{"id":3909,"date":"2018-12-11T13:58:30","date_gmt":"2018-12-11T18:58:30","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/solve-quadratic-equations-using-the-quadratic-formula\/"},"modified":"2018-12-11T13:58:30","modified_gmt":"2018-12-11T18:58:30","slug":"solve-quadratic-equations-using-the-quadratic-formula","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/solve-quadratic-equations-using-the-quadratic-formula\/","title":{"raw":"Solve Quadratic Equations Using the Quadratic Formula","rendered":"Solve Quadratic Equations Using the Quadratic Formula"},"content":{"raw":"\n[latexpage]<div class=\"textbox textbox--learning-objectives\"><h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>By the end of this section, you will be able to: <ul><li>Solve quadratic equations using the Quadratic Formula<\/li><li>Use the discriminant to predict the number and type of solutions of a quadratic equation<\/li><li>Identify the most appropriate method to use to solve a quadratic equation<\/li><\/ul><\/div><div data-type=\"note\" id=\"fs-id1167830093627\" class=\"be-prepared\"><p>Before you get started, take this readiness quiz.<em data-effect=\"italics\"><\/em><\/p><ol id=\"fs-id1167832999312\" type=\"1\"><li>Evaluate \\({b}^{2}-4ab\\) when \\(a=3\\) and \\(b=-2.\\)<div data-type=\"newline\"><br><\/div> If you missed this problem, review <a href=\"\/contents\/05eab039-6d1c-4d80-8c8c-94469164a52c#fs-id1167832053133\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><li>Simplify: \\(\\sqrt{108}.\\)<div data-type=\"newline\"><br><\/div> If you missed this problem, review <a href=\"\/contents\/dbb319b9-f421-4570-9953-ca5a39b933dc#fs-id1169144556679\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><li>Simplify: \\(\\sqrt{50}.\\)<div data-type=\"newline\"><br><\/div> If you missed this problem, review <a href=\"\/contents\/916a2094-3b51-4f1a-803d-95909a359123#fs-id1169145730176\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><\/ol><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167836664862\"><h3 data-type=\"title\">Solve Quadratic Equations Using the Quadratic Formula<\/h3><p id=\"fs-id1167836685641\">When we solved quadratic equations in the last section by completing the square, we took the same steps every time. By the end of the exercise set, you may have been wondering \u2018isn\u2019t there an easier way to do this?\u2019 The answer is \u2018yes\u2019. Mathematicians look for patterns when they do things over and over in order to make their work easier. In this section we will derive and use a formula to find the solution of a quadratic equation.<\/p><p>We have already seen how to solve a formula for a specific variable \u2018in general\u2019, so that we would do the algebraic steps only once, and then use the new formula to find the value of the specific variable. Now we will go through the steps of completing the square using the general form of a quadratic equation to solve a quadratic equation for <em data-effect=\"italics\">x.<\/em><\/p><p id=\"fs-id1167833338980\">We start with the standard form of a quadratic equation and solve it for <em data-effect=\"italics\">x<\/em> by completing the square.<\/p><table id=\"fs-id1167833004917\" class=\"unnumbered unstyled can-break\" summary=\"To develop the Quadratic Formula, start with the standard form of a quadratic equation, a times x squared plus b times x plus c equals 0. Remember that a is not equal to zero. Isolate the variable terms on one side. The new equation is a times x squared plus b x equals negative c. Make the leading coefficient 1 by dividing both sides of the equation by a. We now have the quotient a times x squared divided by a plus the product of the quotient b divided by a and x equals negative c divided by a. Simplified, this becomes x squared plus b divided by a times x equals negative c divided by a. Complete the square on the left side of the equation. Find the square of one half times the quotient b divided by a which simplifies to b squared divided by the product 4 times a squared. Add this value to both sides of the equation. X squared plus b divided by a times x plus b squared divided by the product 4 a squared equals negative c divided by a plus the quotient b squared divided by the product 4 times a squared. Factor the perfect square trinomial on the left side of the equation. The square of x plus the quotient b divided by 2 a equals the quotient negative c divided by a plus the quotient b squared divided by the product 4 times a squared. Find the common denominator of the right side of the equation and write equivalent fractions using the common denominator. Multiply the term negative c divided by a on the right side of the equation by the fraction 4 a divided by 4 a. Rearrange terms on the right side of the equation, and it becomes the square of the sum x plus the quotient b divided by 2 a equals the quotient b squared divided by the product 4 times a squared plus the quotient negative c times 4 a divided by a times 4 a. Combining to one fraction, the square of the sum x plus the quotient b divided by 2 a equals the quotient of the difference b squared minus 4 a c divided by 4 a squared. Use the Square Root Property. X plus the quotient b divded by 2 a equals the positive or negative square root of the quotient of the difference b squared minus 4 a c divided by 4 a squared. Simplify the radical. X plus the quotient b divded by 2 a equals the positive or negative quotient of the square root of the difference b squared minus 4 a c divided by 2 a. Add negative b divided by 2 a to both sides of the equation. So x equals negative b divided by the product 2 a plus or minus the quotient of the square root of the difference b squared minus 4 a c divided by 2 a. Combine the terms on the right side of the equation to get the final form of the Quadratic Formula. X equals the quotient negative b plus or minus the square root of the difference b squared minus 4 a c divided by 2 a.\" 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_03_001a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Isolate the variable terms on one side.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167829942519\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_001b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Make the coefficient of \\({x}^{2}\\) equal to 1, by<div data-type=\"newline\"><br><\/div>dividing by <em data-effect=\"italics\">a<\/em>.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836580108\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_001c_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=\"center\"><span data-type=\"media\" id=\"fs-id1167824736027\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_001d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">To complete the square, find \\({\\left(\\frac{1}{2}\u00b7\\frac{b}{a}\\right)}^{2}\\) and add it to both sides of the equation.<\/td><td><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">\\({\\left(\\frac{1}{2}\\frac{b}{a}\\right)}^{2}=\\frac{{b}^{2}}{4{a}^{2}}\\)<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836690676\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_001e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">The left side is a perfect square, factor it.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167833052027\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_001f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Find the common denominator of the right<div data-type=\"newline\"><br><\/div>side and write equivalent fractions with<div data-type=\"newline\"><br><\/div>the common denominator.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167829620906\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_001g_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=\"center\"><span data-type=\"media\" id=\"fs-id1167829619193\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_001h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Combine to one fraction.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167833237760\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_001i_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-id1167836613501\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_001j_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\" id=\"fs-id1167836713990\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_001k_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Add \\(-\\frac{b}{2a}\\) to both sides of the equation.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167833158651\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_001l_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Combine the terms on the right side.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167833321935\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_001m_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td><\/td><td data-valign=\"top\" data-align=\"left\">This equation is the Quadratic Formula.<\/td><\/tr><\/tbody><\/table><div data-type=\"note\"><div data-type=\"title\">Quadratic Formula<\/div><p id=\"fs-id1167836295293\">The solutions to a <span data-type=\"term\" class=\"no-emphasis\">quadratic equation<\/span> 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, where \\(a\\ne 0\\) are given by the formula:<\/p><div data-type=\"equation\" id=\"fs-id1167829924519\" class=\"unnumbered\" data-label=\"\">\\(x=\\frac{\\text{\u2212}b\u00b1\\sqrt{{b}^{2}-4ac}}{2a}\\)<\/div><\/div><p id=\"fs-id1167836554536\">To use the <span data-type=\"term\" class=\"no-emphasis\">Quadratic Formula<\/span>, we substitute the values of <em data-effect=\"italics\">a<\/em>, <em data-effect=\"italics\">b<\/em>, and <em data-effect=\"italics\">c<\/em> from the standard form into the expression on the right side of the formula. Then we simplify the expression. The result is the pair of solutions to the quadratic equation.<\/p><p id=\"fs-id1167832945838\">Notice the formula is an equation. Make sure you use both sides of the equation.<\/p><div data-type=\"example\" id=\"fs-id1167824740960\" class=\"textbox textbox--examples\"><div data-type=\"title\">How to Solve a Quadratic Equation Using the Quadratic Formula<\/div><div data-type=\"exercise\" id=\"fs-id1167836497688\"><div data-type=\"problem\" id=\"fs-id1167829890840\"><p>Solve by using the Quadratic Formula: \\(2{x}^{2}+9x-5=0.\\)<\/p><\/div><div data-type=\"solution\"><span data-type=\"media\" id=\"fs-id1167829714248\" data-alt=\"Step 1 is to write the quadratic equation in standard form, a times x squared plus b x plus c equals zero, and identify the values a, b, and c. The equation 2 x squared plus 9 x minus 5 equals zero is in standard form. A equals 2, b equals 9, and c equals negative 5.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_002a_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 1 is to write the quadratic equation in standard form, a times x squared plus b x plus c equals zero, and identify the values a, b, and c. The equation 2 x squared plus 9 x minus 5 equals zero is in standard form. A equals 2, b equals 9, and c equals negative 5.\"><\/span><span data-type=\"media\" data-alt=\"Step 2. Write the quadratic formula. Then substitute the values of a, b, and c. Substitute a equals 2, b equals 9, and c equals negative 5 into the equation x equals the quotient negative b plus or minus the square root of the difference b squared minus 4 a c divided by 2 a. So x equals the quotient negative 9 plus or minus the square root of the difference 9 squared minus the product 4 times 2 times negative 5 divided by the product 2 times 2.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_002b_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2. Write the quadratic formula. Then substitute the values of a, b, and c. Substitute a equals 2, b equals 9, and c equals negative 5 into the equation x equals the quotient negative b plus or minus the square root of the difference b squared minus 4 a c divided by 2 a. So x equals the quotient negative 9 plus or minus the square root of the difference 9 squared minus the product 4 times 2 times negative 5 divided by the product 2 times 2.\"><\/span><span data-type=\"media\" id=\"fs-id1167832926037\" data-alt=\"In step 3, simplify the fraction and solve for x. x equals the quotient negative 9 plus or minus the square root of the difference 81 minus negative 40 divided by 4. Simplify the radicand. x equals the quotient negative 9 plus or minus the square root of 121 divided by 4. Simplify the square root. x equals the quotient negative 9 plus or minus 11 divided by 4. Separate into two equations. The first equation is x equals the quotient negative 9 plus 11 divided by 4 which simplifies to 2 divided by 4. The first solution is x equals one half. The second equation is x equals the quotient negative 9 minus 11 divided by 4 which simplifies to negative 20 divided by 4. The second solution is x equals negative 5.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_002c_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"In step 3, simplify the fraction and solve for x. x equals the quotient negative 9 plus or minus the square root of the difference 81 minus negative 40 divided by 4. Simplify the radicand. x equals the quotient negative 9 plus or minus the square root of 121 divided by 4. Simplify the square root. x equals the quotient negative 9 plus or minus 11 divided by 4. Separate into two equations. The first equation is x equals the quotient negative 9 plus 11 divided by 4 which simplifies to 2 divided by 4. The first solution is x equals one half. The second equation is x equals the quotient negative 9 minus 11 divided by 4 which simplifies to negative 20 divided by 4. The second solution is x equals negative 5.\"><\/span><span data-type=\"media\" id=\"fs-id1167836729568\" data-alt=\"The fourth, and final, step is to check the solution. Put each answer into the original equation to check. First, substitute x equals one half into the original equation, 2 x squared plus 9 x minus 5 equals 0. This yields 2 times the square of one half plus nine times one half minus 5. We need to show that this expression equals 0. Simplify the square. 2 times one fourth plus nine times one half minus 5 equals one half plus 9 halves minus 5, or 10 halves minus 5. 5 minus 5 equals 0, so x equals one half is indeed a solution. Next substitute x = negative 5 into the equation 2 x squared plus 9 x minus 5 equals 0. This yields 2 times the square of negative 5 plus 9 times negative 5 minus 5. We need to show that this expression equals 0. Simplify the square. 2 times 25 plus nine times negative 5 minus 5 equals 50 minus 45 minus 5, or 0. x equals negative 5 is a solution as well.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_002d_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The fourth, and final, step is to check the solution. Put each answer into the original equation to check. First, substitute x equals one half into the original equation, 2 x squared plus 9 x minus 5 equals 0. This yields 2 times the square of one half plus nine times one half minus 5. We need to show that this expression equals 0. Simplify the square. 2 times one fourth plus nine times one half minus 5 equals one half plus 9 halves minus 5, or 10 halves minus 5. 5 minus 5 equals 0, so x equals one half is indeed a solution. Next substitute x = negative 5 into the equation 2 x squared plus 9 x minus 5 equals 0. This yields 2 times the square of negative 5 plus 9 times negative 5 minus 5. We need to show that this expression equals 0. Simplify the square. 2 times 25 plus nine times negative 5 minus 5 equals 50 minus 45 minus 5, or 0. x equals negative 5 is a solution as well.\"><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836754953\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167833052817\"><div data-type=\"problem\" id=\"fs-id1167836699334\"><p id=\"fs-id1167829811483\">Solve by using the Quadratic Formula: \\(3{y}^{2}-5y+2=0\\).<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829755644\"><p>\\(y=1,y=\\frac{2}{3}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167833377487\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836530282\"><div data-type=\"problem\" id=\"fs-id1167833018768\"><p id=\"fs-id1167833020111\">Solve by using the Quadratic Formula: \\(4{z}^{2}+2z-6=0\\).<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167830093362\"><p>\\(z=1,z=-\\frac{3}{2}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167829924562\" class=\"howto\"><div data-type=\"title\">Solve a quadratic equation using the quadratic formula.<\/div><ol type=\"1\" class=\"stepwise\"><li>Write the quadratic equation in standard form, <em data-effect=\"italics\">ax<\/em><sup>2<\/sup> + <em data-effect=\"italics\">bx<\/em> + <em data-effect=\"italics\">c<\/em> = 0. Identify the values of <em data-effect=\"italics\">a<\/em>, <em data-effect=\"italics\">b<\/em>, and <em data-effect=\"italics\">c<\/em>.<\/li><li>Write the Quadratic Formula. Then substitute in the values of <em data-effect=\"italics\">a<\/em>, <em data-effect=\"italics\">b<\/em>, and <em data-effect=\"italics\">c<\/em>.<\/li><li>Simplify.<\/li><li>Check the solutions.<\/li><\/ol><\/div><p id=\"fs-id1167833008091\">If you say the formula as you write it in each problem, you\u2019ll have it memorized in no time! And remember, the Quadratic Formula is an EQUATION. Be sure you start with \u201c<em data-effect=\"italics\">x<\/em> =\u201d.<\/p><div data-type=\"example\" id=\"fs-id1167836717133\" class=\"textbox textbox--examples\"><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167829711896\"><p id=\"fs-id1167824735584\">Solve by using the Quadratic Formula: \\({x}^{2}-6x=-5.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829685914\"><table id=\"fs-id1167836407290\" class=\"unnumbered unstyled\" summary=\"Write the equation x squared minus 6 x equals negative 5 in standard form by adding 5 to both sides of the equation. X squared minus 6 x plus 5 equals 0. Identify the values of a, b, and c. The coefficient of x squared is a = 1. The coefficient of x is b equals negative 6. The constant term is c equals 5. Write the quadratic formula, x equals the quotient negative b plus or minus the square root of the difference b squared minus 4 a c divided by 2 a. Then substitute the values for a, b, and c. x equals the quotient of the expression the opposite of negative 6 plus or minus the square root of the difference negative 6 squared minus the product 4 times 1 times 5 divided by the product 2 times 1. Simplify. X equals the quotient of the expression 6 plus or minus the square root of the difference 36 minus 20 divided by 2. This further simplifies to the quotient of 6 plus or minus square root 16 and 2, so x equals the quotient of 6 plus or minus 4 and 2. Rewrite to show two solutions The first is x equals the quotient 6 plus 4 divided by 2, or 10 divided by 2 which equals 5. The second solution is the quotient 6 minus 4 divided by 2, or 2 divided by 2 which equals 1. Check the solutions in the original equation. Substitute x equals 5 into the original equation, x squared minus 6 x plus 5 equals zero to get 5 squared minus 6 times 5 plus 5 on the left side of the equation we must show that this equals 0. Simplifying the expression yields 25 minus 30 plus 5, or 0. So x equals 5 is a solution. Next check x equals 1 in the original equation. X squared minus 6 x plus 5 becomes 1 squared minus 6 times 1 plus 5. We must show this equals 0. 1 minus 6 plus 5 does equal 0, so x = 1 is a solution.\" data-label=\"\"><tbody><tr><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836573724\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_003b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Write the equation in standard form by adding<div data-type=\"newline\"><br><\/div>5 to each side.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829850979\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_003c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">This equation is now in standard form.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836450487\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_003d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Identify the values of \\(a,\\text{\\hspace{0.17em}}\\text{\\hspace{0.17em}}b,\\text{\\hspace{0.17em}}\\text{\\hspace{0.17em}}c.\\)<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829739762\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_003e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Write the Quadratic Formula.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836521083\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_003f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Then substitute in the values of \\(a,\\text{\\hspace{0.17em}}\\text{\\hspace{0.17em}}b,\\text{\\hspace{0.17em}}\\text{\\hspace{0.17em}}c.\\)<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829811936\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_003g_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-id1167824735884\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_003h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><div data-type=\"newline\"><br><\/div><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_003i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167829685990\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_003j_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-id1167833397031\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_003k_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-id1167824732532\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_003l_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-id1167829753751\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_003m_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-id1167833379118\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_003a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span> <\/td><td><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" class=\"try\"><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167829627475\"><p id=\"fs-id1167833142563\">Solve by using the Quadratic Formula: \\({a}^{2}-2a=15\\).<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829714558\"><p id=\"fs-id1167829695255\">\\(a=-3,a=5\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167829878093\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836314565\"><div data-type=\"problem\"><p>Solve by using the Quadratic Formula: \\({b}^{2}+24=-10b\\).<\/p><\/div><div data-type=\"solution\"><p>\\(b=-6,b=-4\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1167836789050\">When we solved quadratic equations by using the Square Root Property, we sometimes got answers that had radicals. That can happen, too, when using the <span data-type=\"term\" class=\"no-emphasis\">Quadratic Formula<\/span>. If we get a <span data-type=\"term\" class=\"no-emphasis\">radical<\/span> as a solution, the final answer must have the radical in its simplified form.<\/p><div data-type=\"example\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167829747416\"><div data-type=\"problem\" id=\"fs-id1167836791948\"><p>Solve by using the Quadratic Formula: \\(2{x}^{2}+10x+11=0.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832950924\"><table id=\"fs-id1167826212241\" class=\"unnumbered unstyled\" summary=\"The equation 2x squared plus 10 x plus 11 equals 0 is already in standard form. Identify the values of a, b, and c. The coefficient of x squared is a = 2. The coefficient of x is b equals 10. The constant term is c equals 11. Write the quadratic formula, x equals the quotient negative b plus or minus the square root of the expression b squared minus 4 a c divided by 2 a. Then substitute the values for a, b, and c. x equals the quotient of the difference negative 10 plus or minus the square root of the difference 10 squared minus the product 4 times 2 times 11 divided by the product 2 times 2. Simplify. X equals the quotient of the expression negative 10 plus or minus the square root of the difference 100 minus 88 divided by 4. This further simplifies to the quotient negative 10 plus or minus square root 12 divided by 4. Simplify the radical. x equals the quotient negative 10 plus or minus 2 times the square root of 3 divided by 2. Factor out the common factor in the numerator. X equals the quotient of 2 times the expression negative 5 plus or minus 2 square root 3 and 4. Remove the common factor to yield x equals the quotient negative 5 plus or minus square root 3 divided by 2. Rewrite to show two solutions The first solution is x equals the quotient of negative 5 plus square root 3 and 2. The second is x equals the quotient of negative 5 minus square root 3 and 2. Remember to check the solutions in the original equation. We leave that to you!\" data-label=\"\"><tbody><tr><td><\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836656566\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_004a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">This equation is in standard form.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167829783604\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_004b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Identify the values of <em data-effect=\"italics\">a<\/em>, <em data-effect=\"italics\">b<\/em>, and <em data-effect=\"italics\">c<\/em>.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167830013853\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_004c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Write the Quadratic Formula.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836790447\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_004d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Then substitute in the values of <em data-effect=\"italics\">a<\/em>, <em data-effect=\"italics\">b<\/em>, and <em data-effect=\"italics\">c<\/em>.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167833021195\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_004e_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=\"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_03_004f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td><\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_004g_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\" id=\"fs-id1167836621035\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_004h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Factor out the common factor in the numerator.