{"id":211,"date":"2023-11-08T16:10:26","date_gmt":"2023-11-08T16:10:26","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/chapter\/read-the-discriminant\/"},"modified":"2026-02-05T11:40:17","modified_gmt":"2026-02-05T11:40:17","slug":"6-3-solutions-of-quadratic-equations","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/chapter\/6-3-solutions-of-quadratic-equations\/","title":{"raw":"6.3 Solutions of Quadratic Equations","rendered":"6.3 Solutions of Quadratic Equations"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning OutcomeS<\/h3>\r\n<ul>\r\n \t<li>Determine the number and type (rational, irrational, or complex) of solutions of a quadratic equation using the discriminant.<\/li>\r\n \t<li>Write a quadratic equation given all integer or rational solutions<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>The Discriminant<\/h2>\r\nThe <strong>quadratic formula<\/strong> not only generates the solutions to a quadratic equation, but also 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], it tells us whether the solutions are real numbers or complex numbers and how many solutions of each type to expect.\r\n\r\nLet us explore how the discriminant affects the evaluation of [latex] \\sqrt{{{b}^{2}}-4ac}[\/latex] in the quadratic formula and how it helps to determine the solution set.\r\n<ul>\r\n \t<li>If [latex]b^{2}-4ac&gt;0[\/latex], then the number underneath the radical will be a positive value. You can always find the square root of a positive number, so evaluating the quadratic formula will result in two real solutions (one by adding the positive square root and one by subtracting it).<\/li>\r\n \t<li>If [latex]b^{2}-4ac=0[\/latex], then you will be taking the square root of\u00a0[latex]0[\/latex], which is\u00a0[latex]0[\/latex]. Since adding and subtracting\u00a0[latex]0[\/latex] both give the same result, the \"[latex]\\pm[\/latex]\" portion of the formula does not matter. There will be one real repeated solution.<\/li>\r\n \t<li>If [latex]b^{2}-4ac&lt;0[\/latex], then the number underneath the radical will be a negative value. Since you cannot find the square root of a negative number using real numbers, there are no real solutions. However, you can use imaginary numbers. You will then have two complex solutions, one by adding the imaginary square root and one by subtracting it.<\/li>\r\n<\/ul>\r\nThe table below summarizes the relationship between the value of the discriminant and the solutions of a quadratic equation.\r\n<table style=\"width: 60%;\" 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 repeated rational 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\">\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<\/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=\"497176\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"497176\"]\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 style=\"text-align: left;\">[latex]{x}^{2}+4x+4=0[\/latex] [latex] \\\\ {b}^{2}-4ac={\\left(4\\right)}^{2}-4\\left(1\\right)\\left(4\\right)=0[\/latex] [latex]\\text{There will be one repeated rational solution.}[\/latex]<\/li>\r\n \t<li style=\"text-align: left;\">[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] [latex]\\text{100 is a perfect square, so there will be two rational solutions.}[\/latex]<\/li>\r\n \t<li style=\"text-align: left;\">[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] [latex]\\text{49 is a perfect square, so there will be two rational solutions.}[\/latex]<\/li>\r\n \t<li style=\"text-align: left;\">[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] [latex]\\text{There will be two complex solutions.}[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nUse the discriminant to determine how many and what kind of solutions the quadratic equation [latex]x^{2}-4x+10=0[\/latex]\u00a0has.\r\n\r\n[reveal-answer q=\"116245\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"116245\"]\r\n\r\nEvaluate [latex]b^{2}-4ac[\/latex]. First note that\u00a0[latex]a=1,b=\u22124[\/latex], and [latex]c=10[\/latex].\r\n\r\n[latex]\\begin{array}{l}b^{2}-4ac=\\left(-4\\right)^{2}-4\\left(1\\right)\\left(10\\right)=16-40=-24\\end{array}[\/latex]\r\n\r\nThe result is a negative number. The discriminant is negative, so the quadratic equation has two complex solutions.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the last example, we will draw a correlation between the number and type of solutions to a quadratic equation and\u00a0the graph of its corresponding function.