{"id":104,"date":"2023-06-21T13:22:33","date_gmt":"2023-06-21T13:22:33","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/completing-the-square-and-the-quadratic-formula\/"},"modified":"2023-09-07T16:13:55","modified_gmt":"2023-09-07T16:13:55","slug":"completing-the-square-and-the-quadratic-formula","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/completing-the-square-and-the-quadratic-formula\/","title":{"raw":"\u25aa   Completing the Square and the Quadratic Formula","rendered":"\u25aa   Completing the Square and the Quadratic Formula"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Complete the square to solve a quadratic equation.<\/li>\r\n \t<li>Use the quadratic formula to solve a quadratic equation.<\/li>\r\n \t<li>Use the discriminant to determine the number and type of solutions to a quadratic equation.<\/li>\r\n<\/ul>\r\n<\/div>\r\nNot all quadratic equations can be factored or can be solved in their original form using the square root property. In these cases, we may use other methods for solving a <strong>quadratic equation.<\/strong>\r\n<h2>Completing the Square<\/h2>\r\nOne method is known as <strong>completing the square<\/strong>. Using this process, we add or subtract terms to both sides of the equation until we have a perfect square trinomial on one side of the equal sign. We then apply the square root property. To complete the square, the leading coefficient, [latex]a[\/latex],\u00a0must equal 1. If it does not, then divide the entire equation by [latex]a[\/latex]. Then, we can use the following procedures to solve a quadratic equation by completing the square.\r\n\r\nWe will use the example [latex]{x}^{2}+4x+1=0[\/latex] to illustrate each step.\r\n<ol>\r\n \t<li>Given a quadratic equation that cannot be factored and with [latex]a=1[\/latex], first add or subtract the constant term to the right sign of the equal sign.\r\n<div style=\"text-align: center;\">[latex]{x}^{2}+4x=-1[\/latex]<\/div><\/li>\r\n \t<li>Multiply the <em>b <\/em>term by [latex]\\frac{1}{2}[\/latex] and square it.\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}\\frac{1}{2}\\left(4\\right)=2\\hfill \\\\ {2}^{2}=4\\hfill \\end{array}[\/latex]<\/div><\/li>\r\n \t<li>Add [latex]{\\left(\\frac{1}{2}b\\right)}^{2}[\/latex] to both sides of the equal sign and simplify the right side. We have\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}{x}^{2}+4x+4=-1+4\\hfill \\\\ {x}^{2}+4x+4=3\\hfill \\end{array}[\/latex]<\/div><\/li>\r\n \t<li>The left side of the equation can now be factored as a perfect square.\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}{x}^{2}+4x+4=3\\hfill \\\\ {\\left(x+2\\right)}^{2}=3\\hfill \\end{array}[\/latex]<\/div><\/li>\r\n \t<li>Use the square root property and solve.\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}\\sqrt{{\\left(x+2\\right)}^{2}}=\\pm \\sqrt{3}\\hfill \\\\ x+2=\\pm \\sqrt{3}\\hfill \\\\ x=-2\\pm \\sqrt{3}\\hfill \\end{array}[\/latex]<\/div><\/li>\r\n \t<li>The solutions are [latex]x=-2+\\sqrt{3}[\/latex], [latex]x=-2-\\sqrt{3}[\/latex].<\/li>\r\n<\/ol>\r\n<div class=\"textbox examples\">\r\n<h3>Properties of Equality and taking the square root of both sides<\/h3>\r\nRemember that we are permitted, by the properties of equality, to add, subtract, multiply, or divide the same amount to both sides of an equation. Doing so won't change the value of the equation but it will enable us to isolate the variable on one side (that is, to solve the equation for the variable).\r\n\r\nThe square root property gives us another operation we can do to both sides of an equation, taking the square root. We just have to remember when taking the square root (or any even root, as we'll see later), to consider both the positive and negative possibilities of the constant.\r\n<p style=\"text-align: center;\">The square root property<\/p>\r\n<p style=\"text-align: center;\">If [latex]\\sqrt{x}=k[\/latex]<\/p>\r\n<p style=\"text-align: center;\">Then [latex]x = \\pm{k}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Solving a Quadratic by Completing the Square<\/h3>\r\nSolve the quadratic equation by completing the square: [latex]{x}^{2}-3x - 5=0[\/latex].\r\n\r\n[reveal-answer q=\"676921\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"676921\"]\r\n\r\nFirst, move the constant term to the right side of the equal sign by adding 5 to both sides of the equation.\r\n<div style=\"text-align: center;\">[latex]{x}^{2}-3x=5[\/latex]<\/div>\r\nThen, take [latex]\\frac{1}{2}[\/latex] of the <em>b <\/em>term and square it.\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}\\frac{1}{2}\\left(-3\\right)=-\\frac{3}{2}\\hfill \\\\ {\\left(-\\frac{3}{2}\\right)}^{2}=\\frac{9}{4}\\hfill \\end{array}[\/latex]<\/div>\r\nAdd the result to both sides of the equal sign.