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167829695938\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_004i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Remove the common factors.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836560610\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_004j_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=\"center\"><span data-type=\"media\" id=\"fs-id1167836532009\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_004k_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> We leave the check for you!<\/td><td><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167829712766\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836554515\"><div data-type=\"problem\" id=\"fs-id1167836502385\"><p>Solve by using the Quadratic Formula: \\(3{m}^{2}+12m+7=0\\).<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167825660450\"><p id=\"fs-id1167833239738\">\\(m=\\frac{-6+\\sqrt{15}}{3},m=\\frac{-6-\\sqrt{15}}{3}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836601096\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167829853072\"><div data-type=\"problem\"><p id=\"fs-id1167836595665\">Solve by using the Quadratic Formula: \\(5{n}^{2}+4n-4=0\\).<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836492623\"><p id=\"fs-id1167836321837\">\\(n=\\frac{-2+2\\sqrt{6}}{5},n=\\frac{-2-2\\sqrt{6}}{5}\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1167836729254\">When we substitute <em data-effect=\"italics\">a<\/em>, <em data-effect=\"italics\">b<\/em>, and <em data-effect=\"italics\">c<\/em> into the Quadratic Formula and the <span data-type=\"term\" class=\"no-emphasis\">radicand<\/span> is negative, the quadratic equation will have imaginary or complex solutions. We will see this in the next example.<\/p><div data-type=\"example\" id=\"fs-id1167826162738\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167836553297\"><div data-type=\"problem\" id=\"fs-id1167836529273\"><p id=\"fs-id1167836539168\">Solve by using the Quadratic Formula: \\(3{p}^{2}+2p+9=0.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829752211\"><table class=\"unnumbered unstyled can-break\" summary=\"The equation 3 p squared plus 2 p plus 9 equals 0 is already in standard form. Identify the values of a, b, and c. The coefficient of p squared is a = 3. The coefficient of p is b equals 2. The constant term is c equals 9. Write the quadratic formula, p equals the quotient negative b plus or minus the square root of the difference b squared minus 4 a c divided by 2 a. Then substitute the values for a, b, and c. p equals the quotient of the expression negative 2 plus or minus the square root of the difference 4 squared minus the product 4 times 3 times 9 divided by the product 2 times 3. Simplify. P equals the quotient of the expression negative 2 plus or minus the square root of the difference 4 minus 108 divided by 6. This further simplifies to p equals the quotient negative 2 plus or minus the square root of negative 104 divided by 6. Simplify the radical using complex numbers. p equals the quotient negative 2 plus or minus square root 104 times I divided by 6. Simplify the radical. p equals the quotient negative 2 plus or minus 2 times square root 104 times I divided by 6. Factor the common factor in the numerator. P equals the quotient of 2 times the expression negative 1 plus or minus square root 26 times I divided by 6. Remove the common factor to yield p equals the quotient negative 1 plus or minus square root 26 times I divided by 3. Rewrite in standard a plus b I form. P equals negative one third plus or minus square root 26 divided by 3 times I. Write as show two solutions The first solution is p equals negative one third plus square root 26 thirds I. The second is p equals negative one third minus square root 26 thirds I.\" data-label=\"\"><tbody><tr><td><\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836630346\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_005a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">This equation is in standard form<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167833051387\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_005b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Identify the values of \\(a,b,c.\\)<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167829812037\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_005c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Write the Quadratic Formula.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836547551\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_005d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Then substitute in the values of \\(a,b,c\\).<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167829586709\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_005e_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=\"center\"><span data-type=\"media\" id=\"fs-id1167836417826\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_005f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td><\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_005g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Simplify the radical using complex numbers.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167826077082\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_005h_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\" id=\"fs-id1167836433941\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_005i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Factor the common factor in the numerator.<\/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_03_005j_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Remove the common factors.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836623945\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_005k_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Rewrite in standard \\(a+bi\\) form.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836547132\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_005l_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Write as two solutions.<\/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_03_005m_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167829906103\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167830123951\"><div data-type=\"problem\" id=\"fs-id1167836393407\"><p id=\"fs-id1167836487107\">Solve by using the Quadratic Formula: \\(4{a}^{2}-2a+8=0\\).<\/p><\/div><div data-type=\"solution\"><p id=\"fs-id1167825829977\">\\(a=\\frac{1}{4}+\\frac{\\sqrt{31}}{4}i,\\phantom{\\rule{0.2em}{0ex}}\\text{}\\phantom{\\rule{0.2em}{0ex}}a=\\frac{1}{4}-\\frac{\\sqrt{31}}{4}i\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167829906736\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836731793\"><div data-type=\"problem\" id=\"fs-id1167836635576\"><p id=\"fs-id1167833396817\">Solve by using the Quadratic Formula: \\(5{b}^{2}+2b+4=0\\).<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829879778\"><p id=\"fs-id1167829597268\">\\(b=-\\frac{1}{5}+\\frac{\\sqrt{19}}{5}i,\\phantom{\\rule{0.2em}{0ex}}\\text{}\\phantom{\\rule{0.2em}{0ex}}b=-\\frac{1}{5}-\\frac{\\sqrt{19}}{5}i\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1167836379181\">Remember, to use the Quadratic Formula, the equation must be written in standard form, <em data-effect=\"italics\">ax<\/em><sup>2<\/sup> + <em data-effect=\"italics\">bx<\/em> + <em data-effect=\"italics\">c<\/em> = 0. Sometimes, we will need to do some algebra to get the equation into standard form before we can use the Quadratic Formula.<\/p><div data-type=\"example\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167829686816\"><div data-type=\"problem\" id=\"fs-id1167836557051\"><p id=\"fs-id1167829721085\">Solve by using the Quadratic Formula: \\(x\\left(x+6\\right)+4=0.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829732080\"><p id=\"fs-id1167829783831\">Our first step is to get the equation in standard form.<\/p><table id=\"fs-id1167836525808\" class=\"unnumbered unstyled can-break\" summary=\"Distribute to rewrite the equation x times the sum of x and 6 plus 4 equals 0 in standard form. The equation becomes x squared plus 6 x plus 4 equals 0. Identify the values of a, b, and c. The coefficient of x squared is a = 1. The coefficient of x is b equals 6. The constant term is c equals 4. Write the quadratic formula, x equals the quotient negative b plus or minus the square root of the difference b squared minus 4 a c divided by 2 a. Then substitute the values for a, b, and c. x equals the quotient of the expression negative 6 plus or minus the square root of the difference 6 squared minus the product 4 times 1 times 4 divided by the product 2 times 1. Simplify. X equals the quotient of the expression negative 6 plus or minus the square root of the difference 36 minus 16 divided by 2. This further simplifies to the quotient of negative 6 plus or minus square root 20 and 2. Simplify the radical. X equals the quotient negative 6 plus or minus 2 square root 5 divided by 2. Factor the common factor in the numerator. X equals the quotient 2 times the expression negative 3 plus or minus 2 square root 5 divided by 2. Remove the common factor, and x equals negative 3 plus or minus 2 square root 5. Rewrite to show two solutions, x equals negative 3 plus 2 square root 5 and x equals negative 3 minus 2 square root 5. Remember to check the solutions in the original equation. We leave that to you!\" data-label=\"\"><tbody><tr><td><\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167829692551\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_006a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Distribute to get the equation in standard form.<\/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_03_006b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">This equation is now in standard form<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167833009856\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_006c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Identify the values of \\(a,b,c.\\)<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836573443\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_006d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Write the Quadratic Formula.<\/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_03_006e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Then substitute in the values of \\(a,b,c\\).<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167829785636\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_006f_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=\"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_03_006g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td><\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836574191\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_006h_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\" id=\"fs-id1167836731465\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_006i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Factor the common factor in the numerator.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167829908075\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_006j_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Remove the common factors.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836520614\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_006k_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Write as two solutions.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836629538\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_006l_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> We leave the check for you!<\/td><td><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167825824208\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836287145\"><div data-type=\"problem\" id=\"fs-id1167829826662\"><p id=\"fs-id1167829596853\">Solve by using the Quadratic Formula: \\(x\\left(x+2\\right)-5=0.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829828288\"><p id=\"fs-id1167836510782\">\\(x=-1+\\sqrt{6},x=-1-\\sqrt{6}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" class=\"try\"><div data-type=\"exercise\"><div data-type=\"problem\"><p id=\"fs-id1167829693355\">Solve by using the Quadratic Formula: \\(3y\\left(y-2\\right)-3=0.\\)<\/p><\/div><div data-type=\"solution\"><p id=\"fs-id1167829811978\">\\(y=1+\\sqrt{2},y=1-\\sqrt{2}\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1167826128252\">When we solved linear equations, if an equation had too many fractions we cleared the fractions by multiplying both sides of the equation by the LCD. This gave us an equivalent equation\u2014without fractions\u2014 to solve. We can use the same strategy with quadratic equations.<\/p><div data-type=\"example\" id=\"fs-id1167836331062\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167836532524\"><div data-type=\"problem\"><p>Solve by using the Quadratic Formula: \\(\\frac{1}{2}{u}^{2}+\\frac{2}{3}u=\\frac{1}{3}.\\)<\/p><\/div><div data-type=\"solution\"><p id=\"fs-id1167833025258\">Our first step is to clear the fractions.<\/p><table id=\"fs-id1167829810567\" class=\"unnumbered unstyled can-break\" summary=\"Write the original equation, one half u squared plus two thirds u equals one third. Multiply both sides of the equation by the LCD, 6, to clear the fractions. 6 times the sum one half u squared plus two thirds u equals 6 times one third. Multiply to yield 3 u squared plus 4 u equals 2. Subtract 2 from both sides of the equation to write it in standard form. 3 u squared plus 4 u minus 2 equals 0. Identify the values of a, b, and c. The coefficient of u squared is a = 3. The coefficient of u is b equals 4. The constant term is c equals negative 2. Write the quadratic formula, x equals the quotient negative b plus or minus the square root of the difference b squared minus 4 a c divided by 2 a. Then substitute the values for a, b, and c. u equals the quotient of the expression negative 4 plus or minus the square root of the difference 4 squared minus the product 4 times 3 times negative 2 divided by the product 2 times 3. Simplify. u equals the quotient of the expression negative 4 plus or minus the square root of the sum 16 plus 24 divided by 6. This further simplifies to the quotient of negative 4 plus or minus square root 40 and 6. Simplify the radical. U equals the quotient of the expression negative 4 plus or minus 2 square root 10 and 6. Factor the common factor in the numerator. U equals the quotient of 2 times the expression negative 2 plus or minus square root 10 and 6. Remove the common factor yielding u equals the quotient of negative 2 plus or minus square root 10 and 3. Rewrite to show two solutions. u equals the quotient negative 2 plus square root 10 divided by 3 and u equals the quotient negative 2 minus square root 10 divided by 3 Check the solutions in the original equation. We leave that to you!\" data-label=\"\"><tbody><tr><td><\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836575024\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_007a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Multiply both sides by the LCD, 6, to clear the fractions.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836389274\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_007b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Multiply.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167824735095\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_007c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Subtract 2 to get the equation in standard form.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836481420\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_007d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Identify the values of <em data-effect=\"italics\">a<\/em>, <em data-effect=\"italics\">b<\/em>, and <em data-effect=\"italics\">c<\/em>.<\/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_03_007e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Write the Quadratic Formula.<\/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_03_007f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Then substitute in the values of <em data-effect=\"italics\">a<\/em>, <em data-effect=\"italics\">b<\/em>, and <em data-effect=\"italics\">c<\/em>.<\/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_03_007g_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=\"center\"><span data-type=\"media\" id=\"fs-id1167829808815\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_007h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td><\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167832999050\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_007i_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\" id=\"fs-id1167833056373\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_007j_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Factor the common factor in the numerator.<\/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_03_007k_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Remove the common factors.<\/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_03_007l_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=\"center\"><span data-type=\"media\" id=\"fs-id1167836387084\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_007m_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> We leave the check for you!<\/td><td><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836511739\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836516031\"><div data-type=\"problem\"><p id=\"fs-id1167836524102\">Solve by using the Quadratic Formula: \\(\\frac{1}{4}{c}^{2}-\\frac{1}{3}c=\\frac{1}{12}\\).<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167824734634\"><p id=\"fs-id1167836530748\">\\(c=\\frac{2+\\sqrt{7}}{3},\\phantom{\\rule{0.2em}{0ex}}\\text{}\\phantom{\\rule{0.2em}{0ex}}c=\\frac{2-\\sqrt{7}}{3}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167829690058\" class=\"try\"><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167836629454\"><p id=\"fs-id1167836484657\">Solve by using the Quadratic Formula: \\(\\frac{1}{9}{d}^{2}-\\frac{1}{2}d=-\\frac{1}{3}\\).<\/p><\/div><div data-type=\"solution\"><p id=\"fs-id1167836545245\">\\(d=\\frac{9+\\sqrt{33}}{4},\\phantom{\\rule{0.2em}{0ex}}\\text{d}=\\frac{9-\\sqrt{33}}{4}\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1167832999563\">Think about the equation (<em data-effect=\"italics\">x<\/em> \u2212 3)<sup>2<\/sup> = 0. We know from the <span data-type=\"term\" class=\"no-emphasis\">Zero Product Property<\/span> that this equation has only one solution,<\/p><div data-type=\"newline\"><br><\/div><em data-effect=\"italics\">x<\/em> = 3.<p id=\"fs-id1167825830172\">We will see in the next example how using the <span data-type=\"term\" class=\"no-emphasis\">Quadratic Formula<\/span> to solve an equation whose standard form is a perfect square <span data-type=\"term\" class=\"no-emphasis\">trinomial<\/span> equal to 0 gives just one solution. Notice that once the radicand is simplified it becomes 0 , which leads to only one solution.<\/p><div data-type=\"example\" id=\"fs-id1167825830135\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167829689050\"><div data-type=\"problem\" id=\"fs-id1167836409726\"><p id=\"fs-id1167833025902\">Solve by using the Quadratic Formula: \\(4{x}^{2}-20x=-25.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836300298\"><table id=\"fs-id1167836619633\" class=\"unnumbered unstyled can-break\" summary=\"Write the equation 4 x squared minus 20 x equals negative 25 in standard form by adding 25 to both sides of the equation. 4 x squared minus 20 x plus 25 equals 0. Identify the values of a, b, and c. The coefficient of x squared is a = 4. The coefficient of x is b equals negative 20. The constant term is c equals 25. Write the quadratic formula, x equals the quotient negative b plus or minus the square root of the difference b squared minus 4 a c divided by 2 a. Then substitute the values for a, b, and c. x equals the quotient of the expression the opposite of negative 20 plus or minus the square root of the difference negative 20 squared minus the product 4 times 4 times 25 divided by the product 2 times 4. Simplify. X equals the quotient of the expression 20 plus or minus the square root of the difference 400 minus 400 divided by 8. This further simplifies to the quotient of 20 plus or minus 0 and 2, so x equals 20 divided by 8 or 5 halves. We leave it to you to check the solution in the original equation.\" data-label=\"\"><tbody><tr><td><\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167829695158\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_008a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Add 25 to get the equation in standard form.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836294437\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_008b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Identify the values of <em data-effect=\"italics\">a<\/em>, <em data-effect=\"italics\">b<\/em>, and <em data-effect=\"italics\">c<\/em>.<\/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_03_008c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Write the quadratic formula.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167826025420\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_008d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Then substitute in the values of <em data-effect=\"italics\">a<\/em>, <em data-effect=\"italics\">b<\/em>, and <em data-effect=\"italics\">c<\/em>.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167824763423\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_008e_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=\"center\"><span data-type=\"media\" id=\"fs-id1167829580517\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_008f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td><\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836511094\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_008g_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\" id=\"fs-id1167836362464\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_008h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Simplify the fraction.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167833082053\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_008i_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> We leave the check for you!<\/td><td><\/td><\/tr><\/tbody><\/table><p id=\"fs-id1167836701179\">Did you recognize that 4<em data-effect=\"italics\">x<\/em><sup>2<\/sup> \u2212 20<em data-effect=\"italics\">x<\/em> + 25 is a perfect square trinomial. It is equivalent to (2<em data-effect=\"italics\">x<\/em> \u2212 5)<sup>2<\/sup>? If you solve<\/p><div data-type=\"newline\"><br><\/div>4<em data-effect=\"italics\">x<\/em><sup>2<\/sup> \u2212 20<em data-effect=\"italics\">x<\/em> + 25 = 0 by factoring and then using the Square Root Property, do you get the same result?<\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836732791\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167829719841\"><div data-type=\"problem\" id=\"fs-id1167825913867\"><p id=\"fs-id1167836363237\">Solve by using the Quadratic Formula: \\({r}^{2}+10r+25=0.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167824734801\"><p id=\"fs-id1167832925269\">\\(r=-5\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167833081911\"><div data-type=\"problem\" id=\"fs-id1167829719514\"><p id=\"fs-id1167829807329\">Solve by using the Quadratic Formula: \\(25{t}^{2}-40t=-16.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832940220\"><p>\\(t=\\frac{4}{5}\\)<\/p><\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167836295270\"><h3 data-type=\"title\">Use the Discriminant to Predict the Number and Type of Solutions of a Quadratic Equation<\/h3><p id=\"fs-id1167836493452\">When we solved the quadratic equations in the previous examples, sometimes we got two real solutions, one real solution, and sometimes two complex solutions. Is there a way to predict the number and type of solutions to a quadratic equation without actually solving the equation?<\/p><p id=\"fs-id1167836625084\">Yes, the expression under the radical of the Quadratic Formula makes it easy for us to determine the number and type of solutions. This expression is called the <span data-type=\"term\">discriminant<\/span>.<\/p><div data-type=\"note\" id=\"fs-id1167836386687\"><div data-type=\"title\">Discriminant<\/div><span data-type=\"media\" id=\"fs-id1167836556733\" data-alt=\"In the Quadratic Formula, x equals the quotient of negative b plus or minus the square root of b squared minus 4 times a times c and 2 a, the value under the radical, b squared minus 4 times a times c, is called the discriminant.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_009_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"In the Quadratic Formula, x equals the quotient of negative b plus or minus the square root of b squared minus 4 times a times c and 2 a, the value under the radical, b squared minus 4 times a times c, is called the discriminant.\"><\/span><\/div><p id=\"fs-id1167826171745\">Let\u2019s look at the discriminant of the equations in some of the examples and the number and type of solutions to those quadratic equations.<\/p><table id=\"fs-id1167824764688\" class=\"unnumbered\" summary=\"This table has four columns and four rows. The first row is a header row and labels the columns \u201cQuadratic Equation (in standard form)\u201d, \u201cDiscriminant b squared minus 4 a c\u201d, \u201cValue of the Discriminant\u201d, and \u201cNumber and Type of solutions\u201d. The second row has the quadratic equation 2 x squared plus 9 x minus 5 equals 0. The discriminant is 9 squared minus the expression 4 times 2 times negative 5, which equals 121. The value of 121 is positive, and there are 2 real solutions. The third row has the quadratic equation 4 x squared minus 20 x plus 25 equals 0. The discriminant is negative twenty squared minus the expression 4 times 4 times 25, which equals 0. The value is 0, and there is 1 real solution. The fourth row has the quadratic equation 3 p squared plus 2 p plus 9 equals 0. The discriminant is 2 squared minus the expression 4 times 3 times 9, which equals negative 104. Th value of negative 104 is negative, and there are 2 complex solutions.