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nUse the following graphs of quadratic functions to determine how many and what type of solutions the corresponding quadratic equation [latex]f(x)=0[\/latex] will have. \u00a0Determine whether the discriminant will be greater than, less than, or equal to zero for each.\r\n\r\na.\r\n\r\n<img class=\"alignnone wp-image-3479 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/08\/04191147\/Screen-Shot-2016-08-04-at-12.10.26-PM.png\" alt=\"Upward facing parabola with vertex in quadrant 1 around (2,1).\" width=\"148\" height=\"135\" \/>\r\n\r\nb.\r\n\r\n<img class=\"alignnone wp-image-3480 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/08\/04191227\/Screen-Shot-2016-08-04-at-12.12.08-PM.png\" alt=\"Upward facing parabola with vertex on x-axis at (2,0).\" width=\"154\" height=\"136\" \/>\r\n\r\nc.\r\n\r\n<img class=\"alignnone wp-image-3481 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/08\/04191433\/Screen-Shot-2016-08-04-at-12.14.18-PM.png\" alt=\"Upward facing parabola with vertex below x-axis at (2, negative 1).\" width=\"170\" height=\"140\" \/>\r\n[reveal-answer q=\"26060\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"26060\"]\r\n\r\na. This quadratic function does not touch or cross the x-axis; therefore, the corresponding equation [latex]f(x)=0[\/latex] will have complex solutions. This implies that [latex]b^{2}-4ac&lt;0[\/latex].\r\n\r\nb. This quadratic function touches the x-axis exactly once which implies there is one repeated solution to the equation [latex]f(x)=0[\/latex]. \u00a0We can then say that\u00a0[latex]b^{2}-4ac=0[\/latex].\r\n\r\nc. In our final graph, the quadratic function crosses the x-axis twice which tells us that there are two real number solutions to the equation [latex]f(x)=0[\/latex], and therefore\u00a0[latex]b^{2}-4ac&gt;0[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nWe can summarize our results as follows:\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>Discriminant<\/td>\r\n<td>Number and Type of Solutions<\/td>\r\n<td>Graph of Quadratic Function<\/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<td>will not cross the x-axis<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]b^{2}-4ac=0[\/latex]<\/td>\r\n<td>one real repeated solution<\/td>\r\n<td>will touch x-axis once<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]b^{2}-4ac&gt;0[\/latex]<\/td>\r\n<td>\u00a0two real solutions<\/td>\r\n<td>\u00a0will cross x-axis twice<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nIn the following video, we show more examples of how to use the discriminant to describe the type of solutions of a quadratic equation.\r\n\r\nhttps:\/\/youtu.be\/hSWs0VUyn1k\r\n<h2>Creating an Equation from Given Solutions<\/h2>\r\nThere are many situations, particularly in modeling applications, where we want to create a quadratic equation that has certain solutions. The key to this process is to work backward from how we usually find solutions to a quadratic equation.\r\n\r\nAs an example, consider the following solution to a factorable quadratic equation:\r\n<p style=\"text-align: center;\">[latex]\\begin{align}x^2+6x-16&amp;=0\\\\\r\n(x+8)(x-2)&amp;=0\\\\\r\nx+8=0 \\textsf{ or } x-2 = 0&amp;\\\\\r\nx=-8 \\textsf{ or } x=2&amp; \\end{align}[\/latex]<\/p>\r\nIf we start with the two solutions, we should be able to easily reverse the steps in the problem. This consists of first writing a factor corresponding to each given solution, then multiplying the factors to create an equation.\u00a0 Here is a quick example with given solutions [latex]x=5[\/latex] and [latex]x=6[\/latex]:\r\n<p style=\"text-align: center;\">[latex]\\begin{align}x=5 \\textsf{ or } x=6&amp;\\\\\r\nx-5=0 \\textsf{ or } x-6 = 0&amp;\\\\\r\n(x-5)(x-6)&amp;=0\\\\\r\nx^2-11x+30&amp;=0 \\end{align}[\/latex]<\/p>\r\nThe first two steps can be skipped by simply writing a factor [latex](x-c)[\/latex] for each given solution [latex]c[\/latex].\r\n<div class=\"textbox exercises\">\r\n<h3>EXAMPLE<\/h3>\r\nFind the equation, in standard form [latex]y=ax^2+bx+c[\/latex], for a quadratic that has roots at [latex]x=-3[\/latex] and [latex]x=7[\/latex], and has leading coefficient [latex]1[\/latex].\r\n\r\n[reveal-answer q=\"657251\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"657251\"]\r\n\r\nLet's begin by discussing some of the words in this problem prompt.