\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}\\text{ }{x}^{2}-3x+{\\left(-\\frac{3}{2}\\right)}^{2}=5+{\\left(-\\frac{3}{2}\\right)}^{2}\\hfill \\\\ {x}^{2}-3x+\\frac{9}{4}=5+\\frac{9}{4}\\hfill \\end{array}[\/latex]<\/div>\r\nFactor the left side as a perfect square and simplify the right side.\r\n<div style=\"text-align: center;\">[latex]{\\left(x-\\frac{3}{2}\\right)}^{2}=\\frac{29}{4}[\/latex]<\/div>\r\nUse the square root property and solve.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\sqrt{{\\left(x-\\frac{3}{2}\\right)}^{2}}=\\pm \\sqrt{\\frac{29}{4}}\\hfill \\\\ \\left(x-\\frac{3}{2}\\right)=\\pm \\frac{\\sqrt{29}}{2}\\hfill \\\\ x=\\frac{3}{2}\\pm \\frac{\\sqrt{29}}{2}\\hfill \\end{array}[\/latex]<\/p>\r\nThe solutions are [latex]x=\\frac{3}{2}+\\frac{\\sqrt{29}}{2}[\/latex], [latex]x=\\frac{3}{2}-\\frac{\\sqrt{29}}{2}[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nSolve by completing the square: [latex]{x}^{2}-6x=13[\/latex].\r\n\r\n[reveal-answer q=\"222291\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"222291\"]\r\n\r\n[latex]x=3\\pm \\sqrt{22}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\nNote that when solving a quadratic by completing the square, a negative value will sometimes arise under the square root symbol. Later, we'll see that this value can be represented by a complex number (as shown in the video help for the problem below). We may also treat this type of solution as <em>unreal<\/em>, stating that no real solutions exist for this equation, by writing <em>DNE<\/em>. We will study complex numbers more thoroughly in a later module.\r\n\r\n[ohm_question]1384[\/ohm_question]\r\n\r\n[ohm_question height=\"310\"]79619[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Using the Quadratic Formula<\/h2>\r\nThe fourth method of solving a <strong>quadratic equation<\/strong> is by using the <strong>quadratic formula<\/strong>, a formula that will solve all quadratic equations. Although the quadratic formula works on any quadratic equation in standard form, it is easy to make errors in substituting the values into the formula. Pay close attention when substituting, and use parentheses when inserting a negative number.\r\n\r\nWe can derive the quadratic formula by <strong>completing the square<\/strong>. We will assume that the leading coefficient is positive; if it is negative, we can multiply the equation by [latex]-1[\/latex] and obtain a positive <em>a<\/em>. Given [latex]a{x}^{2}+bx+c=0[\/latex], [latex]a\\ne 0[\/latex], we will complete the square as follows:\r\n<ol>\r\n \t<li>First, move the constant term to the right side of the equal sign:\r\n<div style=\"text-align: center;\">[latex]a{x}^{2}+bx=-c[\/latex]<\/div><\/li>\r\n \t<li>As we want the leading coefficient to equal 1, divide through by <em>a<\/em>:\r\n<div style=\"text-align: center;\">[latex]{x}^{2}+\\frac{b}{a}x=-\\frac{c}{a}[\/latex]<\/div><\/li>\r\n \t<li>Then, find [latex]\\frac{1}{2}[\/latex] of the middle term, and add [latex]{\\left(\\frac{1}{2}\\frac{b}{a}\\right)}^{2}=\\frac{{b}^{2}}{4{a}^{2}}[\/latex] to both sides of the equal sign:\r\n<div style=\"text-align: center;\">[latex]{x}^{2}+\\frac{b}{a}x+\\frac{{b}^{2}}{4{a}^{2}}=\\frac{{b}^{2}}{4{a}^{2}}-\\frac{c}{a}[\/latex]<\/div><\/li>\r\n \t<li>Next, write the left side as a perfect square. Find the common denominator of the right side and write it as a single fraction:\r\n<div style=\"text-align: center;\">[latex]{\\left(x+\\frac{b}{2a}\\right)}^{2}=\\frac{{b}^{2}-4ac}{4{a}^{2}}[\/latex]<\/div><\/li>\r\n \t<li>Now, use the square root property, which gives\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}x+\\frac{b}{2a}=\\pm \\sqrt{\\frac{{b}^{2}-4ac}{4{a}^{2}}}\\hfill \\\\ x+\\frac{b}{2a}=\\frac{\\pm \\sqrt{{b}^{2}-4ac}}{2a}\\hfill \\end{array}[\/latex]<\/div><\/li>\r\n \t<li>Finally, add [latex]-\\frac{b}{2a}[\/latex] to both sides of the equation and combine the terms on the right side. Thus,\r\n<div style=\"text-align: center;\">[latex]x=\\frac{-b\\pm \\sqrt{{b}^{2}-4ac}}{2a}[\/latex]<\/div><\/li>\r\n<\/ol>\r\n<div class=\"textbox\">\r\n<h3>A General Note: The Quadratic Formula<\/h3>\r\nWritten in standard form, [latex]a{x}^{2}+bx+c=0[\/latex], any quadratic equation can be solved using the <strong>quadratic formula<\/strong>:\r\n<div style=\"text-align: center;\">[latex]x=\\frac{-b\\pm \\sqrt{{b}^{2}-4ac}}{2a}[\/latex]<\/div>\r\nwhere <em>a<\/em>, <em>b<\/em>, and <em>c<\/em> are real numbers and [latex]a\\ne 0[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given a quadratic equation, solve it using the quadratic formula<\/h3>\r\n<ol>\r\n \t<li>Make sure the equation is in standard form: [latex]a{x}^{2}+bx+c=0[\/latex].