\"><thead><tr><th data-valign=\"middle\" data-align=\"center\">Quadratic Equation<div data-type=\"newline\"><br><\/div>(in standard form)<\/th><th data-valign=\"middle\" data-align=\"center\">Discriminant<div data-type=\"newline\"><br><\/div>\\({b}^{2}-4ac\\)<\/th><th data-valign=\"middle\" data-align=\"center\">Value of the Discriminant<\/th><th data-valign=\"middle\" data-align=\"center\">Number and Type of solutions<\/th><\/tr><\/thead><tbody><tr><td data-valign=\"middle\" data-align=\"left\">\\(2{x}^{2}+9x-5=0\\)<\/td><td data-valign=\"middle\" data-align=\"center\">\\(\\begin{array}{c}\\hfill {9}^{2}-4\u00b72\\left(-5\\right)\\hfill \\\\ \\hfill 121\\hfill \\end{array}\\)<\/td><td data-valign=\"middle\" data-align=\"center\">+<\/td><td data-valign=\"middle\" data-align=\"left\">2 real<\/td><\/tr><tr><td data-valign=\"middle\" data-align=\"left\">\\(4{x}^{2}-20x+25=0\\)<\/td><td data-valign=\"middle\" data-align=\"center\">\\(\\begin{array}{c}\\hfill {\\left(-20\\right)}^{2}-4\u00b74\u00b725\\hfill \\\\ \\hfill 0\\hfill \\end{array}\\)<\/td><td data-valign=\"middle\" data-align=\"center\">0<\/td><td data-valign=\"middle\" data-align=\"left\">1 real<\/td><\/tr><tr><td data-valign=\"middle\" data-align=\"left\">\\(3{p}^{2}+2p+9=0\\)<\/td><td data-valign=\"middle\" data-align=\"center\">\\(\\begin{array}{c}\\hfill {2}^{2}-4\u00b73\u00b79\\hfill \\\\ \\\\ \\hfill -104\\hfill \\end{array}\\)<\/td><td data-valign=\"middle\" data-align=\"center\">\u2212<\/td><td data-valign=\"middle\" data-align=\"left\">2 complex<\/td><\/tr><\/tbody><\/table><span data-type=\"media\" id=\"fs-id1167829811343\" data-alt=\"When the value under the radical in the Quadratic Formula, the discriminant, is positive, the equation has two real solutions. When the value under the radical in the Quadratic Formula, the discriminant, is zero, the equation has one real solution. When the value under the radical in the Quadratic Formula, the discriminant, is negative, the equation has two complex solutions.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_010_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"When the value under the radical in the Quadratic Formula, the discriminant, is positive, the equation has two real solutions. When the value under the radical in the Quadratic Formula, the discriminant, is zero, the equation has one real solution. When the value under the radical in the Quadratic Formula, the discriminant, is negative, the equation has two complex solutions.\"><\/span><div data-type=\"note\" id=\"fs-id1167829738475\"><div data-type=\"title\">Using the Discriminant, <em data-effect=\"italics\">b<\/em><sup>2<\/sup> \u2212 4<em data-effect=\"italics\">ac<\/em>, to Determine the Number and Type of Solutions of a Quadratic Equation<\/div><p>For 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, \\(a\\ne 0,\\)<\/p><ul id=\"fs-id1167833135684\" data-bullet-style=\"bullet\"><li>If <em data-effect=\"italics\">b<\/em><sup>2<\/sup> \u2212 4<em data-effect=\"italics\">ac<\/em> &gt; 0, the equation has 2 real solutions.<\/li><li>if <em data-effect=\"italics\">b<\/em><sup>2<\/sup> \u2212 4<em data-effect=\"italics\">ac<\/em> = 0, the equation has 1 real solution.<\/li><li>if <em data-effect=\"italics\">b<\/em><sup>2<\/sup> \u2212 4<em data-effect=\"italics\">ac<\/em> &lt; 0, the equation has 2 complex solutions.<\/li><\/ul><\/div><div data-type=\"example\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167836296394\"><div data-type=\"problem\" id=\"fs-id1167829720123\"><p id=\"fs-id1167836311816\">Determine the number of solutions to each quadratic equation.<\/p><p id=\"fs-id1167836386298\"><span class=\"token\">\u24d0<\/span>\\(3{x}^{2}+7x-9=0\\)<span class=\"token\">\u24d1<\/span>\\(5{n}^{2}+n+4=0\\)<span class=\"token\">\u24d2<\/span>\\(9{y}^{2}-6y+1=0.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832994493\"><p id=\"fs-id1167829832940\">To determine the number of solutions of each quadratic equation, we will look at its discriminant.<\/p><p><span class=\"token\">\u24d0<\/span><\/p><div data-type=\"newline\"><br><\/div>\\(\\begin{array}{cccc}&amp; &amp; &amp; \\hfill 3{x}^{2}+7x-9=0\\hfill \\\\ \\begin{array}{c}\\text{The equation is in standard form, identify}\\hfill \\\\ a,b,\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}c.\\hfill \\end{array}\\hfill &amp; &amp; &amp; \\hfill a=3,\\phantom{\\rule{0.5em}{0ex}}b=7,\\phantom{\\rule{0.2em}{0ex}}c=-9\\hfill \\\\ \\text{Write the discriminant.}\\hfill &amp; &amp; &amp; \\hfill {b}^{2}-4ac\\hfill \\\\ \\text{Substitute in the values of}\\phantom{\\rule{0.2em}{0ex}}a,b,\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}c.\\hfill &amp; &amp; &amp; \\hfill {\\left(7\\right)}^{2}-4\u00b73\u00b7\\left(-9\\right)\\hfill \\\\ \\text{Simplify.}\\hfill &amp; &amp; &amp; \\hfill 49+108\\hfill \\\\ &amp; &amp; &amp; \\hfill 157\\hfill \\end{array}\\)<p id=\"fs-id1167829716779\">Since the discriminant is positive, there are 2 real solutions to the equation.<\/p><p id=\"fs-id1167836319507\"><span class=\"token\">\u24d1<\/span><\/p><div data-type=\"newline\"><br><\/div>\\(\\begin{array}{cccc}&amp; &amp; &amp; \\hfill 5{n}^{2}+n+4=0\\hfill \\\\ \\begin{array}{c}\\text{The equation is in standard form, identify}\\hfill \\\\ a,b,\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}c.\\hfill \\end{array}\\hfill &amp; &amp; &amp; \\hfill a=5,\\phantom{\\rule{0.5em}{0ex}}b=1,\\phantom{\\rule{0.5em}{0ex}}c=4\\hfill \\\\ \\text{Write the discriminant.}\\hfill &amp; &amp; &amp; \\hfill {b}^{2}-4ac\\hfill \\\\ \\text{Substitute in the values of}\\phantom{\\rule{0.2em}{0ex}}a,b,\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}c.\\hfill &amp; &amp; &amp; \\hfill {\\left(1\\right)}^{2}-4\u00b75\u00b74\\hfill \\\\ \\text{Simplify.}\\hfill &amp; &amp; &amp; \\hfill 1-80\\hfill \\\\ &amp; &amp; &amp; \\hfill -79\\hfill \\end{array}\\)<p id=\"fs-id1167829907197\">Since the discriminant is negative, there are 2 complex solutions to the equation.<\/p><p id=\"fs-id1167836628458\"><span class=\"token\">\u24d2<\/span><\/p><div data-type=\"newline\"><br><\/div>\\(\\begin{array}{cccc}&amp; &amp; &amp; \\hfill 9{y}^{2}-6y+1=0\\hfill \\\\ \\begin{array}{c}\\text{The equation is in standard form, identify}\\hfill \\\\ a,b,\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}c.\\hfill \\end{array}\\hfill &amp; &amp; &amp; \\hfill a=9,b=-6,c=1\\hfill \\\\ \\text{Write the discriminant.}\\hfill &amp; &amp; &amp; \\hfill {b}^{2}-4ac\\hfill \\\\ \\text{Substitute in the values of}\\phantom{\\rule{0.2em}{0ex}}a,b,\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}c.\\hfill &amp; &amp; &amp; \\hfill {\\left(-6\\right)}^{2}-4\u00b79\u00b71\\hfill \\\\ \\text{Simplify.}\\hfill &amp; &amp; &amp; \\hfill 36-36\\hfill \\\\ &amp; &amp; &amp; \\hfill 0\\hfill \\end{array}\\)<p id=\"fs-id1167836539865\">Since the discriminant is 0, there is 1 real solution to the equation.<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836560858\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167833142364\"><div data-type=\"problem\" id=\"fs-id1167833137471\"><p>Determine the numberand type of solutions to each quadratic equation.<\/p><p id=\"fs-id1167829693337\"><span class=\"token\">\u24d0<\/span>\\(8{m}^{2}-3m+6=0\\)<span class=\"token\">\u24d1<\/span>\\(5{z}^{2}+6z-2=0\\)<span class=\"token\">\u24d2<\/span>\\(9{w}^{2}+24w+16=0.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836294317\"><p id=\"fs-id1167829586521\"><span class=\"token\">\u24d0<\/span> 2 complex solutions; <span class=\"token\">\u24d1<\/span> 2 real solutions; <span class=\"token\">\u24d2<\/span> 1 real solution<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167829692639\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836375545\"><div data-type=\"problem\" id=\"fs-id1167836599051\"><p id=\"fs-id1167836549518\">Determine the number and type of solutions to each quadratic equation.<\/p><p id=\"fs-id1167824737254\"><span class=\"token\">\u24d0<\/span>\\({b}^{2}+7b-13=0\\)<span class=\"token\">\u24d1<\/span>\\(5{a}^{2}-6a+10=0\\)<span class=\"token\">\u24d2<\/span>\\(4{r}^{2}-20r+25=0.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836706433\"><p id=\"fs-id1167829755685\"><span class=\"token\">\u24d0<\/span> 2 real solutions; <span class=\"token\">\u24d1<\/span> 2 complex solutions; <span class=\"token\">\u24d2<\/span> 1 real solution<\/p><\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167836520180\"><h3 data-type=\"title\">Identify the Most Appropriate Method to Use to Solve a Quadratic Equation<\/h3><p id=\"fs-id1167836732334\">We summarize the four methods that we have used to solve quadratic equations below.<\/p><div data-type=\"note\" id=\"fs-id1165926509814\"><div data-type=\"title\">Methods for Solving Quadratic Equations<\/div><ol type=\"1\"><li>Factoring<\/li><li>Square Root Property<\/li><li>Completing the Square<\/li><li>Quadratic Formula<\/li><\/ol><\/div><p id=\"fs-id1167836449204\">Given that we have four methods to use to solve a quadratic equation, how do you decide which one to use? Factoring is often the quickest method and so we try it first. If the equation is \\(a{x}^{2}=k\\) or \\(a{\\left(x-h\\right)}^{2}=k\\) we use the Square Root Property. For any other equation, it is probably best to use the Quadratic Formula. Remember, you can solve any quadratic equation by using the Quadratic Formula, but that is not always the easiest method.<\/p><p id=\"fs-id1165926666963\">What about the method of Completing the Square? Most people find that method cumbersome and prefer not to use it. We needed to include it in the list of methods because we completed the square in general to derive the Quadratic Formula. You will also use the process of Completing the Square in other areas of algebra.<\/p><div data-type=\"note\" id=\"fs-id1167833136798\" class=\"howto\"><div data-type=\"title\">Identify the most appropriate method to solve a quadratic equation.<\/div><ol id=\"fs-id1167829833515\" type=\"1\" class=\"stepwise\"><li>Try <strong data-effect=\"bold\">Factoring<\/strong> first. If the quadratic factors easily, this method is very quick.<\/li><li>Try the <strong data-effect=\"bold\">Square Root Property<\/strong> next. If the equation fits the form \\(a{x}^{2}=k\\) or \\(a{\\left(x-h\\right)}^{2}=k,\\) it can easily be solved by using the Square Root Property.<\/li><li>Use the <strong data-effect=\"bold\">Quadratic Formula<\/strong>. Any other quadratic equation is best solved by using the Quadratic Formula.<\/li><\/ol><\/div><p>The next example uses this strategy to decide how to solve each quadratic equation.<\/p><div data-type=\"example\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167829644985\"><div data-type=\"problem\" id=\"fs-id1167826132391\"><p id=\"fs-id1167829692811\">Identify the most appropriate method to use to solve each quadratic equation.<\/p><p><span class=\"token\">\u24d0<\/span>\\(5{z}^{2}=17\\)<span class=\"token\">\u24d1<\/span>\\(4{x}^{2}-12x+9=0\\)<span class=\"token\">\u24d2<\/span>\\(8{u}^{2}+6u=11.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836514624\"><p id=\"fs-id1167830077350\"><span class=\"token\">\u24d0<\/span><\/p><div data-type=\"newline\"><br><\/div>\\(\\begin{array}{}\\\\ \\\\ \\\\ \\hfill \\phantom{\\rule{2em}{0ex}}5{z}^{2}=17\\hfill \\end{array}\\)<p id=\"fs-id1165926686347\">Since the equation is in the \\(a{x}^{2}=k,\\) the most appropriate method is to use the Square Root Property.<\/p><p id=\"fs-id1167836509220\"><span class=\"token\">\u24d1<\/span><\/p><div data-type=\"newline\"><br><\/div>\\(\\begin{array}{}\\\\ \\\\ \\\\ \\hfill \\phantom{\\rule{3.5em}{0ex}}4{x}^{2}-12x+9=0\\hfill \\end{array}\\)<p id=\"fs-id1165926667746\">We recognize that the left side of the equation is a perfect square trinomial, and so factoring will be the most appropriate method.<\/p><p id=\"fs-id1167829860668\"><span class=\"token\">\u24d2<\/span><\/p><div data-type=\"newline\"><br><\/div>\\(\\begin{array}{cccccccccc}&amp; &amp; &amp; &amp; &amp; &amp; &amp; \\hfill 8{u}^{2}+6u&amp; =\\hfill &amp; 11\\hfill \\\\ \\text{Put the equation in standard form.}\\hfill &amp; &amp; &amp; &amp; &amp; &amp; &amp; \\hfill 8{u}^{2}+6u-11&amp; =\\hfill &amp; 0\\hfill \\end{array}\\)<p id=\"fs-id1167836320702\">While our first thought may be to try factoring, thinking about all the possibilities for trial and error method leads us to choose the Quadratic Formula as the most appropriate method.<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836310878\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167829791723\"><div data-type=\"problem\" id=\"fs-id1167833047287\"><p id=\"fs-id1167836542264\">Identify the most appropriate method to use to solve each quadratic equation.<\/p><p id=\"fs-id1167836610515\"><span class=\"token\">\u24d0<\/span>\\({x}^{2}+6x+8=0\\)<span class=\"token\">\u24d1<\/span>\\({\\left(n-3\\right)}^{2}=16\\)<span class=\"token\">\u24d2<\/span>\\(5{p}^{2}-6p=9.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836492309\"><p id=\"fs-id1167829984323\"><span class=\"token\">\u24d0<\/span> factoring; <span class=\"token\">\u24d1<\/span> Square Root Property; <span class=\"token\">\u24d2<\/span> Quadratic Formula<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836390589\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836728825\"><div data-type=\"problem\" id=\"fs-id1167833361603\"><p id=\"fs-id1167832982923\">Identify the most appropriate method to use to solve each quadratic equation.<\/p><p id=\"fs-id1167833050685\"><span class=\"token\">\u24d0<\/span>\\(8{a}^{2}+3a-9=0\\)<span class=\"token\">\u24d1<\/span>\\(4{b}^{2}+4b+1=0\\)<span class=\"token\">\u24d2<\/span>\\(5{c}^{2}=125.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836560742\"><p id=\"fs-id1167836619347\"><span class=\"token\">\u24d0<\/span> Quadratic Forumula;<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> Factoring or Square Root Property <span class=\"token\">\u24d2<\/span> Square Root Property<\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836503984\" class=\"media-2\"><p id=\"fs-id1167836512904\">Access these online resources for additional instruction and practice with using the Quadratic Formula.<\/p><ul id=\"fs-id1167833386397\" data-display=\"block\"><li><a href=\"https:\/\/openstax.org\/l\/37QuadForm1\">Using the Quadratic Formula<\/a><\/li><li><a href=\"https:\/\/openstax.org\/l\/37QuadForm2\">Solve a Quadratic Equation Using the Quadratic Formula with Complex Solutions<\/a><\/li><li><a href=\"https:\/\/openstax.org\/l\/37QuadForm3\">Discriminant in Quadratic Formula<\/a><\/li><\/ul><\/div><\/div><div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167833024581\"><h3 data-type=\"title\">Key Concepts<\/h3><ul id=\"fs-id1167826030445\" data-bullet-style=\"bullet\"><li>Quadratic Formula <ul id=\"fs-id1167836388723\" data-bullet-style=\"open-circle\"><li>The solutions to 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, \\(a\\ne 0\\) are given by the formula:<div data-type=\"newline\"><br><\/div> <div data-type=\"equation\" id=\"fs-id1167836311402\" class=\"unnumbered\" data-label=\"\">\\(x=\\frac{\\text{\u2212}b\u00b1\\sqrt{{b}^{2}-4ac}}{2a}\\)<\/div><\/li><\/ul><\/li><li>How to solve a quadratic equation using the Quadratic Formula. <ol id=\"fs-id1167833380659\" type=\"1\" class=\"stepwise\"><li>Write the quadratic equation in standard form, <em data-effect=\"italics\">ax<\/em><sup>2<\/sup> + <em data-effect=\"italics\">bx<\/em> + <em data-effect=\"italics\">c<\/em> = 0. Identify the values of <em data-effect=\"italics\">a<\/em>, <em data-effect=\"italics\">b<\/em>, <em data-effect=\"italics\">c<\/em>.<\/li><li>Write the Quadratic Formula. Then substitute in the values of <em data-effect=\"italics\">a<\/em>, <em data-effect=\"italics\">b<\/em>, <em data-effect=\"italics\">c<\/em>.<\/li><li>Simplify.<\/li><li>Check the solutions.<\/li><\/ol><\/li><li>Using the Discriminant, <em data-effect=\"italics\">b<\/em><sup>2<\/sup> \u2212 4<em data-effect=\"italics\">ac<\/em>, to Determine the Number and Type of Solutions of a Quadratic Equation <ul id=\"fs-id1167836523246\" data-bullet-style=\"open-circle\"><li>For 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, \\(a\\ne 0,\\) <ul id=\"fs-id1167833290846\" data-bullet-style=\"bullet\"><li>If <em data-effect=\"italics\">b<\/em><sup>2<\/sup> \u2212 4<em data-effect=\"italics\">ac<\/em> &gt; 0, the equation has 2 real solutions.<\/li><li>if <em data-effect=\"italics\">b<\/em><sup>2<\/sup> \u2212 4<em data-effect=\"italics\">ac<\/em> = 0, the equation has 1 real solution.<\/li><li>if <em data-effect=\"italics\">b<\/em><sup>2<\/sup> \u2212 4<em data-effect=\"italics\">ac<\/em> &lt; 0, the equation has 2 complex solutions.<\/li><\/ul><\/li><\/ul><\/li><li>Methods to Solve Quadratic Equations: <ul id=\"fs-id1167833284845\" data-bullet-style=\"open-circle\"><li>Factoring<\/li><li>Square Root Property<\/li><li>Completing the Square<\/li><li>Quadratic Formula<\/li><\/ul><\/li><li>How to identify the most appropriate method to solve a quadratic equation. <ol id=\"fs-id1167825830231\" type=\"1\" class=\"stepwise\"><li>Try Factoring first. If the quadratic factors easily, this method is very quick.<\/li><li>Try the <strong data-effect=\"bold\">Square Root Property<\/strong> next. If the equation fits the form <em data-effect=\"italics\">ax<\/em><sup>2<\/sup> = <em data-effect=\"italics\">k<\/em> or <em data-effect=\"italics\">a<\/em>(<em data-effect=\"italics\">x<\/em> \u2212 <em data-effect=\"italics\">h<\/em>)<sup>2<\/sup> = <em data-effect=\"italics\">k<\/em>, it can easily be solved by using the Square Root Property.<\/li><li>Use the <strong data-effect=\"bold\">Quadratic Formula.<\/strong> Any other quadratic equation is best solved by using the Quadratic Formula.<\/li><\/ol><\/li><\/ul><\/div><div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167833196720\"><div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1167836707345\"><h4 data-type=\"title\">Practice Makes Perfect<\/h4><p id=\"fs-id1167836692628\"><strong data-effect=\"bold\">Solve Quadratic Equations Using the Quadratic Formula<\/strong><\/p><p id=\"fs-id1167836481619\">In the following exercises, solve by using the Quadratic Formula.<\/p><div data-type=\"exercise\" id=\"fs-id1167836287708\"><div data-type=\"problem\" id=\"fs-id1167836579406\"><p id=\"fs-id1167836535044\">\\(4{m}^{2}+m-3=0\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833128877\"><p id=\"fs-id1167833008092\">\\(m=-1,m=\\frac{3}{4}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829756068\"><div data-type=\"problem\" id=\"fs-id1167836448251\"><p id=\"fs-id1167823012097\">\\(4{n}^{2}-9n+5=0\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836613568\"><div data-type=\"problem\" id=\"fs-id1167836513341\"><p id=\"fs-id1167836502019\">\\(2{p}^{2}-7p+3=0\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829747937\"><p id=\"fs-id1167829808028\">\\(p=\\frac{1}{3},p=2\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836363363\"><div data-type=\"problem\" id=\"fs-id1167833046856\"><p>\\(3{q}^{2}+8q-3=0\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829785536\"><div data-type=\"problem\" id=\"fs-id1167836508391\"><p id=\"fs-id1167836706624\">\\({p}^{2}+7p+12=0\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829586796\"><p id=\"fs-id1167829737988\">\\(p=-4,p=-3\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829580115\"><div data-type=\"problem\" id=\"fs-id1167833017933\"><p id=\"fs-id1167836547433\">\\({q}^{2}+3q-18=0\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836704308\"><div data-type=\"problem\" id=\"fs-id1167836547994\"><p id=\"fs-id1167836312366\">\\({r}^{2}-8r=33\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836507162\"><p id=\"fs-id1167836359772\">\\(r=-3,r=11\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829831326\"><div data-type=\"problem\"><p id=\"fs-id1167829872216\">\\({t}^{2}+13t=-40\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836574258\"><div data-type=\"problem\" id=\"fs-id1167836533091\"><p>\\(3{u}^{2}+7u-2=0\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836398780\"><p id=\"fs-id1167836541400\">\\(u=\\frac{-7\u00b1\\sqrt{73}}{6}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836433826\"><div data-type=\"problem\" id=\"fs-id1167824704137\"><p id=\"fs-id1167836621198\">\\(2{p}^{2}+8p+5=0\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836534727\"><div data-type=\"problem\" id=\"fs-id1167836550911\"><p>\\(2{a}^{2}-6a+3=0\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833186350\"><p id=\"fs-id1167829595101\">\\(a=\\frac{3\u00b1\\sqrt{3}}{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836550976\"><div data-type=\"problem\" id=\"fs-id1167836391876\"><p id=\"fs-id1167836768482\">\\(5{b}^{2}+2b-4=0\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829747947\"><div data-type=\"problem\" id=\"fs-id1167836512516\"><p id=\"fs-id1167836393334\">\\({x}^{2}+8x-4=0\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836408060\"><p id=\"fs-id1167836492529\">\\(x=-4\u00b12\\sqrt{5}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836450575\"><div data-type=\"problem\" id=\"fs-id1167836514719\"><p id=\"fs-id1167836579737\">\\({y}^{2}+4y-4=0\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836548619\"><div data-type=\"problem\" id=\"fs-id1167833339898\"><p id=\"fs-id1167836286059\">\\(3{y}^{2}+5y-2=0\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836662437\"><p id=\"fs-id1167836513391\">\\(y=-\\frac{2}{3},y=-1\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836352286\"><div data-type=\"problem\" id=\"fs-id1167836595870\"><p id=\"fs-id1167833025364\">\\(6{x}^{2}+2x-20=0\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826077135\"><div data-type=\"problem\" id=\"fs-id1167829651145\"><p id=\"fs-id1167829651147\">\\(2{x}^{2}+3x+3=0\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836550959\"><p id=\"fs-id1167833310564\">\\(x=-\\frac{3}{4}\u00b1\\frac{\\sqrt{15}}{4}i\\)<\/p><\/div><\/div><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167824891816\"><p id=\"fs-id1167830092950\">\\(2{x}^{2}-x+1=0\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167824738968\"><div data-type=\"problem\" id=\"fs-id1167836607323\"><p id=\"fs-id1167836607325\">\\(8{x}^{2}-6x+2=0\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167822916220\"><p id=\"fs-id1167833060460\">\\(x=\\frac{3}{8}\u00b1\\frac{\\sqrt{7}}{8}i\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836753598\"><div data-type=\"problem\" id=\"fs-id1167833340003\"><p id=\"fs-id1167829666500\">\\(8{x}^{2}-4x+1=0\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833385657\"><div data-type=\"problem\" id=\"fs-id1167833007389\"><p id=\"fs-id1167833007391\">\\(\\left(v+1\\right)\\left(v-5\\right)-4=0\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836409136\"><p id=\"fs-id1167836409138\">\\(v=2\u00b12\\sqrt{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836622221\"><div data-type=\"problem\" id=\"fs-id1167836622223\"><p>\\(\\left(x+1\\right)\\left(x-3\\right)=2\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836546600\"><div data-type=\"problem\" id=\"fs-id1167833048411\"><p id=\"fs-id1167836550106\">\\(\\left(y+4\\right)\\left(y-7\\right)=18\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829691005\"><p id=\"fs-id1167829691007\">\\(y=-4,y=7\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829597601\"><div data-type=\"problem\" id=\"fs-id1167826077158\"><p id=\"fs-id1167826077160\">\\(\\left(x+2\\right)\\left(x+6\\right)=21\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836409913\"><div data-type=\"problem\" id=\"fs-id1167836409915\"><p id=\"fs-id1167836415993\">\\(\\frac{1}{3}{m}^{2}+\\frac{1}{12}m=\\frac{1}{4}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829711812\"><p>\\(m=-1,m=\\frac{3}{4}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833338350\"><div data-type=\"problem\" id=\"fs-id1167829627371\"><p id=\"fs-id1167829627373\">\\(\\frac{1}{3}{n}^{2}+n=-\\frac{1}{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167824781360\"><div data-type=\"problem\" id=\"fs-id1167836694411\"><p id=\"fs-id1167836694413\">\\(\\frac{3}{4}{b}^{2}+\\frac{1}{2}b=\\frac{3}{8}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829717481\"><p id=\"fs-id1167829717483\">\\(b=\\frac{-2\u00b1\\sqrt{11}}{6}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829784975\"><div data-type=\"problem\" id=\"fs-id1167829784978\"><p id=\"fs-id1167836415664\">\\(\\frac{1}{9}{c}^{2}+\\frac{2}{3}c=3\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167824748849\"><div data-type=\"problem\" id=\"fs-id1167829738093\"><p id=\"fs-id1167829738095\">\\(16{c}^{2}+24c+9=0\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836518759\"><p id=\"fs-id1167836518761\">\\(c=-\\frac{3}{4}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829952779\"><div data-type=\"problem\" id=\"fs-id1167829952781\"><p id=\"fs-id1167836609912\">\\(25{d}^{2}-60d+36=0\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836511004\"><div data-type=\"problem\" id=\"fs-id1167836387219\"><p id=\"fs-id1167836387221\">\\(25{q}^{2}+30q+9=0\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836516507\"><p>\\(q=-\\frac{3}{5}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836376530\"><div data-type=\"problem\" id=\"fs-id1167836376532\"><p id=\"fs-id1167829692331\">\\(16{y}^{2}+8y+1=0\\)<\/p><\/div><\/div><p id=\"fs-id1167836477084\"><strong data-effect=\"bold\">Use the Discriminant to Predict the Number of Real Solutions of a Quadratic Equation<\/strong><\/p><p id=\"fs-id1167824767094\">In the following exercises, determine the number of real solutions for each quadratic equation.