\r\n\r\nWe are asked for a quadratic equation in two variables, with [latex]y[\/latex] solved on the left side. If this equation is set equal to [latex]0[\/latex], we saw in the previous section that this will solve for the [latex]x[\/latex]-intercepts of the parabola. These [latex]x[\/latex] values are also called <strong>roots<\/strong>. Thus, roots are solutions after setting the equation equal to [latex]0[\/latex].\r\n\r\nWe will explain after solving this example why the phrase \"leading coefficient of [latex]1[\/latex]\" is added.\r\n\r\nEach of the given roots corresponds to a factor in our quadratic equation. Thus we write\r\n<p style=\"text-align: center;\">[latex]\\begin{align}(x-(-3))(x-7)&amp;=0\\\\\r\n(x+3)(x-7)&amp;=0\\\\\r\nx^2-4x-21&amp;=0 \\end{align}[\/latex]<\/p>\r\nThe desired quadratic equation is [latex]y=x^2-4x-21[\/latex]. When set equal to zero, the two given numbers [latex]x=-3[\/latex] and [latex]x=7[\/latex] are the solutions.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe reason we have to specify \"leading coefficient [latex]1[\/latex]\" is because after setting up our factored expression [latex](x+3)(x-7)=0[\/latex], we could easily add any coefficient we choose to the left side and the given solutions will still be the same. For example,\u00a0[latex]2(x+3)(x-7)=0[\/latex] still has solutions [latex]x=-3[\/latex] and [latex]x=7[\/latex], but will have a leading coefficient of [latex]2[\/latex] when multiplied out.\r\n<div class=\"textbox exercises\">\r\n<h3>EXAMPLE<\/h3>\r\nFind the equation, in standard form [latex]y=ax^2+bx+c[\/latex], for a quadratic that has roots at [latex]x=\\dfrac{1}{2}[\/latex] and [latex]x=-\\dfrac{2}{5}[\/latex]. Write an equation with integer coefficients and minimal leading coefficient.\r\n\r\n[reveal-answer q=\"622630\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"622630\"]\r\n\r\nWe can use the same principle as in the previous example, but with some modifications. If we simply multiply factors corresponding to each given root, the resulting equation will have coefficients that are not integers:\r\n<p style=\"text-align: center;\">[latex]\\begin{align}\\left(x-\\dfrac{1}{2}\\right)\\left(x-\\left(-\\dfrac{2}{5}\\right)\\right)&amp;=0\\\\\r\n\\left(x-\\dfrac{1}{2}\\right)\\left(x+\\dfrac{2}{5}\\right)&amp;=0\\\\\r\nx^2+\\dfrac{2}{5}x-\\dfrac{1}{2}x-\\dfrac{2}{10}&amp;=0 \\end{align}[\/latex]<\/p>\r\nWe can fix the fractional coefficients by now multiplying both sides of the equation by the LCM of [latex]10[\/latex] to clear the fractions:\r\n<p style=\"text-align: center;\">[latex]\\begin{align}10\\cdot(x^2+\\dfrac{2}{5}x-\\dfrac{1}{2}x-\\dfrac{2}{10})&amp;=10\\cdot 0\\\\\r\n10x^2+4x-5x-2&amp;=0\\\\\r\n10x^2-x-2&amp;=0 \\end{align}[\/latex]<\/p>\r\nThe desired equation is [latex]y=10x^2-x-2[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Summary<\/h2>\r\nThe discriminant of the quadratic formula is the quantity under the radical, [latex] {{b}^{2}}-4ac[\/latex]. It determines the number and the type of solutions that a quadratic equation has. If the discriminant is positive, there are\u00a0[latex]2[\/latex] real solutions. If it is\u00a0[latex]0[\/latex], there is\u00a0[latex]1[\/latex] real repeated solution. If the discriminant is negative, there are\u00a0[latex]2[\/latex] complex solutions (but no real solutions).\r\n\r\nWe can create a quadratic equation with given solutions by creating a factor [latex](x-c)[\/latex] for each solution [latex]x=c[\/latex] that the equation is required to have. When a quadratic equation in standard form\u00a0[latex]y=ax^2+bx+c[\/latex] is set equal to [latex]0[\/latex], the solutions are called roots. These give the [latex]x[\/latex]-coordinates of the [latex]x[\/latex]-intercepts of the parabola.","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning OutcomeS<\/h3>\n<ul>\n<li>Determine the number and type (rational, irrational, or complex) of solutions of a quadratic equation using the discriminant.<\/li>\n<li>Write a quadratic equation given all integer or rational solutions<\/li>\n<\/ul>\n<\/div>\n<h2>The Discriminant<\/h2>\n<p>The <strong>quadratic formula<\/strong> not only generates the solutions to a quadratic equation, but also 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], it tells us whether the solutions are real numbers or complex numbers and how many solutions of each type to expect.<\/p>\n<p>Let us explore how the discriminant affects the evaluation of [latex]\\sqrt{{{b}^{2}}-4ac}[\/latex] in the quadratic formula and how it helps to determine the solution set.