<\/li>\r\n \t<li>Make note of the values of the coefficients and constant term, [latex]a,b[\/latex], and [latex]c[\/latex].<\/li>\r\n \t<li>Carefully substitute the values noted in step 2 into the equation. To avoid needless errors, use parentheses around each number input into the formula.<\/li>\r\n \t<li>Calculate and solve.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example : Solve A Quadratic Equation Using the Quadratic Formula<\/h3>\r\nSolve the quadratic equation: [latex]{x}^{2}+5x+1=0[\/latex].\r\n\r\n[reveal-answer q=\"641400\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"641400\"]\r\n\r\nIdentify the coefficients: [latex]a=1,b=5,c=1[\/latex]. Then use the quadratic formula.\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}x\\hfill&amp;=\\frac{-\\left(5\\right)\\pm \\sqrt{{\\left(5\\right)}^{2}-4\\left(1\\right)\\left(1\\right)}}{2\\left(1\\right)}\\hfill \\\\ \\hfill&amp;=\\frac{-5\\pm \\sqrt{25 - 4}}{2}\\hfill \\\\ \\hfill&amp;=\\frac{-5\\pm \\sqrt{21}}{2}\\hfill \\end{array}[\/latex]<\/div>\r\n<div>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Recall adding and subtracting fractions<\/h3>\r\nThe form we have used to recall adding and subtracting fractions can help us write the solutions to quadratic equations.\r\n\r\n[latex]\\dfrac{a}{b}\\pm\\dfrac{c}{d} = \\dfrac{ad \\pm bc}{bd}[\/latex]\r\n\r\nEx. The solutions\r\n\r\n[latex]x=\\dfrac{-b\\pm \\sqrt{{b}^{2}-4ac}}{2a}[\/latex]\r\n\r\ncan also be written as two separate fractions\r\n\r\n[latex]x=\\dfrac{-b}{2a} \\pm \\dfrac{\\sqrt{{b}^{2}-4ac}}{2a}[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Solving a Quadratic Equation with the Quadratic Formula<\/h3>\r\nUse the quadratic formula to solve [latex]{x}^{2}+x+2=0[\/latex].\r\n\r\n[reveal-answer q=\"688902\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"688902\"]\r\n\r\nFirst, we identify the coefficients: [latex]a=1,b=1[\/latex], and [latex]c=2[\/latex].\r\n\r\nSubstitute these values into the quadratic formula.\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}x\\hfill&amp;=\\dfrac{-b\\pm \\sqrt{{b}^{2}-4ac}}{2a}\\hfill \\\\\\hfill&amp;=\\dfrac{-\\left(1\\right)\\pm \\sqrt{{\\left(1\\right)}^{2}-\\left(4\\right)\\cdot \\left(1\\right)\\cdot \\left(2\\right)}}{2\\cdot 1}\\hfill \\\\\\hfill&amp;=\\dfrac{-1\\pm \\sqrt{1 - 8}}{2}\\hfill \\\\ \\hfill&amp;=\\dfrac{-1\\pm \\sqrt{-7}}{2}\\hfill \\end{array}[\/latex]<\/div>\r\n<div><\/div>\r\nThe solutions to the equation are <em>unreal <\/em>because the square root of a negative number does not exist in the real numbers. If asked to find all <em>real<\/em> solutions, we would indicate that they do not exist by writing\u00a0<em>DNE<\/em>.\r\n\r\nWe may also express unreal solutions as complex numbers by writing\r\n\r\n[latex]x=\\dfrac{-1\\pm i\\sqrt{7}}{2}[\/latex]\r\n\r\nor\r\n\r\n[latex]x=-\\dfrac{1}{2}+\\dfrac{\\sqrt{7}}{2}i[\/latex] and [latex]x=-\\dfrac{1}{2}-\\dfrac{\\sqrt{7}}{2}i[\/latex]\u00a0,\u00a0where [latex]\\sqrt{-1}[\/latex] is expressed as [latex]i[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nSolve the quadratic equation using the quadratic formula: [latex]9{x}^{2}+3x - 2=0[\/latex].\r\n\r\n[reveal-answer q=\"232269\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"232269\"]\r\n\r\n[latex]x=-\\frac{2}{3}[\/latex], [latex]x=\\frac{1}{3}[\/latex][\/hidden-answer]\r\n\r\n[ohm_question height=\"205\"]4014[\/ohm_question]\r\n\r\n[ohm_question height=\"425\"]35639[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>The Discriminant<\/h2>\r\nThe <strong>quadratic formula<\/strong> not only generates the solutions to a quadratic equation, it tells us about the nature of the solutions when we consider the <strong>discriminant<\/strong>, or the expression under the radical, [latex]{b}^{2}-4ac[\/latex]. The discriminant tells us whether the solutions are real numbers or complex numbers as well as how many solutions of each type to expect. The table below\u00a0relates the value of the discriminant to the solutions of a quadratic equation.\r\n<table summary=\"A table with 5 rows and 2 columns. The entries in the first row are: Value of Discriminant and Results. The entries in the second row are: b squared minus four times a times c equals zero and One rational solution (double solution). The entries in the third row are: b squared minus four times a times c is greater than zero, perfect square and Two rational solutions. The entries in the fourth row are: b squared minus four times a times c is greater than zero, not a perfect square and Two irrational solutions. The entries in the fifth row are: b squared minus four times a times c is less than zero and Two complex solutions.