<\/p><div data-type=\"exercise\" id=\"fs-id1167822916231\"><div data-type=\"problem\" id=\"fs-id1167822916233\"><p id=\"fs-id1167836569038\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(4{x}^{2}-5x+16=0\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(36{y}^{2}+36y+9=0\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\(6{m}^{2}+3m-5=0\\)<\/div><div data-type=\"solution\" id=\"fs-id1167836521669\"><p id=\"fs-id1167836549783\"><span class=\"token\">\u24d0<\/span>\\(\\text{no real solutions}\\)<span class=\"token\">\u24d1<\/span>\\(1\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\(2\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836572983\"><div data-type=\"problem\" id=\"fs-id1167836376107\"><p><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(9{v}^{2}-15v+25=0\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(100{w}^{2}+60w+9=0\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\(5{c}^{2}+7c-10=0\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836328607\"><div data-type=\"problem\" id=\"fs-id1167833053956\"><p id=\"fs-id1167833053958\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\({r}^{2}+12r+36=0\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(8{t}^{2}-11t+5=0\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\(3{v}^{2}-5v-1=0\\)<\/div><div data-type=\"solution\" id=\"fs-id1167825760157\"><p id=\"fs-id1167825760159\"><span class=\"token\">\u24d0<\/span>\\(1\\)<span class=\"token\">\u24d1<\/span>\\(\\text{no real solutions}\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\(2\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829709403\"><div data-type=\"problem\" id=\"fs-id1167833240365\"><p id=\"fs-id1167833240367\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(25{p}^{2}+10p+1=0\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(7{q}^{2}-3q-6=0\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\(7{y}^{2}+2y+8=0\\)<\/div><\/div><p id=\"fs-id1167836530634\"><strong data-effect=\"bold\">Identify the Most Appropriate Method to Use to Solve a Quadratic Equation<\/strong><\/p><p id=\"fs-id1167836525480\">In the following exercises, identify the most appropriate method (Factoring, Square Root, or Quadratic Formula) to use to solve each quadratic equation. Do not solve.<\/p><div data-type=\"exercise\" id=\"fs-id1167836493661\"><div data-type=\"problem\" id=\"fs-id1167836732375\"><p id=\"fs-id1167836732377\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\({x}^{2}-5x-24=0\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\({\\left(y+5\\right)}^{2}=12\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\(14{m}^{2}+3m=11\\)<\/div><div data-type=\"solution\" id=\"fs-id1167833076778\"><p id=\"fs-id1167836666727\"><span class=\"token\">\u24d0<\/span>\\(\\text{factor}\\)<span class=\"token\">\u24d1<\/span>\\(\\text{square root}\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\(\\text{Quadratic Formula}\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167825946794\"><div data-type=\"problem\" id=\"fs-id1167829785715\"><p id=\"fs-id1167829785717\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\({\\left(8v+3\\right)}^{2}=81\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\({w}^{2}-9w-22=0\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\(4{n}^{2}-10=6\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167825949223\"><div data-type=\"problem\" id=\"fs-id1167836530566\"><p id=\"fs-id1167836530569\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(6{a}^{2}+14=20\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\({\\left(x-\\frac{1}{4}\\right)}^{2}=\\frac{5}{16}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\({y}^{2}-2y=8\\)<\/div><div data-type=\"solution\" id=\"fs-id1167833364609\"><p><span class=\"token\">\u24d0<\/span>\\(\\text{Quadratic Formula}\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\text{square root}\\)<span class=\"token\">\u24d2<\/span>\\(\\text{factor}\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833239671\"><div data-type=\"problem\" id=\"fs-id1167833239673\"><p id=\"fs-id1167833053326\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(8{b}^{2}+15b=4\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\frac{5}{9}{v}^{2}-\\frac{2}{3}v=1\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\({\\left(w+\\frac{4}{3}\\right)}^{2}=\\frac{2}{9}\\)<\/div><\/div><\/div><div class=\"writing\" data-depth=\"2\" id=\"fs-id1167836508342\"><h4 data-type=\"title\">Writing Exercises<\/h4><div data-type=\"exercise\" id=\"fs-id1167829790134\"><div data-type=\"problem\" id=\"fs-id1167829790136\"><p id=\"fs-id1167829752402\">Solve the equation \\({x}^{2}+10x=120\\)<\/p><p id=\"fs-id1167836498963\"><span class=\"token\">\u24d0<\/span> by completing the square<\/p><p id=\"fs-id1167833083076\"><span class=\"token\">\u24d1<\/span> using the Quadratic Formula<\/p><p id=\"fs-id1167824755448\"><span class=\"token\">\u24d2<\/span> Which method do you prefer? Why?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167826131790\"><p id=\"fs-id1167826131792\">Answers will vary.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836601151\"><div data-type=\"problem\" id=\"fs-id1167836601154\"><p id=\"fs-id1167836496817\">Solve the equation \\(12{y}^{2}+23y=24\\)<\/p><p id=\"fs-id1167836597657\"><span class=\"token\">\u24d0<\/span> by completing the square<\/p><p id=\"fs-id1167832936319\"><span class=\"token\">\u24d1<\/span> using the Quadratic Formula<\/p><p id=\"fs-id1167836601619\"><span class=\"token\">\u24d2<\/span> Which method do you prefer? Why?<\/p><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167829695326\"><h4 data-type=\"title\">Self Check<\/h4><p id=\"fs-id1167836390295\"><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-id1167829744205\" data-alt=\"This table provides a checklist to evaluate mastery of the objectives of this section. Choose how would you respond to the statement \u201cI can solve quadratic equations using the quadratic formula.\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 use the discriminant to predict the number of solutions of a quadratic equation.\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 identify the most appropriate method to use to solve a quadratic equation.\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_03_201_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This table provides a checklist to evaluate mastery of the objectives of this section. Choose how would you respond to the statement \u201cI can solve quadratic equations using the quadratic formula.\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 use the discriminant to predict the number of solutions of a quadratic equation.\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 identify the most appropriate method to use to solve a quadratic equation.\u201d \u201cConfidently,\u201d \u201cwith some help,\u201d or \u201cNo, I don\u2019t get it.\u201d\"><\/span><p id=\"fs-id1167829893818\"><span class=\"token\">\u24d1<\/span> What does this checklist tell you about your mastery of this section? What steps will you take to improve?<\/p><\/div><\/div><div data-type=\"glossary\" class=\"textbox shaded\"><h3 data-type=\"glossary-title\">Glossary<\/h3><dl id=\"fs-id1167825946740\"><dt>discriminant<\/dt><dd id=\"fs-id1167825946743\">In the Quadratic Formula, \\(x=\\frac{\\text{\u2212}b\u00b1\\sqrt{{b}^{2}-4ac}}{2a},\\) the quantity <em data-effect=\"italics\">b<\/em><sup>2<\/sup> \u2212 4<em data-effect=\"italics\">ac<\/em> is called the discriminant.<\/dd><\/dl><\/div>\n","rendered":"<div class=\"textbox textbox--learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<p>By the end of this section, you will be able to: <\/p>\n<ul>\n<li>Solve quadratic equations using the Quadratic Formula<\/li>\n<li>Use the discriminant to predict the number and type of solutions of a quadratic equation<\/li>\n<li>Identify the most appropriate method to use to solve a quadratic equation<\/li>\n<\/ul>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167830093627\" class=\"be-prepared\">\n<p>Before you get started, take this readiness quiz.<em data-effect=\"italics\"><\/em><\/p>\n<ol id=\"fs-id1167832999312\" type=\"1\">\n<li>Evaluate <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-54486e2bff193d9f8558f90af8dbdddf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#97;&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"63\" style=\"vertical-align: -1px;\" \/> when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9418f76b7fb8efbd61d4b14b3df06bad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-159bea5680db08ee12ac3ee2ac90283c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;&#61;&#45;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"58\" style=\"vertical-align: 0px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p> If you missed this problem, review <a href=\"\/contents\/05eab039-6d1c-4d80-8c8c-94469164a52c#fs-id1167832053133\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<li>Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-47b52239c759d9ae2b5be2cffa70a7a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#48;&#56;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"44\" style=\"vertical-align: -2px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p> If you missed this problem, review <a href=\"\/contents\/dbb319b9-f421-4570-9953-ca5a39b933dc#fs-id1169144556679\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<li>Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2e1276c9ec13d9b52fa79f2459c54a3f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#53;&#48;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"36\" style=\"vertical-align: -2px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p> If you missed this problem, review <a href=\"\/contents\/916a2094-3b51-4f1a-803d-95909a359123#fs-id1169145730176\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<\/ol>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167836664862\">\n<h3 data-type=\"title\">Solve Quadratic Equations Using the Quadratic Formula<\/h3>\n<p id=\"fs-id1167836685641\">When we solved quadratic equations in the last section by completing the square, we took the same steps every time. By the end of the exercise set, you may have been wondering \u2018isn\u2019t there an easier way to do this?\u2019 The answer is \u2018yes\u2019. Mathematicians look for patterns when they do things over and over in order to make their work easier. In this section we will derive and use a formula to find the solution of a quadratic equation.<\/p>\n<p>We have already seen how to solve a formula for a specific variable \u2018in general\u2019, so that we would do the algebraic steps only once, and then use the new formula to find the value of the specific variable. Now we will go through the steps of completing the square using the general form of a quadratic equation to solve a quadratic equation for <em data-effect=\"italics\">x.<\/em><\/p>\n<p id=\"fs-id1167833338980\">We start with the standard form of a quadratic equation and solve it for <em data-effect=\"italics\">x<\/em> by completing the square.<\/p>\n<table id=\"fs-id1167833004917\" class=\"unnumbered unstyled can-break\" summary=\"To develop the Quadratic Formula, start with the standard form of a quadratic equation, a times x squared plus b times x plus c equals 0. Remember that a is not equal to zero. Isolate the variable terms on one side. The new equation is a times x squared plus b x equals negative c. Make the leading coefficient 1 by dividing both sides of the equation by a. We now have the quotient a times x squared divided by a plus the product of the quotient b divided by a and x equals negative c divided by a. Simplified, this becomes x squared plus b divided by a times x equals negative c divided by a. Complete the square on the left side of the equation. Find the square of one half times the quotient b divided by a which simplifies to b squared divided by the product 4 times a squared. Add this value to both sides of the equation. X squared plus b divided by a times x plus b squared divided by the product 4 a squared equals negative c divided by a plus the quotient b squared divided by the product 4 times a squared. Factor the perfect square trinomial on the left side of the equation. The square of x plus the quotient b divided by 2 a equals the quotient negative c divided by a plus the quotient b squared divided by the product 4 times a squared. Find the common denominator of the right side of the equation and write equivalent fractions using the common denominator. Multiply the term negative c divided by a on the right side of the equation by the fraction 4 a divided by 4 a. Rearrange terms on the right side of the equation, and it becomes the square of the sum x plus the quotient b divided by 2 a equals the quotient b squared divided by the product 4 times a squared plus the quotient negative c times 4 a divided by a times 4 a. Combining to one fraction, the square of the sum x plus the quotient b divided by 2 a equals the quotient of the difference b squared minus 4 a c divided by 4 a squared. Use the Square Root Property. X plus the quotient b divded by 2 a equals the positive or negative square root of the quotient of the difference b squared minus 4 a c divided by 4 a squared. Simplify the radical. X plus the quotient b divded by 2 a equals the positive or negative quotient of the square root of the difference b squared minus 4 a c divided by 2 a. Add negative b divided by 2 a to both sides of the equation. So x equals negative b divided by the product 2 a plus or minus the quotient of the square root of the difference b squared minus 4 a c divided by 2 a. Combine the terms on the right side of the equation to get the final form of the Quadratic Formula. X equals the quotient negative b plus or minus the square root of the difference b squared minus 4 a c divided by 2 a.\" 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_03_001a_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 one side.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167829942519\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_001b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Make 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;\" \/> equal to 1, by<\/p>\n<div data-type=\"newline\"><\/div>\n<p>dividing by <em data-effect=\"italics\">a<\/em>.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836580108\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_001c_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=\"center\"><span data-type=\"media\" id=\"fs-id1167824736027\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_001d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">To complete the square, find <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e2b2182c56b202e71ea2e804822b0059_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;&#98;&#125;&#123;&#97;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"43\" style=\"vertical-align: -7px;\" \/> and add it to both sides of the equation.<\/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-50ef5d92bd1aee702aa5161d3f99475b_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;&#102;&#114;&#97;&#99;&#123;&#98;&#125;&#123;&#97;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#52;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"90\" style=\"vertical-align: -7px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836690676\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_001e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">The left side is a perfect square, factor it.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167833052027\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_001f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Find the common denominator of the right<\/p>\n<div data-type=\"newline\"><\/div>\n<p>side and write equivalent fractions with<\/p>\n<div data-type=\"newline\"><\/div>\n<p>the common denominator.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167829620906\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_001g_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=\"center\"><span data-type=\"media\" id=\"fs-id1167829619193\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_001h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Combine to one fraction.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167833237760\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_001i_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-id1167836613501\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_001j_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\" id=\"fs-id1167836713990\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_001k_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/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-7137224bdc6600511c4c03a0a63f370c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#98;&#125;&#123;&#50;&#97;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"30\" style=\"vertical-align: -6px;\" \/> to both sides of the equation.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167833158651\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_001l_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Combine the terms on the right side.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167833321935\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_001m_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\">This equation is the Quadratic Formula.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div data-type=\"note\">\n<div data-type=\"title\">Quadratic Formula<\/div>\n<p id=\"fs-id1167836295293\">The solutions to a <span data-type=\"term\" class=\"no-emphasis\">quadratic equation<\/span> 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, where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-df2afd8a9cb6add6da009531338a6949_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#92;&#110;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"42\" style=\"vertical-align: -4px;\" \/> are given by the formula:<\/p>\n<div data-type=\"equation\" id=\"fs-id1167829924519\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-09ae580202af47cd724e09f8ddbfda6e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#98;&plusmn;&#92;&#115;&#113;&#114;&#116;&#123;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#97;&#99;&#125;&#125;&#123;&#50;&#97;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"97\" style=\"vertical-align: -6px;\" \/><\/div>\n<\/div>\n<p id=\"fs-id1167836554536\">To use the <span data-type=\"term\" class=\"no-emphasis\">Quadratic Formula<\/span>, we substitute the values of <em data-effect=\"italics\">a<\/em>, <em data-effect=\"italics\">b<\/em>, and <em data-effect=\"italics\">c<\/em> from the standard form into the expression on the right side of the formula. Then we simplify the expression. The result is the pair of solutions to the quadratic equation.<\/p>\n<p id=\"fs-id1167832945838\">Notice the formula is an equation. Make sure you use both sides of the equation.<\/p>\n<div data-type=\"example\" id=\"fs-id1167824740960\" class=\"textbox textbox--examples\">\n<div data-type=\"title\">How to Solve a Quadratic Equation Using the Quadratic Formula<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836497688\">\n<div data-type=\"problem\" id=\"fs-id1167829890840\">\n<p>Solve by using the Quadratic Formula: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d554fa73f1227bdecce1b7e9018ac00f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#57;&#120;&#45;&#53;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"135\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\"><span data-type=\"media\" id=\"fs-id1167829714248\" data-alt=\"Step 1 is to write the quadratic equation in standard form, a times x squared plus b x plus c equals zero, and identify the values a, b, and c. The equation 2 x squared plus 9 x minus 5 equals zero is in standard form. A equals 2, b equals 9, and c equals negative 5.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_002a_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 1 is to write the quadratic equation in standard form, a times x squared plus b x plus c equals zero, and identify the values a, b, and c. The equation 2 x squared plus 9 x minus 5 equals zero is in standard form. A equals 2, b equals 9, and c equals negative 5.\" \/><\/span><span data-type=\"media\" data-alt=\"Step 2. Write the quadratic formula. Then substitute the values of a, b, and c. Substitute a equals 2, b equals 9, and c equals negative 5 into the equation x equals the quotient negative b plus or minus the square root of the difference b squared minus 4 a c divided by 2 a. So x equals the quotient negative 9 plus or minus the square root of the difference 9 squared minus the product 4 times 2 times negative 5 divided by the product 2 times 2.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_002b_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2. Write the quadratic formula. Then substitute the values of a, b, and c. Substitute a equals 2, b equals 9, and c equals negative 5 into the equation x equals the quotient negative b plus or minus the square root of the difference b squared minus 4 a c divided by 2 a. So x equals the quotient negative 9 plus or minus the square root of the difference 9 squared minus the product 4 times 2 times negative 5 divided by the product 2 times 2.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167832926037\" data-alt=\"In step 3, simplify the fraction and solve for x. x equals the quotient negative 9 plus or minus the square root of the difference 81 minus negative 40 divided by 4. Simplify the radicand. x equals the quotient negative 9 plus or minus the square root of 121 divided by 4. Simplify the square root. x equals the quotient negative 9 plus or minus 11 divided by 4. Separate into two equations. The first equation is x equals the quotient negative 9 plus 11 divided by 4 which simplifies to 2 divided by 4. The first solution is x equals one half. The second equation is x equals the quotient negative 9 minus 11 divided by 4 which simplifies to negative 20 divided by 4. The second solution is x equals negative 5.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_002c_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"In step 3, simplify the fraction and solve for x. x equals the quotient negative 9 plus or minus the square root of the difference 81 minus negative 40 divided by 4. Simplify the radicand. x equals the quotient negative 9 plus or minus the square root of 121 divided by 4. Simplify the square root. x equals the quotient negative 9 plus or minus 11 divided by 4. Separate into two equations. The first equation is x equals the quotient negative 9 plus 11 divided by 4 which simplifies to 2 divided by 4. The first solution is x equals one half. The second equation is x equals the quotient negative 9 minus 11 divided by 4 which simplifies to negative 20 divided by 4. The second solution is x equals negative 5.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167836729568\" data-alt=\"The fourth, and final, step is to check the solution. Put each answer into the original equation to check. First, substitute x equals one half into the original equation, 2 x squared plus 9 x minus 5 equals 0. This yields 2 times the square of one half plus nine times one half minus 5. We need to show that this expression equals 0. Simplify the square. 2 times one fourth plus nine times one half minus 5 equals one half plus 9 halves minus 5, or 10 halves minus 5. 5 minus 5 equals 0, so x equals one half is indeed a solution. Next substitute x = negative 5 into the equation 2 x squared plus 9 x minus 5 equals 0. This yields 2 times the square of negative 5 plus 9 times negative 5 minus 5. We need to show that this expression equals 0. Simplify the square. 2 times 25 plus nine times negative 5 minus 5 equals 50 minus 45 minus 5, or 0. x equals negative 5 is a solution as well.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_002d_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The fourth, and final, step is to check the solution. Put each answer into the original equation to check. First, substitute x equals one half into the original equation, 2 x squared plus 9 x minus 5 equals 0. This yields 2 times the square of one half plus nine times one half minus 5. We need to show that this expression equals 0. Simplify the square. 2 times one fourth plus nine times one half minus 5 equals one half plus 9 halves minus 5, or 10 halves minus 5. 5 minus 5 equals 0, so x equals one half is indeed a solution. Next substitute x = negative 5 into the equation 2 x squared plus 9 x minus 5 equals 0. This yields 2 times the square of negative 5 plus 9 times negative 5 minus 5. We need to show that this expression equals 0. Simplify the square. 2 times 25 plus nine times negative 5 minus 5 equals 50 minus 45 minus 5, or 0. x equals negative 5 is a solution as well.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836754953\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167833052817\">\n<div data-type=\"problem\" id=\"fs-id1167836699334\">\n<p id=\"fs-id1167829811483\">Solve by using the Quadratic Formula: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a9c6f724e421835fb290d8d40a2ce95c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#121;&#43;&#50;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"129\" style=\"vertical-align: -4px;\" \/>.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829755644\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-314d0dd007ff8c886e7a3caacf21002a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#49;&#44;&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"92\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167833377487\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836530282\">\n<div data-type=\"problem\" id=\"fs-id1167833018768\">\n<p id=\"fs-id1167833020111\">Solve by using the Quadratic Formula: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-75df51946ac41999679a039d83a74e38_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#122;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#122;&#45;&#54;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"128\" style=\"vertical-align: -2px;\" \/>.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167830093362\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-149fd4d65fe4de26fad998e606c2d35b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#122;&#61;&#49;&#44;&#122;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"105\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167829924562\" class=\"howto\">\n<div data-type=\"title\">Solve a quadratic equation using the quadratic formula.<\/div>\n<ol type=\"1\" class=\"stepwise\">\n<li>Write the quadratic equation in standard form, <em data-effect=\"italics\">ax<\/em><sup>2<\/sup> + <em data-effect=\"italics\">bx<\/em> + <em data-effect=\"italics\">c<\/em> = 0. Identify the values of <em data-effect=\"italics\">a<\/em>, <em data-effect=\"italics\">b<\/em>, and <em data-effect=\"italics\">c<\/em>.<\/li>\n<li>Write the Quadratic Formula. Then substitute in the values of <em data-effect=\"italics\">a<\/em>, <em data-effect=\"italics\">b<\/em>, and <em data-effect=\"italics\">c<\/em>.<\/li>\n<li>Simplify.<\/li>\n<li>Check the solutions.<\/li>\n<\/ol>\n<\/div>\n<p id=\"fs-id1167833008091\">If you say the formula as you write it in each problem, you\u2019ll have it memorized in no time! And remember, the Quadratic Formula is an EQUATION. Be sure you start with \u201c<em data-effect=\"italics\">x<\/em> =\u201d.