<\/p>\n<ul>\n<li>If [latex]b^{2}-4ac>0[\/latex], then the number underneath the radical will be a positive value. You can always find the square root of a positive number, so evaluating the quadratic formula will result in two real solutions (one by adding the positive square root and one by subtracting it).<\/li>\n<li>If [latex]b^{2}-4ac=0[\/latex], then you will be taking the square root of\u00a0[latex]0[\/latex], which is\u00a0[latex]0[\/latex]. Since adding and subtracting\u00a0[latex]0[\/latex] both give the same result, the &#8220;[latex]\\pm[\/latex]&#8221; portion of the formula does not matter. There will be one real repeated solution.<\/li>\n<li>If [latex]b^{2}-4ac<0[\/latex], then the number underneath the radical will be a negative value. Since you cannot find the square root of a negative number using real numbers, there are no real solutions. However, you can use imaginary numbers. You will then have two complex solutions, one by adding the imaginary square root and one by subtracting it.<\/li>\n<\/ul>\n<p>The table below summarizes the relationship between the value of the discriminant and the solutions of a quadratic equation.<\/p>\n<table style=\"width: 60%;\" 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 repeated rational 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\">\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<\/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=\"q497176\">Show Solution<\/span><\/p>\n<div id=\"q497176\" 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 style=\"text-align: left;\">[latex]{x}^{2}+4x+4=0[\/latex] [latex]\\\\ {b}^{2}-4ac={\\left(4\\right)}^{2}-4\\left(1\\right)\\left(4\\right)=0[\/latex] [latex]\\text{There will be one repeated rational solution.}[\/latex]<\/li>\n<li style=\"text-align: left;\">[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] [latex]\\text{100 is a perfect square, so there will be two rational solutions.}[\/latex]<\/li>\n<li style=\"text-align: left;\">[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] [latex]\\text{49 is a perfect square, so there will be two rational solutions.}[\/latex]<\/li>\n<li style=\"text-align: left;\">[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] [latex]\\text{There will be two complex solutions.}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Use the discriminant to determine how many and what kind of solutions the quadratic equation [latex]x^{2}-4x+10=0[\/latex]\u00a0has.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q116245\">Show Solution<\/span><\/p>\n<div id=\"q116245\" class=\"hidden-answer\" style=\"display: none\">\n<p>Evaluate [latex]b^{2}-4ac[\/latex]. First note that\u00a0[latex]a=1,b=\u22124[\/latex], and [latex]c=10[\/latex].<\/p>\n<p>[latex]\\begin{array}{l}b^{2}-4ac=\\left(-4\\right)^{2}-4\\left(1\\right)\\left(10\\right)=16-40=-24\\end{array}[\/latex]<\/p>\n<p>The result is a negative number. The discriminant is negative, so the quadratic equation has two complex solutions.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the last example, we will draw a correlation between the number and type of solutions to a quadratic equation and\u00a0the graph of its corresponding function.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Use the following graphs of quadratic functions to determine how many and what type of solutions the corresponding quadratic equation [latex]f(x)=0[\/latex] will have. \u00a0Determine whether the discriminant will be greater than, less than, or equal to zero for each.<\/p>\n<p>a.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-3479 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/08\/04191147\/Screen-Shot-2016-08-04-at-12.10.26-PM.png\" alt=\"Upward facing parabola with vertex in quadrant 1 around (2,1).\" width=\"148\" height=\"135\" \/><\/p>\n<p>b.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-3480 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/08\/04191227\/Screen-Shot-2016-08-04-at-12.12.08-PM.png\" alt=\"Upward facing parabola with vertex on x-axis at (2,0).\" width=\"154\" height=\"136\" \/><\/p>\n<p>c.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-3481 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/08\/04191433\/Screen-Shot-2016-08-04-at-12.14.18-PM.png\" alt=\"Upward facing parabola with vertex below x-axis at (2, negative 1).