\">\r\n<thead>\r\n<tr>\r\n<th>Value of Discriminant<\/th>\r\n<th>Results<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>[latex]{b}^{2}-4ac=0[\/latex]<\/td>\r\n<td>One rational solution (double solution)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]{b}^{2}-4ac&gt;0[\/latex], perfect square<\/td>\r\n<td>Two rational solutions<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]{b}^{2}-4ac&gt;0[\/latex], not a perfect square<\/td>\r\n<td>Two irrational solutions<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]{b}^{2}-4ac&lt;0[\/latex]<\/td>\r\n<td>Two complex solutions<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"textbox examples\">\r\n<h3>recall rational and irrational numbers are real numbers<\/h3>\r\n<ul>\r\n \t<li><strong>Rational numbers<\/strong>: A rational number is a number that can be expressed as a ratio of integers (a fraction with an integer numerator and a positive, non-zero integer denominator). The square root of a perfect square, such as [latex]\\sqrt{25}[\/latex] is rational because it can be expressed as [latex]\\dfrac{5}{1} = 5[\/latex]. Solutions containing only rational numbers are themselves rational solutions.<\/li>\r\n \t<li><strong>Irrational numbers<\/strong>: All the real numbers that are not rational are called irrational numbers. These numbers cannot be expressed as a fraction of integers. As quadratic solutions, irrational numbers arise in the form of a square root of a numbers that is not a perfect square. For example, [latex]\\sqrt{6}[\/latex] cannot be expressed as a ratio of two integers. [latex]\\sqrt{6}[\/latex] is an irrational number. Solutions containing an irrational number are themselves irrational solutions.<\/li>\r\n \t<li>Both rational and irrational numbers are real numbers. Quadratic solutions that are either rational or irrational are real solutions.<\/li>\r\n \t<li>Quadratic solutions that are unreal (they have a negative under the square root symbol, the <em>radical\u00a0<\/em>symbol) are called <em>complex<\/em> solutions.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>A General Note: The Discriminant<\/h3>\r\nFor [latex]a{x}^{2}+bx+c=0[\/latex], where [latex]a[\/latex], [latex]b[\/latex], and [latex]c[\/latex] are real numbers, the <strong>discriminant<\/strong> is the expression under the radical in the quadratic formula: [latex]{b}^{2}-4ac[\/latex]. It tells us whether the solutions are real numbers or complex numbers and how many solutions of each type to expect.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Using the Discriminant to Find the Nature of the Solutions to a Quadratic Equation<\/h3>\r\nUse the discriminant to find the nature of the solutions to the following quadratic equations:\r\n<ol>\r\n \t<li>[latex]{x}^{2}+4x+4=0[\/latex]<\/li>\r\n \t<li>[latex]8{x}^{2}+14x+3=0[\/latex]<\/li>\r\n \t<li>[latex]3{x}^{2}-5x - 2=0[\/latex]<\/li>\r\n \t<li>[latex]3{x}^{2}-10x+15=0[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"229118\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"229118\"]\r\n\r\nCalculate the discriminant [latex]{b}^{2}-4ac[\/latex] for each equation and state the expected type of solutions.\r\n<ol>\r\n \t<li>[latex]{x}^{2}+4x+4=0[\/latex]: [latex]{b}^{2}-4ac={\\left(4\\right)}^{2}-4\\left(1\\right)\\left(4\\right)=0[\/latex]. There will be one rational double solution.<\/li>\r\n \t<li>[latex]8{x}^{2}+14x+3=0[\/latex]: [latex]{b}^{2}-4ac={\\left(14\\right)}^{2}-4\\left(8\\right)\\left(3\\right)=100[\/latex]. As [latex]100[\/latex] is a perfect square, there will be two rational solutions.<\/li>\r\n \t<li>[latex]3{x}^{2}-5x - 2=0[\/latex]: [latex]{b}^{2}-4ac={\\left(-5\\right)}^{2}-4\\left(3\\right)\\left(-2\\right)=49[\/latex]. As [latex]49[\/latex] is a perfect square, there will be two rational solutions.<\/li>\r\n \t<li>[latex]3{x}^{2}-10x+15=0[\/latex]: [latex]{b}^{2}-4ac={\\left(-10\\right)}^{2}-4\\left(3\\right)\\left(15\\right)=-80[\/latex]. There will be two complex solutions.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question height=\"235\"]35145[\/ohm_question]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Complete the square to solve a quadratic equation.<\/li>\n<li>Use the quadratic formula to solve a quadratic equation.<\/li>\n<li>Use the discriminant to determine the number and type of solutions to a quadratic equation.<\/li>\n<\/ul>\n<\/div>\n<p>Not all quadratic equations can be factored or can be solved in their original form using the square root property. In these cases, we may use other methods for solving a <strong>quadratic equation.