<\/p>\n<div data-type=\"example\" id=\"fs-id1167836717133\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167829711896\">\n<p id=\"fs-id1167824735584\">Solve by using the Quadratic Formula: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dd9b3817e3467b7ccf840eb9cded3dd7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#120;&#61;&#45;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"109\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829685914\">\n<table id=\"fs-id1167836407290\" class=\"unnumbered unstyled\" summary=\"Write the equation x squared minus 6 x equals negative 5 in standard form by adding 5 to both sides of the equation. X squared minus 6 x plus 5 equals 0. Identify the values of a, b, and c. The coefficient of x squared is a = 1. The coefficient of x is b equals negative 6. The constant term is c equals 5. Write the quadratic formula, x equals the quotient negative b plus or minus the square root of the difference b squared minus 4 a c divided by 2 a. Then substitute the values for a, b, and c. x equals the quotient of the expression the opposite of negative 6 plus or minus the square root of the difference negative 6 squared minus the product 4 times 1 times 5 divided by the product 2 times 1. Simplify. X equals the quotient of the expression 6 plus or minus the square root of the difference 36 minus 20 divided by 2. This further simplifies to the quotient of 6 plus or minus square root 16 and 2, so x equals the quotient of 6 plus or minus 4 and 2. Rewrite to show two solutions The first is x equals the quotient 6 plus 4 divided by 2, or 10 divided by 2 which equals 5. The second solution is the quotient 6 minus 4 divided by 2, or 2 divided by 2 which equals 1. Check the solutions in the original equation. Substitute x equals 5 into the original equation, x squared minus 6 x plus 5 equals zero to get 5 squared minus 6 times 5 plus 5 on the left side of the equation we must show that this equals 0. Simplifying the expression yields 25 minus 30 plus 5, or 0. So x equals 5 is a solution. Next check x equals 1 in the original equation. X squared minus 6 x plus 5 becomes 1 squared minus 6 times 1 plus 5. We must show this equals 0. 1 minus 6 plus 5 does equal 0, so x = 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-id1167836573724\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_003b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Write the equation in standard form by adding<\/p>\n<div data-type=\"newline\"><\/div>\n<p>5 to each side.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829850979\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_003c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">This equation is now in standard form.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836450487\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_003d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Identify the values of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-909ffdde580d2a41dd373150836d0b44_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#92;&#104;&#115;&#112;&#97;&#99;&#101;&#123;&#48;&#46;&#49;&#55;&#101;&#109;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#92;&#104;&#115;&#112;&#97;&#99;&#101;&#123;&#48;&#46;&#49;&#55;&#101;&#109;&#125;&#125;&#98;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#92;&#104;&#115;&#112;&#97;&#99;&#101;&#123;&#48;&#46;&#49;&#55;&#101;&#109;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#92;&#104;&#115;&#112;&#97;&#99;&#101;&#123;&#48;&#46;&#49;&#55;&#101;&#109;&#125;&#125;&#99;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"57\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829739762\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_003e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Write the Quadratic Formula.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836521083\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_003f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Then substitute in the values of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-909ffdde580d2a41dd373150836d0b44_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#92;&#104;&#115;&#112;&#97;&#99;&#101;&#123;&#48;&#46;&#49;&#55;&#101;&#109;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#92;&#104;&#115;&#112;&#97;&#99;&#101;&#123;&#48;&#46;&#49;&#55;&#101;&#109;&#125;&#125;&#98;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#92;&#104;&#115;&#112;&#97;&#99;&#101;&#123;&#48;&#46;&#49;&#55;&#101;&#109;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#92;&#104;&#115;&#112;&#97;&#99;&#101;&#123;&#48;&#46;&#49;&#55;&#101;&#109;&#125;&#125;&#99;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"57\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829811936\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_003g_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-id1167824735884\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_003h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_003i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167829685990\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_003j_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-id1167833397031\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_003k_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-id1167824732532\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_003l_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-id1167829753751\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_003m_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-id1167833379118\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_003a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span> <\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"try\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167829627475\">\n<p id=\"fs-id1167833142563\">Solve by using the Quadratic Formula: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3e1837c9668bdb2cc0a621eb2f890a7a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#97;&#61;&#49;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"98\" style=\"vertical-align: -1px;\" \/>.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829714558\">\n<p id=\"fs-id1167829695255\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1f89ce59f6215fae818ac7884e062bde_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&#45;&#51;&#44;&#97;&#61;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"105\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167829878093\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836314565\">\n<div data-type=\"problem\">\n<p>Solve by using the Quadratic Formula: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1c346c32eefc67ca0148b917bf1a5dba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#52;&#61;&#45;&#49;&#48;&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"118\" 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-6f8c5739a2c9e38a75ff1b930d473a50_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;&#61;&#45;&#54;&#44;&#98;&#61;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"116\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836789050\">When we solved quadratic equations by using the Square Root Property, we sometimes got answers that had radicals. That can happen, too, when using the <span data-type=\"term\" class=\"no-emphasis\">Quadratic Formula<\/span>. If we get a <span data-type=\"term\" class=\"no-emphasis\">radical<\/span> as a solution, the final answer must have the radical in its simplified form.<\/p>\n<div data-type=\"example\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167829747416\">\n<div data-type=\"problem\" id=\"fs-id1167836791948\">\n<p>Solve by using the Quadratic Formula: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5b21680152ef9b2b001f8cf3cbc243da_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#48;&#120;&#43;&#49;&#49;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"152\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832950924\">\n<table id=\"fs-id1167826212241\" class=\"unnumbered unstyled\" summary=\"The equation 2x squared plus 10 x plus 11 equals 0 is already in standard form. Identify the values of a, b, and c. The coefficient of x squared is a = 2. The coefficient of x is b equals 10. The constant term is c equals 11. Write the quadratic formula, x equals the quotient negative b plus or minus the square root of the expression b squared minus 4 a c divided by 2 a. Then substitute the values for a, b, and c. x equals the quotient of the difference negative 10 plus or minus the square root of the difference 10 squared minus the product 4 times 2 times 11 divided by the product 2 times 2. Simplify. X equals the quotient of the expression negative 10 plus or minus the square root of the difference 100 minus 88 divided by 4. This further simplifies to the quotient negative 10 plus or minus square root 12 divided by 4. Simplify the radical. x equals the quotient negative 10 plus or minus 2 times the square root of 3 divided by 2. Factor out the common factor in the numerator. X equals the quotient of 2 times the expression negative 5 plus or minus 2 square root 3 and 4. Remove the common factor to yield x equals the quotient negative 5 plus or minus square root 3 divided by 2. Rewrite to show two solutions The first solution is x equals the quotient of negative 5 plus square root 3 and 2. The second is x equals the quotient of negative 5 minus square root 3 and 2. Remember to check the solutions in the original equation. We leave that to you!\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836656566\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_004a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">This equation is in standard form.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167829783604\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_004b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Identify the values of <em data-effect=\"italics\">a<\/em>, <em data-effect=\"italics\">b<\/em>, and <em data-effect=\"italics\">c<\/em>.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167830013853\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_004c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Write the Quadratic Formula.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836790447\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_004d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Then substitute in the values of <em data-effect=\"italics\">a<\/em>, <em data-effect=\"italics\">b<\/em>, and <em data-effect=\"italics\">c<\/em>.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167833021195\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_004e_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=\"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_03_004f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_004g_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\" id=\"fs-id1167836621035\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_004h_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 common factor in the numerator.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167829695938\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_004i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Remove the common factors.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836560610\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_004j_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=\"center\"><span data-type=\"media\" id=\"fs-id1167836532009\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_004k_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<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-id1167829712766\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836554515\">\n<div data-type=\"problem\" id=\"fs-id1167836502385\">\n<p>Solve by using the Quadratic Formula: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-573c6143ff58ecc4a202e424a68ea8fa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#50;&#109;&#43;&#55;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"150\" style=\"vertical-align: -2px;\" \/>.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167825660450\">\n<p id=\"fs-id1167833239738\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1c0bb1d808028950a1d3dd587998d167_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#54;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#53;&#125;&#125;&#123;&#51;&#125;&#44;&#109;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#54;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#53;&#125;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"200\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836601096\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167829853072\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167836595665\">Solve by using the Quadratic Formula: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-45ea7fa78d8ff3abaf0192df26440d0f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#110;&#45;&#52;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"132\" style=\"vertical-align: -2px;\" \/>.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836492623\">\n<p id=\"fs-id1167836321837\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ea1980f67ab1a15e6e90fe09215fe9ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#50;&#43;&#50;&#92;&#115;&#113;&#114;&#116;&#123;&#54;&#125;&#125;&#123;&#53;&#125;&#44;&#110;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#50;&#45;&#50;&#92;&#115;&#113;&#114;&#116;&#123;&#54;&#125;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"190\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836729254\">When we substitute <em data-effect=\"italics\">a<\/em>, <em data-effect=\"italics\">b<\/em>, and <em data-effect=\"italics\">c<\/em> into the Quadratic Formula and the <span data-type=\"term\" class=\"no-emphasis\">radicand<\/span> is negative, the quadratic equation will have imaginary or complex solutions. We will see this in the next example.<\/p>\n<div data-type=\"example\" id=\"fs-id1167826162738\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167836553297\">\n<div data-type=\"problem\" id=\"fs-id1167836529273\">\n<p id=\"fs-id1167836539168\">Solve by using the Quadratic Formula: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-72511c760f5fca5149283a04e7d16678_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#112;&#43;&#57;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"132\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829752211\">\n<table class=\"unnumbered unstyled can-break\" summary=\"The equation 3 p squared plus 2 p plus 9 equals 0 is already in standard form. Identify the values of a, b, and c. The coefficient of p squared is a = 3. The coefficient of p is b equals 2. The constant term is c equals 9. Write the quadratic formula, p equals the quotient negative b plus or minus the square root of the difference b squared minus 4 a c divided by 2 a. Then substitute the values for a, b, and c. p equals the quotient of the expression negative 2 plus or minus the square root of the difference 4 squared minus the product 4 times 3 times 9 divided by the product 2 times 3. Simplify. P equals the quotient of the expression negative 2 plus or minus the square root of the difference 4 minus 108 divided by 6. This further simplifies to p equals the quotient negative 2 plus or minus the square root of negative 104 divided by 6. Simplify the radical using complex numbers. p equals the quotient negative 2 plus or minus square root 104 times I divided by 6. Simplify the radical. p equals the quotient negative 2 plus or minus 2 times square root 104 times I divided by 6. Factor the common factor in the numerator. P equals the quotient of 2 times the expression negative 1 plus or minus square root 26 times I divided by 6. Remove the common factor to yield p equals the quotient negative 1 plus or minus square root 26 times I divided by 3. Rewrite in standard a plus b I form. P equals negative one third plus or minus square root 26 divided by 3 times I. Write as show two solutions The first solution is p equals negative one third plus square root 26 thirds I. The second is p equals negative one third minus square root 26 thirds I.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836630346\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_005a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">This equation is in standard form<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167833051387\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_005b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Identify the values of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-355a73bbce4f63a5ff319af1c86e003b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#44;&#98;&#44;&#99;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"45\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167829812037\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_005c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Write the Quadratic Formula.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836547551\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_005d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Then substitute in the values of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-58797fcd980ddcdad97f6b6f5260b5fa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#44;&#98;&#44;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"41\" style=\"vertical-align: -4px;\" \/>.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167829586709\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_005e_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=\"center\"><span data-type=\"media\" id=\"fs-id1167836417826\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_005f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_005g_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 using complex numbers.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167826077082\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_005h_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\" id=\"fs-id1167836433941\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_005i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Factor the common factor in the numerator.<\/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_03_005j_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Remove the common factors.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836623945\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_005k_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Rewrite in standard <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4ffcd7d918a83e37de98a41b94384c07_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#43;&#98;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"45\" style=\"vertical-align: -2px;\" \/> form.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836547132\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_005l_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Write as two solutions.<\/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_03_005m_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-id1167829906103\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167830123951\">\n<div data-type=\"problem\" id=\"fs-id1167836393407\">\n<p id=\"fs-id1167836487107\">Solve by using the Quadratic Formula: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-95d7f38719c827da8f7090a57b0169b5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#97;&#43;&#56;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"129\" style=\"vertical-align: -2px;\" \/>.<\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167825829977\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5fa9b6a3065790536c8399218ff4f938_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#49;&#125;&#125;&#123;&#52;&#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;&#97;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#49;&#125;&#125;&#123;&#52;&#125;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"216\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167829906736\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836731793\">\n<div data-type=\"problem\" id=\"fs-id1167836635576\">\n<p id=\"fs-id1167833396817\">Solve by using the Quadratic Formula: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-12fae6cf9125f0bfbda677a297480d7e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#98;&#43;&#52;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"125\" style=\"vertical-align: -2px;\" \/>.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829879778\">\n<p id=\"fs-id1167829597268\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-77c00d0f3127afe8011ac393f4e9665d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#53;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#57;&#125;&#125;&#123;&#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;&#98;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#53;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#57;&#125;&#125;&#123;&#53;&#125;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"240\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836379181\">Remember, to use the Quadratic Formula, the equation must be written in standard form, <em data-effect=\"italics\">ax<\/em><sup>2<\/sup> + <em data-effect=\"italics\">bx<\/em> + <em data-effect=\"italics\">c<\/em> = 0. Sometimes, we will need to do some algebra to get the equation into standard form before we can use the Quadratic Formula.<\/p>\n<div data-type=\"example\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167829686816\">\n<div data-type=\"problem\" id=\"fs-id1167836557051\">\n<p id=\"fs-id1167829721085\">Solve by using the Quadratic Formula: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dab606ae772de011c84043e851cefa1a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#52;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"135\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829732080\">\n<p id=\"fs-id1167829783831\">Our first step is to get the equation in standard form.<\/p>\n<table id=\"fs-id1167836525808\" class=\"unnumbered unstyled can-break\" summary=\"Distribute to rewrite the equation x times the sum of x and 6 plus 4 equals 0 in standard form. The equation becomes x squared plus 6 x plus 4 equals 0. Identify the values of a, b, and c. The coefficient of x squared is a = 1. The coefficient of x is b equals 6. The constant term is c equals 4. Write the quadratic formula, x equals the quotient negative b plus or minus the square root of the difference b squared minus 4 a c divided by 2 a. Then substitute the values for a, b, and c. x equals the quotient of the expression negative 6 plus or minus the square root of the difference 6 squared minus the product 4 times 1 times 4 divided by the product 2 times 1. Simplify. X equals the quotient of the expression negative 6 plus or minus the square root of the difference 36 minus 16 divided by 2. This further simplifies to the quotient of negative 6 plus or minus square root 20 and 2. Simplify the radical. X equals the quotient negative 6 plus or minus 2 square root 5 divided by 2. Factor the common factor in the numerator. X equals the quotient 2 times the expression negative 3 plus or minus 2 square root 5 divided by 2. Remove the common factor, and x equals negative 3 plus or minus 2 square root 5. Rewrite to show two solutions, x equals negative 3 plus 2 square root 5 and x equals negative 3 minus 2 square root 5. Remember to check the solutions in the original equation. We leave that to you!\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167829692551\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_006a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Distribute to get the equation in standard form.<\/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_03_006b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">This equation is now in standard form<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167833009856\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_006c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Identify the values of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-355a73bbce4f63a5ff319af1c86e003b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#44;&#98;&#44;&#99;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"45\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836573443\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_006d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Write the Quadratic Formula.<\/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_03_006e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Then substitute in the values of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-58797fcd980ddcdad97f6b6f5260b5fa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#44;&#98;&#44;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"41\" style=\"vertical-align: -4px;\" \/>.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167829785636\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_006f_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=\"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_03_006g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836574191\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_006h_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\" id=\"fs-id1167836731465\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_006i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Factor the common factor in the numerator.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167829908075\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_006j_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Remove the common factors.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836520614\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_006k_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Write as two solutions.