\" width=\"170\" height=\"140\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q26060\">Show Solution<\/span><\/p>\n<div id=\"q26060\" class=\"hidden-answer\" style=\"display: none\">\n<p>a. This quadratic function does not touch or cross the x-axis; therefore, the corresponding equation [latex]f(x)=0[\/latex] will have complex solutions. This implies that [latex]b^{2}-4ac<0[\/latex].\n\nb. This quadratic function touches the x-axis exactly once which implies there is one repeated solution to the equation [latex]f(x)=0[\/latex]. \u00a0We can then say that\u00a0[latex]b^{2}-4ac=0[\/latex].\n\nc. In our final graph, the quadratic function crosses the x-axis twice which tells us that there are two real number solutions to the equation [latex]f(x)=0[\/latex], and therefore\u00a0[latex]b^{2}-4ac>0[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>We can summarize our results as follows:<\/p>\n<table>\n<tbody>\n<tr>\n<td>Discriminant<\/td>\n<td>Number and Type of Solutions<\/td>\n<td>Graph of Quadratic Function<\/td>\n<\/tr>\n<tr>\n<td>[latex]b^{2}-4ac<0[\/latex]<\/td>\n<td>two complex solutions<\/td>\n<td>will not cross the x-axis<\/td>\n<\/tr>\n<tr>\n<td>[latex]b^{2}-4ac=0[\/latex]<\/td>\n<td>one real repeated solution<\/td>\n<td>will touch x-axis once<\/td>\n<\/tr>\n<tr>\n<td>[latex]b^{2}-4ac>0[\/latex]<\/td>\n<td>\u00a0two real solutions<\/td>\n<td>\u00a0will cross x-axis twice<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>In the following video, we show more examples of how to use the discriminant to describe the type of solutions of a quadratic equation.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex:  The Discriminant\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/hSWs0VUyn1k?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Creating an Equation from Given Solutions<\/h2>\n<p>There are many situations, particularly in modeling applications, where we want to create a quadratic equation that has certain solutions. The key to this process is to work backward from how we usually find solutions to a quadratic equation.<\/p>\n<p>As an example, consider the following solution to a factorable quadratic equation:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}x^2+6x-16&=0\\\\  (x+8)(x-2)&=0\\\\  x+8=0 \\textsf{ or } x-2 = 0&\\\\  x=-8 \\textsf{ or } x=2& \\end{align}[\/latex]<\/p>\n<p>If we start with the two solutions, we should be able to easily reverse the steps in the problem. This consists of first writing a factor corresponding to each given solution, then multiplying the factors to create an equation.\u00a0 Here is a quick example with given solutions [latex]x=5[\/latex] and [latex]x=6[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}x=5 \\textsf{ or } x=6&\\\\  x-5=0 \\textsf{ or } x-6 = 0&\\\\  (x-5)(x-6)&=0\\\\  x^2-11x+30&=0 \\end{align}[\/latex]<\/p>\n<p>The first two steps can be skipped by simply writing a factor [latex](x-c)[\/latex] for each given solution [latex]c[\/latex].<\/p>\n<div class=\"textbox exercises\">\n<h3>EXAMPLE<\/h3>\n<p>Find the equation, in standard form [latex]y=ax^2+bx+c[\/latex], for a quadratic that has roots at [latex]x=-3[\/latex] and [latex]x=7[\/latex], and has leading coefficient [latex]1[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q657251\">Show Solution<\/span><\/p>\n<div id=\"q657251\" class=\"hidden-answer\" style=\"display: none\">\n<p>Let&#8217;s begin by discussing some of the words in this problem prompt.<\/p>\n<p>We are asked for a quadratic equation in two variables, with [latex]y[\/latex] solved on the left side. If this equation is set equal to [latex]0[\/latex], we saw in the previous section that this will solve for the [latex]x[\/latex]-intercepts of the parabola. These [latex]x[\/latex] values are also called <strong>roots<\/strong>. Thus, roots are solutions after setting the equation equal to [latex]0[\/latex].<\/p>\n<p>We will explain after solving this example why the phrase &#8220;leading coefficient of [latex]1[\/latex]&#8221; is added.<\/p>\n<p>Each of the given roots corresponds to a factor in our quadratic equation. Thus we write<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}(x-(-3))(x-7)&=0\\\\  (x+3)(x-7)&=0\\\\  x^2-4x-21&=0 \\end{align}[\/latex]<\/p>\n<p>The desired quadratic equation is [latex]y=x^2-4x-21[\/latex]. When set equal to zero, the two given numbers [latex]x=-3[\/latex] and [latex]x=7[\/latex] are the solutions.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>The reason we have to specify &#8220;leading coefficient [latex]1[\/latex]&#8221; is because after setting up our factored expression [latex](x+3)(x-7)=0[\/latex], we could easily add any coefficient we choose to the left side and the given solutions will still be the same. For example,\u00a0[latex]2(x+3)(x-7)=0[\/latex] still has solutions [latex]x=-3[\/latex] and [latex]x=7[\/latex], but will have a leading coefficient of [latex]2[\/latex] when multiplied out.