<\/strong><\/p>\n<h2>Completing the Square<\/h2>\n<p>One method is known as <strong>completing the square<\/strong>. Using this process, we add or subtract terms to both sides of the equation until we have a perfect square trinomial on one side of the equal sign. We then apply the square root property. To complete the square, the leading coefficient, [latex]a[\/latex],\u00a0must equal 1. If it does not, then divide the entire equation by [latex]a[\/latex]. Then, we can use the following procedures to solve a quadratic equation by completing the square.<\/p>\n<p>We will use the example [latex]{x}^{2}+4x+1=0[\/latex] to illustrate each step.<\/p>\n<ol>\n<li>Given a quadratic equation that cannot be factored and with [latex]a=1[\/latex], first add or subtract the constant term to the right sign of the equal sign.\n<div style=\"text-align: center;\">[latex]{x}^{2}+4x=-1[\/latex]<\/div>\n<\/li>\n<li>Multiply the <em>b <\/em>term by [latex]\\frac{1}{2}[\/latex] and square it.\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}\\frac{1}{2}\\left(4\\right)=2\\hfill \\\\ {2}^{2}=4\\hfill \\end{array}[\/latex]<\/div>\n<\/li>\n<li>Add [latex]{\\left(\\frac{1}{2}b\\right)}^{2}[\/latex] to both sides of the equal sign and simplify the right side. We have\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}{x}^{2}+4x+4=-1+4\\hfill \\\\ {x}^{2}+4x+4=3\\hfill \\end{array}[\/latex]<\/div>\n<\/li>\n<li>The left side of the equation can now be factored as a perfect square.\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}{x}^{2}+4x+4=3\\hfill \\\\ {\\left(x+2\\right)}^{2}=3\\hfill \\end{array}[\/latex]<\/div>\n<\/li>\n<li>Use the square root property and solve.\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}\\sqrt{{\\left(x+2\\right)}^{2}}=\\pm \\sqrt{3}\\hfill \\\\ x+2=\\pm \\sqrt{3}\\hfill \\\\ x=-2\\pm \\sqrt{3}\\hfill \\end{array}[\/latex]<\/div>\n<\/li>\n<li>The solutions are [latex]x=-2+\\sqrt{3}[\/latex], [latex]x=-2-\\sqrt{3}[\/latex].<\/li>\n<\/ol>\n<div class=\"textbox examples\">\n<h3>Properties of Equality and taking the square root of both sides<\/h3>\n<p>Remember that we are permitted, by the properties of equality, to add, subtract, multiply, or divide the same amount to both sides of an equation. Doing so won&#8217;t change the value of the equation but it will enable us to isolate the variable on one side (that is, to solve the equation for the variable).<\/p>\n<p>The square root property gives us another operation we can do to both sides of an equation, taking the square root. We just have to remember when taking the square root (or any even root, as we&#8217;ll see later), to consider both the positive and negative possibilities of the constant.<\/p>\n<p style=\"text-align: center;\">The square root property<\/p>\n<p style=\"text-align: center;\">If [latex]\\sqrt{x}=k[\/latex]<\/p>\n<p style=\"text-align: center;\">Then [latex]x = \\pm{k}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Solving a Quadratic by Completing the Square<\/h3>\n<p>Solve the quadratic equation by completing the square: [latex]{x}^{2}-3x - 5=0[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q676921\">Show Solution<\/span><\/p>\n<div id=\"q676921\" class=\"hidden-answer\" style=\"display: none\">\n<p>First, move the constant term to the right side of the equal sign by adding 5 to both sides of the equation.<\/p>\n<div style=\"text-align: center;\">[latex]{x}^{2}-3x=5[\/latex]<\/div>\n<p>Then, take [latex]\\frac{1}{2}[\/latex] of the <em>b <\/em>term and square it.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}\\frac{1}{2}\\left(-3\\right)=-\\frac{3}{2}\\hfill \\\\ {\\left(-\\frac{3}{2}\\right)}^{2}=\\frac{9}{4}\\hfill \\end{array}[\/latex]<\/div>\n<p>Add the result to both sides of the equal sign.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}\\text{ }{x}^{2}-3x+{\\left(-\\frac{3}{2}\\right)}^{2}=5+{\\left(-\\frac{3}{2}\\right)}^{2}\\hfill \\\\ {x}^{2}-3x+\\frac{9}{4}=5+\\frac{9}{4}\\hfill \\end{array}[\/latex]<\/div>\n<p>Factor the left side as a perfect square and simplify the right side.<\/p>\n<div style=\"text-align: center;\">[latex]{\\left(x-\\frac{3}{2}\\right)}^{2}=\\frac{29}{4}[\/latex]<\/div>\n<p>Use the square root property and solve.