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836629538\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_006l_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<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-id1167825824208\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836287145\">\n<div data-type=\"problem\" id=\"fs-id1167829826662\">\n<p id=\"fs-id1167829596853\">Solve by using the Quadratic Formula: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2449fd379408b1f3cf13678d5d8f4d96_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#53;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"135\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829828288\">\n<p id=\"fs-id1167836510782\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-94d376b9fa396c148f43fc3b9cd441de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#49;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#54;&#125;&#44;&#120;&#61;&#45;&#49;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"211\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"try\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167829693355\">Solve by using the Quadratic Formula: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5a8ee9a330b1564310c1002150a469f4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#121;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#51;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"142\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167829811978\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3867b4602c294690625124441ed6b82c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#49;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#125;&#44;&#121;&#61;&#49;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"182\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167826128252\">When we solved linear equations, if an equation had too many fractions we cleared the fractions by multiplying both sides of the equation by the LCD. This gave us an equivalent equation\u2014without fractions\u2014 to solve. We can use the same strategy with quadratic equations.<\/p>\n<div data-type=\"example\" id=\"fs-id1167836331062\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167836532524\">\n<div data-type=\"problem\">\n<p>Solve by using the Quadratic Formula: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bd27d7d50608a4617fc504ac6eb6182d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#123;&#117;&#125;&#94;&#123;&#50;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#117;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"108\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167833025258\">Our first step is to clear the fractions.<\/p>\n<table id=\"fs-id1167829810567\" class=\"unnumbered unstyled can-break\" summary=\"Write the original equation, one half u squared plus two thirds u equals one third. Multiply both sides of the equation by the LCD, 6, to clear the fractions. 6 times the sum one half u squared plus two thirds u equals 6 times one third. Multiply to yield 3 u squared plus 4 u equals 2. Subtract 2 from both sides of the equation to write it in standard form. 3 u squared plus 4 u minus 2 equals 0. Identify the values of a, b, and c. The coefficient of u squared is a = 3. The coefficient of u is b equals 4. The constant term is c equals negative 2. Write the quadratic formula, x equals the quotient negative b plus or minus the square root of the difference b squared minus 4 a c divided by 2 a. Then substitute the values for a, b, and c. u equals the quotient of the expression negative 4 plus or minus the square root of the difference 4 squared minus the product 4 times 3 times negative 2 divided by the product 2 times 3. Simplify. u equals the quotient of the expression negative 4 plus or minus the square root of the sum 16 plus 24 divided by 6. This further simplifies to the quotient of negative 4 plus or minus square root 40 and 6. Simplify the radical. U equals the quotient of the expression negative 4 plus or minus 2 square root 10 and 6. Factor the common factor in the numerator. U equals the quotient of 2 times the expression negative 2 plus or minus square root 10 and 6. Remove the common factor yielding u equals the quotient of negative 2 plus or minus square root 10 and 3. Rewrite to show two solutions. u equals the quotient negative 2 plus square root 10 divided by 3 and u equals the quotient negative 2 minus square root 10 divided by 3 Check the solutions in the original equation. We leave that to you!\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836575024\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_007a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Multiply both sides by the LCD, 6, to clear the fractions.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836389274\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_007b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Multiply.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167824735095\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_007c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Subtract 2 to get the equation in standard form.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836481420\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_007d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Identify the values of <em data-effect=\"italics\">a<\/em>, <em data-effect=\"italics\">b<\/em>, and <em data-effect=\"italics\">c<\/em>.<\/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_03_007e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Write the Quadratic Formula.<\/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_03_007f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Then substitute in the values of <em data-effect=\"italics\">a<\/em>, <em data-effect=\"italics\">b<\/em>, and <em data-effect=\"italics\">c<\/em>.<\/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_03_007g_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=\"center\"><span data-type=\"media\" id=\"fs-id1167829808815\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_007h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167832999050\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_007i_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\" id=\"fs-id1167833056373\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_007j_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Factor the common factor in the numerator.<\/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_03_007k_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Remove the common factors.<\/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_03_007l_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=\"center\"><span data-type=\"media\" id=\"fs-id1167836387084\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_007m_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<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-id1167836511739\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836516031\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167836524102\">Solve by using the Quadratic Formula: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-efbf65b9412cee623c681872cec0b675_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#123;&#99;&#125;&#94;&#123;&#50;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#99;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"104\" style=\"vertical-align: -7px;\" \/>.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167824734634\">\n<p id=\"fs-id1167836530748\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-74ba703e120aba1b594f8253fdade436_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#55;&#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;&#99;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#55;&#125;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"155\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167829690058\" class=\"try\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167836629454\">\n<p id=\"fs-id1167836484657\">Solve by using the Quadratic Formula: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-223384d23dd58fc0f8b364c99a6535fc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#57;&#125;&#123;&#100;&#125;&#94;&#123;&#50;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#100;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"114\" style=\"vertical-align: -6px;\" \/>.<\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167836545245\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2729cdb7ccb4b67ef5024ec6c487bb70_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#100;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#51;&#125;&#125;&#123;&#52;&#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;&#100;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#51;&#125;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"169\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167832999563\">Think about the equation (<em data-effect=\"italics\">x<\/em> \u2212 3)<sup>2<\/sup> = 0. We know from the <span data-type=\"term\" class=\"no-emphasis\">Zero Product Property<\/span> that this equation has only one solution,<\/p>\n<div data-type=\"newline\"><\/div>\n<p><em data-effect=\"italics\">x<\/em> = 3.<\/p>\n<p id=\"fs-id1167825830172\">We will see in the next example how using the <span data-type=\"term\" class=\"no-emphasis\">Quadratic Formula<\/span> to solve an equation whose standard form is a perfect square <span data-type=\"term\" class=\"no-emphasis\">trinomial<\/span> equal to 0 gives just one solution. Notice that once the radicand is simplified it becomes 0 , which leads to only one solution.<\/p>\n<div data-type=\"example\" id=\"fs-id1167825830135\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167829689050\">\n<div data-type=\"problem\" id=\"fs-id1167836409726\">\n<p id=\"fs-id1167833025902\">Solve by using the Quadratic Formula: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d75be324d71906e0014fbbee300ec048_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#48;&#120;&#61;&#45;&#50;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"136\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836300298\">\n<table id=\"fs-id1167836619633\" class=\"unnumbered unstyled can-break\" summary=\"Write the equation 4 x squared minus 20 x equals negative 25 in standard form by adding 25 to both sides of the equation. 4 x squared minus 20 x plus 25 equals 0. Identify the values of a, b, and c. The coefficient of x squared is a = 4. The coefficient of x is b equals negative 20. The constant term is c equals 25. Write the quadratic formula, x equals the quotient negative b plus or minus the square root of the difference b squared minus 4 a c divided by 2 a. Then substitute the values for a, b, and c. x equals the quotient of the expression the opposite of negative 20 plus or minus the square root of the difference negative 20 squared minus the product 4 times 4 times 25 divided by the product 2 times 4. Simplify. X equals the quotient of the expression 20 plus or minus the square root of the difference 400 minus 400 divided by 8. This further simplifies to the quotient of 20 plus or minus 0 and 2, so x equals 20 divided by 8 or 5 halves. We leave it to you to check the solution in the original equation.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167829695158\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_008a_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 get the equation in standard form.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836294437\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_008b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Identify the values of <em data-effect=\"italics\">a<\/em>, <em data-effect=\"italics\">b<\/em>, and <em data-effect=\"italics\">c<\/em>.<\/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_03_008c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Write the quadratic formula.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167826025420\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_008d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Then substitute in the values of <em data-effect=\"italics\">a<\/em>, <em data-effect=\"italics\">b<\/em>, and <em data-effect=\"italics\">c<\/em>.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167824763423\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_008e_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=\"center\"><span data-type=\"media\" id=\"fs-id1167829580517\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_008f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167836511094\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_008g_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\" id=\"fs-id1167836362464\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_008h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Simplify the fraction.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167833082053\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_008i_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<p> We leave the check for you!<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167836701179\">Did you recognize that 4<em data-effect=\"italics\">x<\/em><sup>2<\/sup> \u2212 20<em data-effect=\"italics\">x<\/em> + 25 is a perfect square trinomial. It is equivalent to (2<em data-effect=\"italics\">x<\/em> \u2212 5)<sup>2<\/sup>? If you solve<\/p>\n<div data-type=\"newline\"><\/div>\n<p>4<em data-effect=\"italics\">x<\/em><sup>2<\/sup> \u2212 20<em data-effect=\"italics\">x<\/em> + 25 = 0 by factoring and then using the Square Root Property, do you get the same result?<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836732791\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167829719841\">\n<div data-type=\"problem\" id=\"fs-id1167825913867\">\n<p id=\"fs-id1167836363237\">Solve by using the Quadratic Formula: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a16273ffb04d2023639d03d3195b4b3d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#114;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#48;&#114;&#43;&#50;&#53;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"140\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167824734801\">\n<p id=\"fs-id1167832925269\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fc452a7e92693c274e671dae17cf59f0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#114;&#61;&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"54\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167833081911\">\n<div data-type=\"problem\" id=\"fs-id1167829719514\">\n<p id=\"fs-id1167829807329\">Solve by using the Quadratic Formula: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d5a27ce32379e8350e47bfd98c7e9aa8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#53;&#123;&#116;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#48;&#116;&#61;&#45;&#49;&#54;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"137\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832940220\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-80c5b6a816aff7c222091fc4d715f916_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"39\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167836295270\">\n<h3 data-type=\"title\">Use the Discriminant to Predict the Number and Type of Solutions of a Quadratic Equation<\/h3>\n<p id=\"fs-id1167836493452\">When we solved the quadratic equations in the previous examples, sometimes we got two real solutions, one real solution, and sometimes two complex solutions. Is there a way to predict the number and type of solutions to a quadratic equation without actually solving the equation?<\/p>\n<p id=\"fs-id1167836625084\">Yes, the expression under the radical of the Quadratic Formula makes it easy for us to determine the number and type of solutions. This expression is called the <span data-type=\"term\">discriminant<\/span>.<\/p>\n<div data-type=\"note\" id=\"fs-id1167836386687\">\n<div data-type=\"title\">Discriminant<\/div>\n<p><span data-type=\"media\" id=\"fs-id1167836556733\" data-alt=\"In the Quadratic Formula, x equals the quotient of negative b plus or minus the square root of b squared minus 4 times a times c and 2 a, the value under the radical, b squared minus 4 times a times c, is called the discriminant.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_009_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"In the Quadratic Formula, x equals the quotient of negative b plus or minus the square root of b squared minus 4 times a times c and 2 a, the value under the radical, b squared minus 4 times a times c, is called the discriminant.\" \/><\/span><\/div>\n<p id=\"fs-id1167826171745\">Let\u2019s look at the discriminant of the equations in some of the examples and the number and type of solutions to those quadratic equations.<\/p>\n<table id=\"fs-id1167824764688\" class=\"unnumbered\" summary=\"This table has four columns and four rows. The first row is a header row and labels the columns \u201cQuadratic Equation (in standard form)\u201d, \u201cDiscriminant b squared minus 4 a c\u201d, \u201cValue of the Discriminant\u201d, and \u201cNumber and Type of solutions\u201d. The second row has the quadratic equation 2 x squared plus 9 x minus 5 equals 0. The discriminant is 9 squared minus the expression 4 times 2 times negative 5, which equals 121. The value of 121 is positive, and there are 2 real solutions. The third row has the quadratic equation 4 x squared minus 20 x plus 25 equals 0. The discriminant is negative twenty squared minus the expression 4 times 4 times 25, which equals 0. The value is 0, and there is 1 real solution. The fourth row has the quadratic equation 3 p squared plus 2 p plus 9 equals 0. The discriminant is 2 squared minus the expression 4 times 3 times 9, which equals negative 104. Th value of negative 104 is negative, and there are 2 complex solutions.\">\n<thead>\n<tr>\n<th data-valign=\"middle\" data-align=\"center\">Quadratic Equation<\/p>\n<div data-type=\"newline\"><\/div>\n<p>(in standard form)<\/th>\n<th data-valign=\"middle\" data-align=\"center\">Discriminant<\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fcee70366e83c8bc6ee77df344dbd11e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#97;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"63\" style=\"vertical-align: -1px;\" \/><\/th>\n<th data-valign=\"middle\" data-align=\"center\">Value of the Discriminant<\/th>\n<th data-valign=\"middle\" data-align=\"center\">Number and Type of solutions<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td data-valign=\"middle\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6848a8cc9e1bcebe049569c48b9c0d6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#57;&#120;&#45;&#53;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"131\" style=\"vertical-align: -2px;\" \/><\/td>\n<td data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-278fe1e9c3a2d7cd21aa3dd75ff75108_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;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#57;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&middot;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#50;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"95\" style=\"vertical-align: -12px;\" \/><\/td>\n<td data-valign=\"middle\" data-align=\"center\">+<\/td>\n<td data-valign=\"middle\" data-align=\"left\">2 real<\/td>\n<\/tr>\n<tr>\n<td data-valign=\"middle\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-712a57b4c219c64bd47cf4ba240b17c3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#48;&#120;&#43;&#50;&#53;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"148\" style=\"vertical-align: -2px;\" \/><\/td>\n<td data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-04a1c1bf7bbcf2f2e4a04d70145e606d_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;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&middot;&#52;&middot;&#50;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"109\" style=\"vertical-align: -12px;\" \/><\/td>\n<td data-valign=\"middle\" data-align=\"center\">0<\/td>\n<td data-valign=\"middle\" data-align=\"left\">1 real<\/td>\n<\/tr>\n<tr>\n<td data-valign=\"middle\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-75e95676f4dbd09429550a5a49b45d7b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#112;&#43;&#57;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"128\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dd0f0730241d616e5bda904cd3a26358_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;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#50;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&middot;&#51;&middot;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#48;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"60\" width=\"65\" style=\"vertical-align: -23px;\" \/><\/td>\n<td data-valign=\"middle\" data-align=\"center\">\u2212<\/td>\n<td data-valign=\"middle\" data-align=\"left\">2 complex<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><span data-type=\"media\" id=\"fs-id1167829811343\" data-alt=\"When the value under the radical in the Quadratic Formula, the discriminant, is positive, the equation has two real solutions. When the value under the radical in the Quadratic Formula, the discriminant, is zero, the equation has one real solution. When the value under the radical in the Quadratic Formula, the discriminant, is negative, the equation has two complex solutions.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_03_010_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"When the value under the radical in the Quadratic Formula, the discriminant, is positive, the equation has two real solutions. When the value under the radical in the Quadratic Formula, the discriminant, is zero, the equation has one real solution. When the value under the radical in the Quadratic Formula, the discriminant, is negative, the equation has two complex solutions.\" \/><\/span><\/p>\n<div data-type=\"note\" id=\"fs-id1167829738475\">\n<div data-type=\"title\">Using the Discriminant, <em data-effect=\"italics\">b<\/em><sup>2<\/sup> \u2212 4<em data-effect=\"italics\">ac<\/em>, to Determine the Number and Type of Solutions of a Quadratic Equation<\/div>\n<p>For 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, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6d28cb478bd8b6abe7d9573551313d6d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#92;&#110;&#101;&#32;&#48;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"46\" style=\"vertical-align: -4px;\" \/><\/p>\n<ul id=\"fs-id1167833135684\" data-bullet-style=\"bullet\">\n<li>If <em data-effect=\"italics\">b<\/em><sup>2<\/sup> \u2212 4<em data-effect=\"italics\">ac<\/em> &gt; 0, the equation has 2 real solutions.<\/li>\n<li>if <em data-effect=\"italics\">b<\/em><sup>2<\/sup> \u2212 4<em data-effect=\"italics\">ac<\/em> = 0, the equation has 1 real solution.<\/li>\n<li>if <em data-effect=\"italics\">b<\/em><sup>2<\/sup> \u2212 4<em data-effect=\"italics\">ac<\/em> &lt; 0, the equation has 2 complex solutions.<\/li>\n<\/ul>\n<\/div>\n<div data-type=\"example\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167836296394\">\n<div data-type=\"problem\" id=\"fs-id1167829720123\">\n<p id=\"fs-id1167836311816\">Determine the number of solutions to each quadratic equation.<\/p>\n<p id=\"fs-id1167836386298\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4382fc8576c81055ff2dae52a094bf1d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#55;&#120;&#45;&#57;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"131\" style=\"vertical-align: -2px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2d193266eba29021861a32ba2b3979d9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#43;&#110;&#43;&#52;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"123\" 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-87da413b45fcb63f3825d6695fda4394_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#121;&#43;&#49;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"133\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832994493\">\n<p id=\"fs-id1167829832940\">To determine the number of solutions of each quadratic equation, we will look at its discriminant.<\/p>\n<p><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f9d9410ab581b73cd70bf2d8334f41a0_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;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#55;&#120;&#45;&#57;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#104;&#101;&#32;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#32;&#105;&#115;&#32;&#105;&#110;&#32;&#115;&#116;&#97;&#110;&#100;&#97;&#114;&#100;&#32;&#102;&#111;&#114;&#109;&#44;&#32;&#105;&#100;&#101;&#110;&#116;&#105;&#102;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#97;&#44;&#98;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#99;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#97;&#61;&#51;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#98;&#61;&#55;&#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;&#99;&#61;&#45;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#87;&#114;&#105;&#116;&#101;&#32;&#116;&#104;&#101;&#32;&#100;&#105;&#115;&#99;&#114;&#105;&#109;&#105;&#110;&#97;&#110;&#116;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#97;&#99;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#117;&#98;&#115;&#116;&#105;&#116;&#117;&#116;&#101;&#32;&#105;&#110;&#32;&#116;&#104;&#101;&#32;&#118;&#97;&#108;&#117;&#101;&#115;&#32;&#111;&#102;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#97;&#44;&#98;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#99;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&middot;&#51;&middot;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#105;&#109;&#112;&#108;&#105;&#102;&#121;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#52;&#57;&#43;&#49;&#48;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#53;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"150\" width=\"556\" style=\"vertical-align: -68px;\" \/><\/p>\n<p id=\"fs-id1167829716779\">Since the discriminant is positive, there are 2 real solutions to the equation.