<\/p>\n<div class=\"textbox exercises\">\n<h3>EXAMPLE<\/h3>\n<p>Find the equation, in standard form [latex]y=ax^2+bx+c[\/latex], for a quadratic that has roots at [latex]x=\\dfrac{1}{2}[\/latex] and [latex]x=-\\dfrac{2}{5}[\/latex]. Write an equation with integer coefficients and minimal leading coefficient.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q622630\">Show Solution<\/span><\/p>\n<div id=\"q622630\" class=\"hidden-answer\" style=\"display: none\">\n<p>We can use the same principle as in the previous example, but with some modifications. If we simply multiply factors corresponding to each given root, the resulting equation will have coefficients that are not integers:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}\\left(x-\\dfrac{1}{2}\\right)\\left(x-\\left(-\\dfrac{2}{5}\\right)\\right)&=0\\\\  \\left(x-\\dfrac{1}{2}\\right)\\left(x+\\dfrac{2}{5}\\right)&=0\\\\  x^2+\\dfrac{2}{5}x-\\dfrac{1}{2}x-\\dfrac{2}{10}&=0 \\end{align}[\/latex]<\/p>\n<p>We can fix the fractional coefficients by now multiplying both sides of the equation by the LCM of [latex]10[\/latex] to clear the fractions:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}10\\cdot(x^2+\\dfrac{2}{5}x-\\dfrac{1}{2}x-\\dfrac{2}{10})&=10\\cdot 0\\\\  10x^2+4x-5x-2&=0\\\\  10x^2-x-2&=0 \\end{align}[\/latex]<\/p>\n<p>The desired equation is [latex]y=10x^2-x-2[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Summary<\/h2>\n<p>The discriminant of the quadratic formula is the quantity under the radical, [latex]{{b}^{2}}-4ac[\/latex]. It determines the number and the type of solutions that a quadratic equation has. If the discriminant is positive, there are\u00a0[latex]2[\/latex] real solutions. If it is\u00a0[latex]0[\/latex], there is\u00a0[latex]1[\/latex] real repeated solution. If the discriminant is negative, there are\u00a0[latex]2[\/latex] complex solutions (but no real solutions).<\/p>\n<p>We can create a quadratic equation with given solutions by creating a factor [latex](x-c)[\/latex] for each solution [latex]x=c[\/latex] that the equation is required to have. When a quadratic equation in standard form\u00a0[latex]y=ax^2+bx+c[\/latex] is set equal to [latex]0[\/latex], the solutions are called roots. These give the [latex]x[\/latex]-coordinates of the [latex]x[\/latex]-intercepts of the parabola.<\/p>\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-211\">\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>Ex: The Discriminant. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/hSWs0VUyn1k\">https:\/\/youtu.be\/hSWs0VUyn1k<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><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@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 <\/section>","protected":false},"author":395986,"menu_order":3,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Ex: The Discriminant\",\"author\":\"James Sousa (Mathispower4u.com) \",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/hSWs0VUyn1k\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"Abramson, Jay et al.\",\"organization\":\"OpenStax\",\"url\":\" http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"c8cd579d-2e52-4396-a652-bc5f09475271","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-211","chapter","type-chapter","status-publish","hentry"],"part":199,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/pressbooks\/v2\/chapters\/211","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/wp\/v2\/users\/395986"}],"version-history":[{"count":12,"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/pressbooks\/v2\/chapters\/211\/revisions"}],"predecessor-version":[{"id":2134,"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/pressbooks\/v2\/chapters\/211\/revisions\/2134"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/pressbooks\/v2\/parts\/199"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/pressbooks\/v2\/chapters\/211\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/wp\/v2\/media?parent=211"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=211"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/wp\/v2\/contributor?post=211"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/wp\/v2\/license?post=211"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}