836731358\">\n<div data-type=\"problem\" id=\"fs-id1167836662626\">\n<p id=\"fs-id1167836662628\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f60474411ce448ee29146262f5be144a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#120;&#61;&#45;&#49;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"123\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167829783759\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a496cf15759b03b7ab0a7be26fe3b867_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#49;&#48;&#125;&plusmn;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#57;&#49;&#125;&#125;&#123;&#49;&#48;&#125;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"93\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167824578910\">\n<div data-type=\"problem\" id=\"fs-id1167824578912\">\n<p id=\"fs-id1167824578914\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4d509d678054b50059dca377f358ecc1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#120;&#61;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"114\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"writing\" data-depth=\"2\" id=\"fs-id1167836619200\">\n<h4 data-type=\"title\">Writing Exercises<\/h4>\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167833274696\">\n<p>Solve the equation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-84df8870330ae368b893ca6bfe3dfebb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#48;&#120;&#61;&#45;&#50;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"122\" style=\"vertical-align: -2px;\" \/><\/p>\n<p id=\"fs-id1167836665108\"><span class=\"token\">\u24d0<\/span> by using the Square Root Property<\/p>\n<p id=\"fs-id1167829753427\"><span class=\"token\">\u24d1<\/span> by Completing the Square<\/p>\n<p id=\"fs-id1167829753433\"><span class=\"token\">\u24d2<\/span> Which method do you prefer? Why?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836310437\">\n<p id=\"fs-id1167836310440\">Answers will vary.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836730436\">\n<div data-type=\"problem\" id=\"fs-id1167836730438\">\n<p id=\"fs-id1167836730440\">Solve the equation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-00d02a42ecf152897cca90857aaa655e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#56;&#121;&#61;&#52;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"99\" style=\"vertical-align: -4px;\" \/> by completing the square and explain all your steps.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167832945882\">\n<h4 data-type=\"title\">Self Check<\/h4>\n<p id=\"fs-id1167829827835\"><span class=\"token\">\u24d0<\/span> After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167833290114\" 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 complete the square of a binomial expression.\u201d \u201cConfidently,\u201d \u201cwith some help,\u201d or \u201cNo, I don\u2019t get it.\u201d Choose how would you respond to the statement \u201cI can solve quadratic equations of the form x squared plus b times x plus c equals 0 by completing the square.\u201d \u201cConfidently,\u201d \u201cwith some help,\u201d or \u201cNo, I don\u2019t get it.\u201d Choose how would you respond to the statement \u201cI can solve quadratic equations of the form a times x squared plus b times x plus c equals 0 by completing the square.\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_02_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 complete the square of a binomial expression.\u201d \u201cConfidently,\u201d \u201cwith some help,\u201d or \u201cNo, I don\u2019t get it.\u201d Choose how would you respond to the statement \u201cI can solve quadratic equations of the form x squared plus b times x plus c equals 0 by completing the square.\u201d \u201cConfidently,\u201d \u201cwith some help,\u201d or \u201cNo, I don\u2019t get it.\u201d Choose how would you respond to the statement \u201cI can solve quadratic equations of the form a times x squared plus b times x plus c equals 0 by completing the square.\u201d \u201cConfidently,\u201d \u201cwith some help,\u201d or \u201cNo, I don\u2019t get it.\u201d\" \/><\/span><\/p>\n<p id=\"fs-id1167833290123\"><span class=\"token\">\u24d1<\/span> After reviewing this checklist, what will you do to become confident for all objectives?<\/p>\n<\/div>\n<\/div>\n","protected":false},"author":103,"menu_order":3,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-3817","chapter","type-chapter","status-publish","hentry"],"part":3677,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/3817","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\/3817\/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\/3817\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/media?parent=3817"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapter-type?post=3817"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/contributor?post=3817"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/license?post=3817"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}