<\/p>\n<p id=\"fs-id1167836319507\"><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-793a11132455a116c59c068ac7ae98ec_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;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#53;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#43;&#110;&#43;&#52;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#104;&#101;&#32;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#32;&#105;&#115;&#32;&#105;&#110;&#32;&#115;&#116;&#97;&#110;&#100;&#97;&#114;&#100;&#32;&#102;&#111;&#114;&#109;&#44;&#32;&#105;&#100;&#101;&#110;&#116;&#105;&#102;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#97;&#44;&#98;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#99;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#97;&#61;&#53;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#98;&#61;&#49;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#99;&#61;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#87;&#114;&#105;&#116;&#101;&#32;&#116;&#104;&#101;&#32;&#100;&#105;&#115;&#99;&#114;&#105;&#109;&#105;&#110;&#97;&#110;&#116;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#97;&#99;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#117;&#98;&#115;&#116;&#105;&#116;&#117;&#116;&#101;&#32;&#105;&#110;&#32;&#116;&#104;&#101;&#32;&#118;&#97;&#108;&#117;&#101;&#115;&#32;&#111;&#102;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#97;&#44;&#98;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#99;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&middot;&#53;&middot;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#105;&#109;&#112;&#108;&#105;&#102;&#121;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#45;&#56;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#55;&#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=\"149\" width=\"547\" style=\"vertical-align: -67px;\" \/><\/p>\n<p id=\"fs-id1167829907197\">Since the discriminant is negative, there are 2 complex solutions to the equation.<\/p>\n<p id=\"fs-id1167836628458\"><span class=\"token\">\u24d2<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-331390c226ee2fc3e7e61f2a33012d7d_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;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#57;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#121;&#43;&#49;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#104;&#101;&#32;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#32;&#105;&#115;&#32;&#105;&#110;&#32;&#115;&#116;&#97;&#110;&#100;&#97;&#114;&#100;&#32;&#102;&#111;&#114;&#109;&#44;&#32;&#105;&#100;&#101;&#110;&#116;&#105;&#102;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#97;&#44;&#98;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#99;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#97;&#61;&#57;&#44;&#98;&#61;&#45;&#54;&#44;&#99;&#61;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#87;&#114;&#105;&#116;&#101;&#32;&#116;&#104;&#101;&#32;&#100;&#105;&#115;&#99;&#114;&#105;&#109;&#105;&#110;&#97;&#110;&#116;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#97;&#99;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#117;&#98;&#115;&#116;&#105;&#116;&#117;&#116;&#101;&#32;&#105;&#110;&#32;&#116;&#104;&#101;&#32;&#118;&#97;&#108;&#117;&#101;&#115;&#32;&#111;&#102;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#97;&#44;&#98;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#99;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&middot;&#57;&middot;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#105;&#109;&#112;&#108;&#105;&#102;&#121;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#54;&#45;&#51;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"149\" width=\"542\" style=\"vertical-align: -67px;\" \/><\/p>\n<p id=\"fs-id1167836539865\">Since the discriminant is 0, there is 1 real solution to the equation.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836560858\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167833142364\">\n<div data-type=\"problem\" id=\"fs-id1167833137471\">\n<p>Determine the numberand type of solutions to each quadratic equation.<\/p>\n<p id=\"fs-id1167829693337\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-198a0d100663f996dea4992036948f03_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#109;&#43;&#54;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"141\" style=\"vertical-align: -2px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8eb32c0af8a41b8300bd26ac5eed2d41_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#123;&#122;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#122;&#45;&#50;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"128\" 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-23a9af72876321d4ddcfd21ca3119a66_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#123;&#119;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#52;&#119;&#43;&#49;&#54;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"158\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836294317\">\n<p id=\"fs-id1167829586521\"><span class=\"token\">\u24d0<\/span> 2 complex solutions; <span class=\"token\">\u24d1<\/span> 2 real solutions; <span class=\"token\">\u24d2<\/span> 1 real solution<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167829692639\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836375545\">\n<div data-type=\"problem\" id=\"fs-id1167836599051\">\n<p id=\"fs-id1167836549518\">Determine the number and type of solutions to each quadratic equation.<\/p>\n<p id=\"fs-id1167824737254\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-62189d75b50240476b21c06111fbba8c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#43;&#55;&#98;&#45;&#49;&#51;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"125\" style=\"vertical-align: -2px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-90072f184bd06d8128ae08681d40425d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#97;&#43;&#49;&#48;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"138\" 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-22cb44631d6eb8cd6e129dffa31a3ef7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#114;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#48;&#114;&#43;&#50;&#53;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"149\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836706433\">\n<p id=\"fs-id1167829755685\"><span class=\"token\">\u24d0<\/span> 2 real solutions; <span class=\"token\">\u24d1<\/span> 2 complex solutions; <span class=\"token\">\u24d2<\/span> 1 real solution<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167836520180\">\n<h3 data-type=\"title\">Identify the Most Appropriate Method to Use to Solve a Quadratic Equation<\/h3>\n<p id=\"fs-id1167836732334\">We summarize the four methods that we have used to solve quadratic equations below.<\/p>\n<div data-type=\"note\" id=\"fs-id1165926509814\">\n<div data-type=\"title\">Methods for Solving Quadratic Equations<\/div>\n<ol type=\"1\">\n<li>Factoring<\/li>\n<li>Square Root Property<\/li>\n<li>Completing the Square<\/li>\n<li>Quadratic Formula<\/li>\n<\/ol>\n<\/div>\n<p id=\"fs-id1167836449204\">Given that we have four methods to use to solve a quadratic equation, how do you decide which one to use? Factoring is often the quickest method and so we try it first. If the equation is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-149f616ba7a3e69badbb87621e79d7c9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#61;&#107;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"60\" style=\"vertical-align: 0px;\" \/> or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-680768cf5d8298df19f7c58a2ffbdf32_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#104;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#107;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"105\" style=\"vertical-align: -4px;\" \/> we use the Square Root Property. For any other equation, it is probably best to use the Quadratic Formula. Remember, you can solve any quadratic equation by using the Quadratic Formula, but that is not always the easiest method.<\/p>\n<p id=\"fs-id1165926666963\">What about the method of Completing the Square? Most people find that method cumbersome and prefer not to use it. We needed to include it in the list of methods because we completed the square in general to derive the Quadratic Formula. You will also use the process of Completing the Square in other areas of algebra.<\/p>\n<div data-type=\"note\" id=\"fs-id1167833136798\" class=\"howto\">\n<div data-type=\"title\">Identify the most appropriate method to solve a quadratic equation.<\/div>\n<ol id=\"fs-id1167829833515\" type=\"1\" class=\"stepwise\">\n<li>Try <strong data-effect=\"bold\">Factoring<\/strong> first. If the quadratic factors easily, this method is very quick.<\/li>\n<li>Try the <strong data-effect=\"bold\">Square Root Property<\/strong> next. If the equation fits the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-149f616ba7a3e69badbb87621e79d7c9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#61;&#107;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"60\" style=\"vertical-align: 0px;\" \/> or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-17811027cccd99153718f8017dc5bb16_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#104;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#107;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"110\" style=\"vertical-align: -4px;\" \/> it can easily be solved by using the Square Root Property.<\/li>\n<li>Use the <strong data-effect=\"bold\">Quadratic Formula<\/strong>. Any other quadratic equation is best solved by using the Quadratic Formula.<\/li>\n<\/ol>\n<\/div>\n<p>The next example uses this strategy to decide how to solve each quadratic equation.<\/p>\n<div data-type=\"example\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167829644985\">\n<div data-type=\"problem\" id=\"fs-id1167826132391\">\n<p id=\"fs-id1167829692811\">Identify the most appropriate method to use to solve each quadratic equation.<\/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-37c64970d0eb6f181ac04dd853057a7a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#123;&#122;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"67\" 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-06644290cc36ced212ed3d7f705f363b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#50;&#120;&#43;&#57;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"139\" 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-585cff88597e1fee905f1d18f90e5e07_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#123;&#117;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#117;&#61;&#49;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"113\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836514624\">\n<p id=\"fs-id1167830077350\"><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c48161aa25f9209e4dea1baca0b5fb21_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;&#125;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#53;&#123;&#122;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"67\" style=\"vertical-align: -34px;\" \/><\/p>\n<p id=\"fs-id1165926686347\">Since the equation is in the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0b32345d798ecf2c09a6fb3e3d4eaf88_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#61;&#107;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"65\" style=\"vertical-align: -4px;\" \/> the most appropriate method is to use the Square Root Property.<\/p>\n<p id=\"fs-id1167836509220\"><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d0df6376833156b6a1373a134f69e38a_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;&#125;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#50;&#120;&#43;&#57;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"139\" style=\"vertical-align: -35px;\" \/><\/p>\n<p id=\"fs-id1165926667746\">We recognize that the left side of the equation is a perfect square trinomial, and so factoring will be the most appropriate method.<\/p>\n<p id=\"fs-id1167829860668\"><span class=\"token\">\u24d2<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0e369eba59ade1c33ca6da45206140bf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#56;&#123;&#117;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#117;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#49;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#80;&#117;&#116;&#32;&#116;&#104;&#101;&#32;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#32;&#105;&#110;&#32;&#115;&#116;&#97;&#110;&#100;&#97;&#114;&#100;&#32;&#102;&#111;&#114;&#109;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#56;&#123;&#117;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#117;&#45;&#49;&#49;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"557\" style=\"vertical-align: -14px;\" \/><\/p>\n<p id=\"fs-id1167836320702\">While our first thought may be to try factoring, thinking about all the possibilities for trial and error method leads us to choose the Quadratic Formula as the most appropriate method.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836310878\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167829791723\">\n<div data-type=\"problem\" id=\"fs-id1167833047287\">\n<p id=\"fs-id1167836542264\">Identify the most appropriate method to use to solve each quadratic equation.<\/p>\n<p id=\"fs-id1167836610515\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-17e70ee48772925df350396bc18c4cd6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#120;&#43;&#56;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"122\" style=\"vertical-align: -2px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a86a5f782e7ec8688d1003aee3071db3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#110;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"103\" 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-0a426bac0c88a73daeb4ccb7ef34dd9c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#112;&#61;&#57;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"102\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836492309\">\n<p id=\"fs-id1167829984323\"><span class=\"token\">\u24d0<\/span> factoring; <span class=\"token\">\u24d1<\/span> Square Root Property; <span class=\"token\">\u24d2<\/span> Quadratic Formula<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836390589\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836728825\">\n<div data-type=\"problem\" id=\"fs-id1167833361603\">\n<p id=\"fs-id1167832982923\">Identify the most appropriate method to use to solve each quadratic equation.<\/p>\n<p id=\"fs-id1167833050685\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-197f0e412c7fb798347f93cb2d8bef03_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#97;&#45;&#57;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"129\" style=\"vertical-align: -2px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5aedec0777a76c57577afa24de6a4d60_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#98;&#43;&#49;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"125\" 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-b7a2ee308a3ee899d117a2f8cef28a3f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#123;&#99;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#50;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"79\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836560742\">\n<p id=\"fs-id1167836619347\"><span class=\"token\">\u24d0<\/span> Quadratic Forumula;<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> Factoring or Square Root Property <span class=\"token\">\u24d2<\/span> Square Root Property<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836503984\" class=\"media-2\">\n<p id=\"fs-id1167836512904\">Access these online resources for additional instruction and practice with using the Quadratic Formula.<\/p>\n<ul id=\"fs-id1167833386397\" data-display=\"block\">\n<li><a href=\"https:\/\/openstax.org\/l\/37QuadForm1\">Using the Quadratic Formula<\/a><\/li>\n<li><a href=\"https:\/\/openstax.org\/l\/37QuadForm2\">Solve a Quadratic Equation Using the Quadratic Formula with Complex Solutions<\/a><\/li>\n<li><a href=\"https:\/\/openstax.org\/l\/37QuadForm3\">Discriminant in Quadratic Formula<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167833024581\">\n<h3 data-type=\"title\">Key Concepts<\/h3>\n<ul id=\"fs-id1167826030445\" data-bullet-style=\"bullet\">\n<li>Quadratic Formula\n<ul id=\"fs-id1167836388723\" data-bullet-style=\"open-circle\">\n<li>The solutions to 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, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-df2afd8a9cb6add6da009531338a6949_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#92;&#110;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"42\" style=\"vertical-align: -4px;\" \/> are given by the formula:\n<div data-type=\"newline\"><\/div>\n<div data-type=\"equation\" id=\"fs-id1167836311402\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-09ae580202af47cd724e09f8ddbfda6e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#98;&plusmn;&#92;&#115;&#113;&#114;&#116;&#123;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#97;&#99;&#125;&#125;&#123;&#50;&#97;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"97\" style=\"vertical-align: -6px;\" \/><\/div>\n<\/li>\n<\/ul>\n<\/li>\n<li>How to solve a quadratic equation using the Quadratic Formula.\n<ol id=\"fs-id1167833380659\" type=\"1\" class=\"stepwise\">\n<li>Write the quadratic equation in standard form, <em data-effect=\"italics\">ax<\/em><sup>2<\/sup> + <em data-effect=\"italics\">bx<\/em> + <em data-effect=\"italics\">c<\/em> = 0. Identify the values of <em data-effect=\"italics\">a<\/em>, <em data-effect=\"italics\">b<\/em>, <em data-effect=\"italics\">c<\/em>.<\/li>\n<li>Write the Quadratic Formula. Then substitute in the values of <em data-effect=\"italics\">a<\/em>, <em data-effect=\"italics\">b<\/em>, <em data-effect=\"italics\">c<\/em>.<\/li>\n<li>Simplify.<\/li>\n<li>Check the solutions.<\/li>\n<\/ol>\n<\/li>\n<li>Using the Discriminant, <em data-effect=\"italics\">b<\/em><sup>2<\/sup> \u2212 4<em data-effect=\"italics\">ac<\/em>, to Determine the Number and Type of Solutions of a Quadratic Equation\n<ul id=\"fs-id1167836523246\" data-bullet-style=\"open-circle\">\n<li>For 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, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6d28cb478bd8b6abe7d9573551313d6d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#92;&#110;&#101;&#32;&#48;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"46\" style=\"vertical-align: -4px;\" \/>\n<ul id=\"fs-id1167833290846\" data-bullet-style=\"bullet\">\n<li>If <em data-effect=\"italics\">b<\/em><sup>2<\/sup> \u2212 4<em data-effect=\"italics\">ac<\/em> &gt; 0, the equation has 2 real solutions.<\/li>\n<li>if <em data-effect=\"italics\">b<\/em><sup>2<\/sup> \u2212 4<em data-effect=\"italics\">ac<\/em> = 0, the equation has 1 real solution.<\/li>\n<li>if <em data-effect=\"italics\">b<\/em><sup>2<\/sup> \u2212 4<em data-effect=\"italics\">ac<\/em> &lt; 0, the equation has 2 complex solutions.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li>Methods to Solve Quadratic Equations:\n<ul id=\"fs-id1167833284845\" data-bullet-style=\"open-circle\">\n<li>Factoring<\/li>\n<li>Square Root Property<\/li>\n<li>Completing the Square<\/li>\n<li>Quadratic Formula<\/li>\n<\/ul>\n<\/li>\n<li>How to identify the most appropriate method to solve a quadratic equation.\n<ol id=\"fs-id1167825830231\" type=\"1\" class=\"stepwise\">\n<li>Try Factoring first. If the quadratic factors easily, this method is very quick.<\/li>\n<li>Try the <strong data-effect=\"bold\">Square Root Property<\/strong> next. If the equation fits the form <em data-effect=\"italics\">ax<\/em><sup>2<\/sup> = <em data-effect=\"italics\">k<\/em> or <em data-effect=\"italics\">a<\/em>(<em data-effect=\"italics\">x<\/em> \u2212 <em data-effect=\"italics\">h<\/em>)<sup>2<\/sup> = <em data-effect=\"italics\">k<\/em>, it can easily be solved by using the Square Root Property.<\/li>\n<li>Use the <strong data-effect=\"bold\">Quadratic Formula.<\/strong> Any other quadratic equation is best solved by using the Quadratic Formula.<\/li>\n<\/ol>\n<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167833196720\">\n<div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1167836707345\">\n<h4 data-type=\"title\">Practice Makes Perfect<\/h4>\n<p id=\"fs-id1167836692628\"><strong data-effect=\"bold\">Solve Quadratic Equations Using the Quadratic Formula<\/strong><\/p>\n<p id=\"fs-id1167836481619\">In the following exercises, solve by using the Quadratic Formula.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167836287708\">\n<div data-type=\"problem\" id=\"fs-id1167836579406\">\n<p id=\"fs-id1167836535044\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-532df2a0d76b158efc16b3bd6619e4cf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#43;&#109;&#45;&#51;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"133\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833128877\">\n<p id=\"fs-id1167833008092\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2bea017780be8f922f8fc43d3f7d6b70_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#45;&#49;&#44;&#109;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"118\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829756068\">\n<div data-type=\"problem\" id=\"fs-id1167836448251\">\n<p id=\"fs-id1167823012097\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-09f2825710febc0521d838cdb6cb9c2d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#45;&#57;&#110;&#43;&#53;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"132\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836613568\">\n<div data-type=\"problem\" id=\"fs-id1167836513341\">\n<p id=\"fs-id1167836502019\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-88246f374fa929a725a9fd3fdb9a957b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#45;&#55;&#112;&#43;&#51;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"128\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829747937\">\n<p id=\"fs-id1167829808028\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-974abaf08cfa09b184e10a992868b75e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#112;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#44;&#112;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"93\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836363363\">\n<div data-type=\"problem\" id=\"fs-id1167833046856\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-715f8efb3662e29fd9123c7fbdf7cba2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#123;&#113;&#125;&#94;&#123;&#50;&#125;&#43;&#56;&#113;&#45;&#51;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"127\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829785536\">\n<div data-type=\"problem\" id=\"fs-id1167836508391\">\n<p id=\"fs-id1167836706624\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4650409e470d88807fb735878aa0be3d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#43;&#55;&#112;&#43;&#49;&#50;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"129\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829586796\">\n<p id=\"fs-id1167829737988\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-96095c0d41a30f886ba25b89ad97cf85_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#112;&#61;&#45;&#52;&#44;&#112;&#61;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"120\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829580115\">\n<div data-type=\"problem\" id=\"fs-id1167833017933\">\n<p id=\"fs-id1167836547433\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d244c39bb4f762e4a8999601e058b353_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#113;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#113;&#45;&#49;&#56;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"127\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836704308\">\n<div data-type=\"problem\" id=\"fs-id1167836547994\">\n<p id=\"fs-id1167836312366\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8ac1af47344fb20be788e207e472d662_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#114;&#125;&#94;&#123;&#50;&#125;&#45;&#56;&#114;&#61;&#51;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"97\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836507162\">\n<p id=\"fs-id1167836359772\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7280cc7c6520a9820f61763499ed393b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#114;&#61;&#45;&#51;&#44;&#114;&#61;&#49;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"112\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829831326\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167829872216\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b552ba41ee3b748f0e201722801fbf20_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#116;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#51;&#116;&#61;&#45;&#52;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"116\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836574258\">\n<div data-type=\"problem\" id=\"fs-id1167836533091\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ebadcea6dc7945a481ff6a3eb4832835_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#123;&#117;&#125;&#94;&#123;&#50;&#125;&#43;&#55;&#117;&#45;&#50;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"131\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836398780\">\n<p id=\"fs-id1167836541400\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-71ef3085caa49ef47dc084e4c6030565_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#55;&plusmn;&#92;&#115;&#113;&#114;&#116;&#123;&#55;&#51;&#125;&#125;&#123;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"79\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836433826\">\n<div data-type=\"problem\" id=\"fs-id1167824704137\">\n<p id=\"fs-id1167836621198\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0049c037e976d78f3bd1834acaaccb0d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#43;&#56;&#112;&#43;&#53;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"128\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836534727\">\n<div data-type=\"problem\" id=\"fs-id1167836550911\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1d328bf9fd0a8af25e436039e44a3552_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#97;&#43;&#51;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"129\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833186350\">\n<p id=\"fs-id1167829595101\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b05346d77196e97ab038a0442ee0e084_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&plusmn;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#125;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"61\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836550976\">\n<div data-type=\"problem\" id=\"fs-id1167836391876\">\n<p id=\"fs-id1167836768482\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b914046eb0218fb7b2ff79fb3a813406_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#98;&#45;&#52;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"125\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829747947\">\n<div data-type=\"problem\" id=\"fs-id1167836512516\">\n<p id=\"fs-id1167836393334\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9a1bf9bd602c0e42d6f3678551215d76_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#56;&#120;&#45;&#52;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"122\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836408060\">\n<p