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\sqrt{{\\left(x-\\frac{3}{2}\\right)}^{2}}=\\pm \\sqrt{\\frac{29}{4}}\\hfill \\\\ \\left(x-\\frac{3}{2}\\right)=\\pm \\frac{\\sqrt{29}}{2}\\hfill \\\\ x=\\frac{3}{2}\\pm \\frac{\\sqrt{29}}{2}\\hfill \\end{array}[\/latex]<\/p>\n<p>The solutions are [latex]x=\\frac{3}{2}+\\frac{\\sqrt{29}}{2}[\/latex], [latex]x=\\frac{3}{2}-\\frac{\\sqrt{29}}{2}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Solve by completing the square: [latex]{x}^{2}-6x=13[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q222291\">Show Solution<\/span><\/p>\n<div id=\"q222291\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]x=3\\pm \\sqrt{22}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p>Note that when solving a quadratic by completing the square, a negative value will sometimes arise under the square root symbol. Later, we&#8217;ll see that this value can be represented by a complex number (as shown in the video help for the problem below). We may also treat this type of solution as <em>unreal<\/em>, stating that no real solutions exist for this equation, by writing <em>DNE<\/em>. We will study complex numbers more thoroughly in a later module.<\/p>\n<p><iframe loading=\"lazy\" id=\"ohm1384\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=1384&theme=oea&iframe_resize_id=ohm1384&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm79619\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=79619&theme=oea&iframe_resize_id=ohm79619&show_question_numbers\" width=\"100%\" height=\"310\"><\/iframe><\/p>\n<\/div>\n<h2>Using the Quadratic Formula<\/h2>\n<p>The fourth method of solving a <strong>quadratic equation<\/strong> is by using the <strong>quadratic formula<\/strong>, a formula that will solve all quadratic equations. Although the quadratic formula works on any quadratic equation in standard form, it is easy to make errors in substituting the values into the formula. Pay close attention when substituting, and use parentheses when inserting a negative number.<\/p>\n<p>We can derive the quadratic formula by <strong>completing the square<\/strong>. We will assume that the leading coefficient is positive; if it is negative, we can multiply the equation by [latex]-1[\/latex] and obtain a positive <em>a<\/em>. Given [latex]a{x}^{2}+bx+c=0[\/latex], [latex]a\\ne 0[\/latex], we will complete the square as follows:<\/p>\n<ol>\n<li>First, move the constant term to the right side of the equal sign:\n<div style=\"text-align: center;\">[latex]a{x}^{2}+bx=-c[\/latex]<\/div>\n<\/li>\n<li>As we want the leading coefficient to equal 1, divide through by <em>a<\/em>:\n<div style=\"text-align: center;\">[latex]{x}^{2}+\\frac{b}{a}x=-\\frac{c}{a}[\/latex]<\/div>\n<\/li>\n<li>Then, find [latex]\\frac{1}{2}[\/latex] of the middle term, and add [latex]{\\left(\\frac{1}{2}\\frac{b}{a}\\right)}^{2}=\\frac{{b}^{2}}{4{a}^{2}}[\/latex] to both sides of the equal sign:\n<div style=\"text-align: center;\">[latex]{x}^{2}+\\frac{b}{a}x+\\frac{{b}^{2}}{4{a}^{2}}=\\frac{{b}^{2}}{4{a}^{2}}-\\frac{c}{a}[\/latex]<\/div>\n<\/li>\n<li>Next, write the left side as a perfect square. Find the common denominator of the right side and write it as a single fraction:\n<div style=\"text-align: center;\">[latex]{\\left(x+\\frac{b}{2a}\\right)}^{2}=\\frac{{b}^{2}-4ac}{4{a}^{2}}[\/latex]<\/div>\n<\/li>\n<li>Now, use the square root property, which gives\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}x+\\frac{b}{2a}=\\pm \\sqrt{\\frac{{b}^{2}-4ac}{4{a}^{2}}}\\hfill \\\\ x+\\frac{b}{2a}=\\frac{\\pm \\sqrt{{b}^{2}-4ac}}{2a}\\hfill \\end{array}[\/latex]<\/div>\n<\/li>\n<li>Finally, add [latex]-\\frac{b}{2a}[\/latex] to both sides of the equation and combine the terms on the right side. Thus,\n<div style=\"text-align: center;\">[latex]x=\\frac{-b\\pm \\sqrt{{b}^{2}-4ac}}{2a}[\/latex]<\/div>\n<\/li>\n<\/ol>\n<div class=\"textbox\">\n<h3>A General Note: The Quadratic Formula<\/h3>\n<p>Written in standard form, [latex]a{x}^{2}+bx+c=0[\/latex], any quadratic equation can be solved using the <strong>quadratic formula<\/strong>:<\/p>\n<div style=\"text-align: center;\">[latex]x=\\frac{-b\\pm \\sqrt{{b}^{2}-4ac}}{2a}[\/latex]<\/div>\n<p>where <em>a<\/em>, <em>b<\/em>, and <em>c<\/em> are real numbers and [latex]a\\ne 0[\/latex].<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a quadratic equation, solve it using the quadratic formula<\/h3>\n<ol>\n<li>Make sure the equation is in standard form: [latex]a{x}^{2}+bx+c=0[\/latex].<\/li>\n<li>Make note of the values of the coefficients and constant term, [latex]a,b[\/latex], and [latex]c[\/latex].<\/li>\n<li>Carefully substitute the values noted in step 2 into the equation. To avoid needless errors, use parentheses around each number input into the formula.<\/li>\n<li>Calculate and solve.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example : Solve A Quadratic Equation Using the Quadratic Formula<\/h3>\n<p>Solve the quadratic equation: [latex]{x}^{2}+5x+1=0[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q641400\">Show Solution<\/span><\/p>\n<div id=\"q641400\" class=\"hidden-answer\" style=\"display: none\">\n<p>Identify the coefficients: [latex]a=1,b=5,c=1[\/latex]. Then use the quadratic formula.