id=\"fs-id1167836492529\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8463248510e42765606d320b50f4cd38_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#52;&plusmn;&#50;&#92;&#115;&#113;&#114;&#116;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"89\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836450575\">\n<div data-type=\"problem\" id=\"fs-id1167836514719\">\n<p id=\"fs-id1167836579737\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1490f4a5cd9e4646356856c0429a60f3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#121;&#45;&#52;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"120\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836548619\">\n<div data-type=\"problem\" id=\"fs-id1167833339898\">\n<p id=\"fs-id1167836286059\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c4cc218159e553d9a796934f65f20243_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#53;&#121;&#45;&#50;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"129\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836662437\">\n<p id=\"fs-id1167836513391\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5e1ad01a675660a613852328bced935c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#44;&#121;&#61;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"120\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836352286\">\n<div data-type=\"problem\" id=\"fs-id1167836595870\">\n<p id=\"fs-id1167833025364\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-73416fb9bbfcacf6179a63ee4253f52b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#120;&#45;&#50;&#48;&#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-id1167826077135\">\n<div data-type=\"problem\" id=\"fs-id1167829651145\">\n<p id=\"fs-id1167829651147\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6bd4091bc9ebac20d8d85cc4fce8d22d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#120;&#43;&#51;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"131\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836550959\">\n<p id=\"fs-id1167833310564\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-24a21c87879143e9cfb0fa014cf48703_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&plusmn;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#53;&#125;&#125;&#123;&#52;&#125;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"93\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167824891816\">\n<p id=\"fs-id1167830092950\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-44ca539f8af66d4f179226256f872ef6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#120;&#43;&#49;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"122\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167824738968\">\n<div data-type=\"problem\" id=\"fs-id1167836607323\">\n<p id=\"fs-id1167836607325\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e07d13565f7f56d4d5d4234712a734f9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#120;&#43;&#50;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"131\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167822916220\">\n<p id=\"fs-id1167833060460\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-99fd6ad4e009413f03039fb2d6b75a6e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#56;&#125;&plusmn;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#55;&#125;&#125;&#123;&#56;&#125;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"73\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836753598\">\n<div data-type=\"problem\" id=\"fs-id1167833340003\">\n<p id=\"fs-id1167829666500\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7ed6c64cef42c65e76dc48fb1ac435f3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#120;&#43;&#49;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"131\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833385657\">\n<div data-type=\"problem\" id=\"fs-id1167833007389\">\n<p id=\"fs-id1167833007391\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2616c21e0f3d27258e6d6d88f231a943_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#118;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#118;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#52;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"172\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836409136\">\n<p id=\"fs-id1167836409138\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-689ba57147efc21f62ef40ebb269624d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;&#61;&#50;&plusmn;&#50;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"74\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836622221\">\n<div data-type=\"problem\" id=\"fs-id1167836622223\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ce478f6b1078ee0db06fd09ca5e30d71_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"142\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836546600\">\n<div data-type=\"problem\" id=\"fs-id1167833048411\">\n<p id=\"fs-id1167836550106\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-30f222d87d09cadf333bd32ffde7f5b1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#49;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"151\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829691005\">\n<p id=\"fs-id1167829691007\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-639ce86103c9a2e17b201642577972de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#52;&#44;&#121;&#61;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"105\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829597601\">\n<div data-type=\"problem\" id=\"fs-id1167826077158\">\n<p id=\"fs-id1167826077160\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-30c78ac275633a5a6ae2b39b7d207473_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;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#49;\" 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-id1167836409913\">\n<div data-type=\"problem\" id=\"fs-id1167836409915\">\n<p id=\"fs-id1167836415993\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1f9b636cceaa91641af6a09b49bc7d8f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#50;&#125;&#109;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"119\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829711812\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2bea017780be8f922f8fc43d3f7d6b70_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#45;&#49;&#44;&#109;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"118\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833338350\">\n<div data-type=\"problem\" id=\"fs-id1167829627371\">\n<p id=\"fs-id1167829627373\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9eb0f516b78f5263cd601ad865a6fe92_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#43;&#110;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"106\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167824781360\">\n<div data-type=\"problem\" id=\"fs-id1167836694411\">\n<p id=\"fs-id1167836694413\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-152f45d1b506c5d383d28113d80a57da_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#98;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"97\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829717481\">\n<p id=\"fs-id1167829717483\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-aa6644b72c6005b2a97b0505ba38f9d9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#50;&plusmn;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#49;&#125;&#125;&#123;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"76\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829784975\">\n<div data-type=\"problem\" id=\"fs-id1167829784978\">\n<p id=\"fs-id1167836415664\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-20c07698df2c59ce01ef0156b628344b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#57;&#125;&#123;&#99;&#125;&#94;&#123;&#50;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#99;&#61;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"97\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167824748849\">\n<div data-type=\"problem\" id=\"fs-id1167829738093\">\n<p id=\"fs-id1167829738095\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-93fdccd4d30232214ca361f18ee50d24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#54;&#123;&#99;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#52;&#99;&#43;&#57;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"142\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836518759\">\n<p id=\"fs-id1167836518761\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-81240219f16f804047c3da57515b210a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"54\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829952779\">\n<div data-type=\"problem\" id=\"fs-id1167829952781\">\n<p id=\"fs-id1167836609912\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4b900a8ad2f71260b24e580935630797_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#53;&#123;&#100;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#48;&#100;&#43;&#51;&#54;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"155\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836511004\">\n<div data-type=\"problem\" id=\"fs-id1167836387219\">\n<p id=\"fs-id1167836387221\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bd6f6ea2675a18a8e4de6d390179467e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#53;&#123;&#113;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#48;&#113;&#43;&#57;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"145\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836516507\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a73233a3b4bb6f67ea9d3288b22ff7d6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#113;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"55\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836376530\">\n<div data-type=\"problem\" id=\"fs-id1167836376532\">\n<p id=\"fs-id1167829692331\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-17059520b40e1a1f4c430db64813e6a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#54;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#56;&#121;&#43;&#49;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"137\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836477084\"><strong data-effect=\"bold\">Use the Discriminant to Predict the Number of Real Solutions of a Quadratic Equation<\/strong><\/p>\n<p id=\"fs-id1167824767094\">In the following exercises, determine the number of real solutions for each quadratic equation.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167822916231\">\n<div data-type=\"problem\" id=\"fs-id1167822916233\">\n<p id=\"fs-id1167836569038\">\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-d1232d10066cd83c7db09372a4eb9f6d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#120;&#43;&#49;&#54;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"139\" style=\"vertical-align: -2px;\" \/><\/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-5f52531d2a57f7de77b4c330fdb47d7b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#54;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#54;&#121;&#43;&#57;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"147\" 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-8cbd09025c0bbf2d624d310bd13a3f82_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#109;&#45;&#53;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"141\" style=\"vertical-align: -2px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1167836521669\">\n<p id=\"fs-id1167836549783\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d6d7ed88a8716d7d645863274ddf6ff5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#110;&#111;&#32;&#114;&#101;&#97;&#108;&#32;&#115;&#111;&#108;&#117;&#116;&#105;&#111;&#110;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"128\" 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-4868771cbc422b5818f85500909ce433_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"7\" 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-e584dd0bab4e6c8efc164939c28db757_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: 0px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836572983\">\n<div data-type=\"problem\" id=\"fs-id1167836376107\">\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-73282d4d7d3ef808e49ecb2ad95c9226_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#123;&#118;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#53;&#118;&#43;&#50;&#53;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"146\" style=\"vertical-align: -2px;\" \/><\/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-e0377bfc4f5d7f1641334f314e96b589_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#48;&#48;&#123;&#119;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#48;&#119;&#43;&#57;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"162\" style=\"vertical-align: -2px;\" \/><\/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-f60417a00ec115b6901fd36dbe72f65c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#123;&#99;&#125;&#94;&#123;&#50;&#125;&#43;&#55;&#99;&#45;&#49;&#48;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"134\" style=\"vertical-align: -2px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836328607\">\n<div data-type=\"problem\" id=\"fs-id1167833053956\">\n<p id=\"fs-id1167833053958\">\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-b0dcb7a9d1c874ab3d9dce4d47626641_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#114;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#50;&#114;&#43;&#51;&#54;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"136\" style=\"vertical-align: -2px;\" \/><\/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-05850cbccf771063ff7c87dfe0f0742b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#123;&#116;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#49;&#116;&#43;&#53;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"132\" style=\"vertical-align: -2px;\" \/><\/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-6817f4f92d39a8f3967009c4c5b23039_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#123;&#118;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#118;&#45;&#49;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"129\" style=\"vertical-align: -1px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1167825760157\">\n<p id=\"fs-id1167825760159\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4868771cbc422b5818f85500909ce433_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"7\" 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-d6d7ed88a8716d7d645863274ddf6ff5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#110;&#111;&#32;&#114;&#101;&#97;&#108;&#32;&#115;&#111;&#108;&#117;&#116;&#105;&#111;&#110;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"128\" 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-e584dd0bab4e6c8efc164939c28db757_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: 0px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829709403\">\n<div data-type=\"problem\" id=\"fs-id1167833240365\">\n<p id=\"fs-id1167833240367\">\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-1169dca6c1e642a2ff623488fb30182f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#53;&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#48;&#112;&#43;&#49;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"146\" 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-67f51589b5ce6003fb98f79317753494_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#123;&#113;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#113;&#45;&#54;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"127\" 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-68d0b1b171a3e6c17f6f405aa9cececd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#121;&#43;&#56;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"129\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<p id=\"fs-id1167836530634\"><strong data-effect=\"bold\">Identify the Most Appropriate Method to Use to Solve a Quadratic Equation<\/strong><\/p>\n<p id=\"fs-id1167836525480\">In the following exercises, identify the most appropriate method (Factoring, Square Root, or Quadratic Formula) to use to solve each quadratic equation. Do not solve.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167836493661\">\n<div data-type=\"problem\" id=\"fs-id1167836732375\">\n<p id=\"fs-id1167836732377\">\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-619f3d17e1bcd90ee5320008e7fc85d1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#120;&#45;&#50;&#52;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"131\" 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-b455ab0cde1f182b57cbd4395f27b203_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"101\" 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-1a03a7cc1be0af02fc1226d9ba05c410_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#52;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#109;&#61;&#49;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"127\" style=\"vertical-align: -2px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1167833076778\">\n<p id=\"fs-id1167836666727\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5a6cc845d71d2c453386b252ae835025_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#102;&#97;&#99;&#116;&#111;&#114;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"45\" 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-d14f21c8b4ebeb4e667fcd30474b4a2b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#113;&#117;&#97;&#114;&#101;&#32;&#114;&#111;&#111;&#116;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"87\" style=\"vertical-align: -3px;\" \/><\/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-616cd8c4b130d193cb4be564edc070c5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#81;&#117;&#97;&#100;&#114;&#97;&#116;&#105;&#99;&#32;&#70;&#111;&#114;&#109;&#117;&#108;&#97;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"148\" style=\"vertical-align: -3px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167825946794\">\n<div data-type=\"problem\" id=\"fs-id1167829785715\">\n<p id=\"fs-id1167829785717\">\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-e29fc9e70b5e4ab358cad2da42ef3946_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#118;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#56;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"110\" 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-ee197698a8f5ffea9173c8a47c5c84ab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#119;&#125;&#94;&#123;&#50;&#125;&#45;&#57;&#119;&#45;&#50;&#50;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"137\" 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-2e7b6817c4ed5655d3269c1713c5990c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#48;&#61;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"99\" style=\"vertical-align: -1px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167825949223\">\n<div data-type=\"problem\" id=\"fs-id1167836530566\">\n<p id=\"fs-id1167836530569\">\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-353bbd0ae8fbaffe5f7d7c2c169affe9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#52;&#61;&#50;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"107\" style=\"vertical-align: -2px;\" \/><\/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-b27efc499f7f6acc34f09a723663f0de_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;&#125;&#123;&#52;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#49;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"104\" 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-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;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1167833364609\">\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-616cd8c4b130d193cb4be564edc070c5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#81;&#117;&#97;&#100;&#114;&#97;&#116;&#105;&#99;&#32;&#70;&#111;&#114;&#109;&#117;&#108;&#97;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"148\" style=\"vertical-align: -3px;\" \/><\/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-d14f21c8b4ebeb4e667fcd30474b4a2b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#113;&#117;&#97;&#114;&#101;&#32;&#114;&#111;&#111;&#116;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"87\" style=\"vertical-align: -3px;\" \/><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5a6cc845d71d2c453386b252ae835025_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#102;&#97;&#99;&#116;&#111;&#114;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"45\" style=\"vertical-align: -1px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833239671\">\n<div data-type=\"problem\" id=\"fs-id1167833239673\">\n<p id=\"fs-id1167833053326\">\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-7659661091ac964089bffc1a1d4b2270_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#53;&#98;&#61;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"104\" style=\"vertical-align: -2px;\" \/><\/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-853cd1ecd99291fccbe5c22fdede505e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#57;&#125;&#123;&#118;&#125;&#94;&#123;&#50;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#118;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"99\" style=\"vertical-align: -6px;\" \/><\/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-d2eca43960c7fbc4eae45048f4281738_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#119;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"100\" style=\"vertical-align: -7px;\" \/><\/div>\n<\/div>\n<\/div>\n<div class=\"writing\" data-depth=\"2\" id=\"fs-id1167836508342\">\n<h4 data-type=\"title\">Writing Exercises<\/h4>\n<div data-type=\"exercise\" id=\"fs-id1167829790134\">\n<div data-type=\"problem\" id=\"fs-id1167829790136\">\n<p id=\"fs-id1167829752402\">Solve the equation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3030bb84eb857157c135d0472d823c2c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#48;&#120;&#61;&#49;&#50;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"118\" style=\"vertical-align: -2px;\" \/><\/p>\n<p id=\"fs-id1167836498963\"><span class=\"token\">\u24d0<\/span> by completing the square<\/p>\n<p id=\"fs-id1167833083076\"><span class=\"token\">\u24d1<\/span> using the Quadratic Formula<\/p>\n<p id=\"fs-id1167824755448\"><span class=\"token\">\u24d2<\/span> Which method do you prefer? Why?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826131790\">\n<p id=\"fs-id1167826131792\">Answers will vary.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836601151\">\n<div data-type=\"problem\" id=\"fs-id1167836601154\">\n<p id=\"fs-id1167836496817\">Solve the equation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-179e32074d13ab87da61d42a81b7ce65_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#50;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#51;&#121;&#61;&#50;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"124\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"fs-id1167836597657\"><span class=\"token\">\u24d0<\/span> by completing the square<\/p>\n<p id=\"fs-id1167832936319\"><span class=\"token\">\u24d1<\/span> using the Quadratic Formula<\/p>\n<p id=\"fs-id1167836601619\"><span class=\"token\">\u24d2<\/span> Which method do you prefer? Why?<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167829695326\">\n<h4 data-type=\"title\">Self Check<\/h4>\n<p id=\"fs-id1167836390295\"><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-id1167829744205\" data-alt=\"This table provides a checklist to evaluate mastery of the objectives of this section. Choose how would you respond to the statement \u201cI can solve quadratic equations using the quadratic formula.\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 use the discriminant to predict the number of solutions of a quadratic equation.\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 identify the most appropriate method to use to solve a quadratic equation.\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_03_201_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This table provides a checklist to evaluate mastery of the objectives of this section. Choose how would you respond to the statement \u201cI can solve quadratic equations using the quadratic formula.\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 use the discriminant to predict the number of solutions of a quadratic equation.\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 identify the most appropriate method to use to solve a quadratic equation.\u201d \u201cConfidently,\u201d \u201cwith some help,\u201d or \u201cNo, I don\u2019t get it.\u201d\" \/><\/span><\/p>\n<p id=\"fs-id1167829893818\"><span class=\"token\">\u24d1<\/span> What does this checklist tell you about your mastery of this section? What steps will you take to improve?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"glossary\" class=\"textbox shaded\">\n<h3 data-type=\"glossary-title\">Glossary<\/h3>\n<dl id=\"fs-id1167825946740\">\n<dt>discriminant<\/dt>\n<dd id=\"fs-id1167825946743\">In the Quadratic Formula, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4a9ea4ea9a72969cc868edd6492b88ca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#98;&plusmn;&#92;&#115;&#113;&#114;&#116;&#123;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#97;&#99;&#125;&#125;&#123;&#50;&#97;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"102\" style=\"vertical-align: -6px;\" \/> the quantity <em data-effect=\"italics\">b<\/em><sup>2<\/sup> \u2212 4<em data-effect=\"italics\">ac<\/em> is called the discriminant.<\/dd>\n<\/dl>\n<\/div>\n","protected":false},"author":103,"menu_order":4,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-3909","chapter","type-chapter","status-publish","hentry"],"part":3677,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/3909","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\/3909\/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\/3909\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/media?parent=3909"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapter-type?post=3909"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/contributor?post=3909"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/license?post=3909"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}