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}x\\hfill&=\\frac{-\\left(5\\right)\\pm \\sqrt{{\\left(5\\right)}^{2}-4\\left(1\\right)\\left(1\\right)}}{2\\left(1\\right)}\\hfill \\\\ \\hfill&=\\frac{-5\\pm \\sqrt{25 - 4}}{2}\\hfill \\\\ \\hfill&=\\frac{-5\\pm \\sqrt{21}}{2}\\hfill \\end{array}[\/latex]<\/div>\n<div><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Recall adding and subtracting fractions<\/h3>\n<p>The form we have used to recall adding and subtracting fractions can help us write the solutions to quadratic equations.<\/p>\n<p>[latex]\\dfrac{a}{b}\\pm\\dfrac{c}{d} = \\dfrac{ad \\pm bc}{bd}[\/latex]<\/p>\n<p>Ex. The solutions<\/p>\n<p>[latex]x=\\dfrac{-b\\pm \\sqrt{{b}^{2}-4ac}}{2a}[\/latex]<\/p>\n<p>can also be written as two separate fractions<\/p>\n<p>[latex]x=\\dfrac{-b}{2a} \\pm \\dfrac{\\sqrt{{b}^{2}-4ac}}{2a}[\/latex].<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Solving a Quadratic Equation with the Quadratic Formula<\/h3>\n<p>Use the quadratic formula to solve [latex]{x}^{2}+x+2=0[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q688902\">Show Solution<\/span><\/p>\n<div id=\"q688902\" class=\"hidden-answer\" style=\"display: none\">\n<p>First, we identify the coefficients: [latex]a=1,b=1[\/latex], and [latex]c=2[\/latex].<\/p>\n<p>Substitute these values into the quadratic formula.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}x\\hfill&=\\dfrac{-b\\pm \\sqrt{{b}^{2}-4ac}}{2a}\\hfill \\\\\\hfill&=\\dfrac{-\\left(1\\right)\\pm \\sqrt{{\\left(1\\right)}^{2}-\\left(4\\right)\\cdot \\left(1\\right)\\cdot \\left(2\\right)}}{2\\cdot 1}\\hfill \\\\\\hfill&=\\dfrac{-1\\pm \\sqrt{1 - 8}}{2}\\hfill \\\\ \\hfill&=\\dfrac{-1\\pm \\sqrt{-7}}{2}\\hfill \\end{array}[\/latex]<\/div>\n<div><\/div>\n<p>The solutions to the equation are <em>unreal <\/em>because the square root of a negative number does not exist in the real numbers. If asked to find all <em>real<\/em> solutions, we would indicate that they do not exist by writing\u00a0<em>DNE<\/em>.<\/p>\n<p>We may also express unreal solutions as complex numbers by writing<\/p>\n<p>[latex]x=\\dfrac{-1\\pm i\\sqrt{7}}{2}[\/latex]<\/p>\n<p>or<\/p>\n<p>[latex]x=-\\dfrac{1}{2}+\\dfrac{\\sqrt{7}}{2}i[\/latex] and [latex]x=-\\dfrac{1}{2}-\\dfrac{\\sqrt{7}}{2}i[\/latex]\u00a0,\u00a0where [latex]\\sqrt{-1}[\/latex] is expressed as [latex]i[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Solve the quadratic equation using the quadratic formula: [latex]9{x}^{2}+3x - 2=0[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q232269\">Show Solution<\/span><\/p>\n<div id=\"q232269\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]x=-\\frac{2}{3}[\/latex], [latex]x=\\frac{1}{3}[\/latex]<\/p><\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm4014\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=4014&theme=oea&iframe_resize_id=ohm4014&show_question_numbers\" width=\"100%\" height=\"205\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm35639\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=35639&theme=oea&iframe_resize_id=ohm35639&show_question_numbers\" width=\"100%\" height=\"425\"><\/iframe><\/p>\n<\/div>\n<h2>The Discriminant<\/h2>\n<p>The <strong>quadratic formula<\/strong> not only generates the solutions to a quadratic equation, it tells us about the nature of the solutions when we consider the <strong>discriminant<\/strong>, or the expression under the radical, [latex]{b}^{2}-4ac[\/latex]. The discriminant tells us whether the solutions are real numbers or complex numbers as well as how many solutions of each type to expect. The table below\u00a0relates the value of the discriminant to the solutions of a quadratic equation.<\/p>\n<table summary=\"A table with 5 rows and 2 columns. The entries in the first row are: Value of Discriminant and Results. The entries in the second row are: b squared minus four times a times c equals zero and One rational solution (double solution). The entries in the third row are: b squared minus four times a times c is greater than zero, perfect square and Two rational solutions. The entries in the fourth row are: b squared minus four times a times c is greater than zero, not a perfect square and Two irrational solutions. The entries in the fifth row are: b squared minus four times a times c is less than zero and Two complex solutions.\">\n<thead>\n<tr>\n<th>Value of Discriminant<\/th>\n<th>Results<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]{b}^{2}-4ac=0[\/latex]<\/td>\n<td>One rational solution (double solution)<\/td>\n<\/tr>\n<tr>\n<td>[latex]{b}^{2}-4ac>0[\/latex], perfect square<\/td>\n<td>Two rational solutions<\/td>\n<\/tr>\n<tr>\n<td>[latex]{b}^{2}-4ac>0[\/latex], not a perfect square<\/td>\n<td>Two irrational solutions<\/td>\n<\/tr>\n<tr>\n<td>[latex]{b}^{2}-4ac<0[\/latex]<\/td>\n<td>Two complex solutions<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"textbox examples\">\n<h3>recall rational and irrational numbers are real numbers<\/h3>\n<ul>\n<li><strong>Rational numbers<\/strong>: A rational number is a number that can be expressed as a ratio of integers (a fraction with an integer numerator and a positive, non-zero integer denominator). The square root of a perfect square, such as [latex]\\sqrt{25}[\/latex] is rational because it can be expressed as [latex]\\dfrac{5}{1} = 5[\/latex]. Solutions containing only rational numbers are themselves rational solutions.<\/li>\n<li><strong>Irrational numbers<\/strong>: All the real numbers that are not rational are called irrational numbers. These numbers cannot be expressed as a fraction of integers. As quadratic solutions, irrational numbers arise in the form of a square root of a numbers that is not a perfect square. For example, [latex]\\sqrt{6}[\/latex] cannot be expressed as a ratio of two integers. [latex]\\sqrt{6}[\/latex] is an irrational number. Solutions containing an irrational number are themselves irrational solutions.<\/li>\n<li>Both rational and irrational numbers are real numbers. Quadratic solutions that are either rational or irrational are real solutions.<\/li>\n<li>Quadratic solutions that are unreal (they have a negative under the square root symbol, the <em>radical\u00a0<\/em>symbol) are called <em>complex<\/em> solutions.<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox\">\n<h3>A General Note: The Discriminant<\/h3>\n<p>For [latex]a{x}^{2}+bx+c=0[\/latex], where [latex]a[\/latex], [latex]b[\/latex], and [latex]c[\/latex] are real numbers, the <strong>discriminant<\/strong> is the expression under the radical in the quadratic formula: [latex]{b}^{2}-4ac[\/latex]. It tells us whether the solutions are real numbers or complex numbers and how many solutions of each type to expect.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Using the Discriminant to Find the Nature of the Solutions to a Quadratic Equation<\/h3>\n<p>Use the discriminant to find the nature of the solutions to the following quadratic equations:<\/p>\n<ol>\n<li>[latex]{x}^{2}+4x+4=0[\/latex]<\/li>\n<li>[latex]8{x}^{2}+14x+3=0[\/latex]<\/li>\n<li>[latex]3{x}^{2}-5x - 2=0[\/latex]<\/li>\n<li>[latex]3{x}^{2}-10x+15=0[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q229118\">Show Solution<\/span><\/p>\n<div id=\"q229118\" class=\"hidden-answer\" style=\"display: none\">\n<p>Calculate the discriminant [latex]{b}^{2}-4ac[\/latex] for each equation and state the expected type of solutions.<\/p>\n<ol>\n<li>[latex]{x}^{2}+4x+4=0[\/latex]: [latex]{b}^{2}-4ac={\\left(4\\right)}^{2}-4\\left(1\\right)\\left(4\\right)=0[\/latex]. There will be one rational double solution.<\/li>\n<li>[latex]8{x}^{2}+14x+3=0[\/latex]: [latex]{b}^{2}-4ac={\\left(14\\right)}^{2}-4\\left(8\\right)\\left(3\\right)=100[\/latex]. As [latex]100[\/latex] is a perfect square, there will be two rational solutions.<\/li>\n<li>[latex]3{x}^{2}-5x - 2=0[\/latex]: [latex]{b}^{2}-4ac={\\left(-5\\right)}^{2}-4\\left(3\\right)\\left(-2\\right)=49[\/latex]. As [latex]49[\/latex] is a perfect square, there will be two rational solutions.<\/li>\n<li>[latex]3{x}^{2}-10x+15=0[\/latex]: [latex]{b}^{2}-4ac={\\left(-10\\right)}^{2}-4\\left(3\\right)\\left(15\\right)=-80[\/latex]. There will be two complex solutions.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm35145\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=35145&theme=oea&iframe_resize_id=ohm35145&show_question_numbers\" width=\"100%\" height=\"235\"><\/iframe><\/p>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-104\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/li><li>Question ID 1384. <strong>Authored by<\/strong>: WebWork-Rochester. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC- BY + GPL<\/li><li>Question ID 79619. <strong>Authored by<\/strong>: Edward Wicks. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC- BY + GPL<\/li><li>Question ID 35145. <strong>Authored by<\/strong>: Jim Smart. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC- BY + GPL<\/li><li>Question ID 41014. <strong>Authored by<\/strong>: Tyler Wallace. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC- BY + GPL<\/li><li>Question ID 35639. <strong>Authored by<\/strong>: Christina Hughes Martin. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC- BY + GPL<\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>College Algebra. <strong>Authored by<\/strong>: OpenStax College Algebra. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t 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