{"id":4978,"date":"2018-12-11T14:07:14","date_gmt":"2018-12-11T19:07:14","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/solve-quadratic-inequalities\/"},"modified":"2018-12-11T14:10:10","modified_gmt":"2018-12-11T19:10:10","slug":"solve-quadratic-inequalities","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/solve-quadratic-inequalities\/","title":{"raw":"Solve Quadratic Inequalities","rendered":"Solve Quadratic Inequalities"},"content":{"raw":"[latexpage]\r\n<div class=\"textbox textbox--learning-objectives\">\r\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\r\nBy the end of this section, you will be able to:\r\n<ul>\r\n \t<li>Solve quadratic inequalities graphically<\/li>\r\n \t<li>Solve quadratic inequalities algebraically<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1169148924848\" class=\"be-prepared\">\r\n<p id=\"fs-id1169146648302\">Before you get started, take this readiness quiz.<\/p>\r\n\r\n<ol type=\"1\">\r\n \t<li>Solve: \\(2x-3=0.\\)\r\n<div data-type=\"newline\"><\/div>\r\nIf you missed this problem, review <a href=\"\/contents\/9f100e8f-6d15-4cae-bc22-c306e9d7d55c#fs-id1167836432956\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\r\n \t<li>Solve: \\(2{y}^{2}+y=15\\).\r\n<div data-type=\"newline\"><\/div>\r\nIf you missed this problem, review <a href=\"\/contents\/da8478b4-93bc-4919-81a1-5e3267050e7e#fs-id1167836625705\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\r\n \t<li>Solve \\(\\frac{1}{{x}^{2}+2x-8}&gt;0\\)\r\n<div data-type=\"newline\"><\/div>\r\nIf you missed this problem, review <a href=\"\/contents\/a68b06f6-2833-4512-b24f-c0da889a8759#fs-id1167835534361\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<p id=\"fs-id1169148924000\">We have learned how to solve linear inequalities and rational inequalities previously. Some of the techniques we used to solve them were the same and some were different.<\/p>\r\n<p id=\"fs-id1165926586447\">We will now learn to solve inequalities that have a quadratic expression. We will use some of the techniques from solving linear and rational inequalities as well as quadratic equations.<\/p>\r\n<p id=\"fs-id1169149008053\">We will solve quadratic inequalities two ways\u2014both graphically and algebraically.<\/p>\r\n\r\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1169144377740\">\r\n<h3 data-type=\"title\">Solve Quadratic Inequalities Graphically<\/h3>\r\n<p id=\"fs-id1169149115761\">A <span data-type=\"term\" class=\"no-emphasis\">quadratic equation<\/span> is in standard form when written as <em data-effect=\"italics\">ax<\/em><sup>2<\/sup> + <em data-effect=\"italics\">bx<\/em> + <em data-effect=\"italics\">c<\/em> = 0. If we replace the equal sign with an inequality sign, we have a <span data-type=\"term\">quadratic inequality<\/span> in standard form.<\/p>\r\n\r\n<div data-type=\"note\" id=\"fs-id1169148991490\">\r\n<div data-type=\"title\">Quadratic Inequality<\/div>\r\nA <strong data-effect=\"bold\">quadratic inequality<\/strong> is an inequality that contains a quadratic expression.\r\n<p id=\"fs-id1169148916649\">The standard form of a quadratic inequality is written:<\/p>\r\n\r\n<div data-type=\"equation\" id=\"fs-id1169148962762\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{cccccc}a{x}^{2}+bx+c&lt;0\\hfill &amp; &amp; &amp; &amp; &amp; a{x}^{2}+bx+c\\le 0\\hfill \\\\ a{x}^{2}+bx+c&gt;0\\hfill &amp; &amp; &amp; &amp; &amp; a{x}^{2}+bx+c\\ge 0\\hfill \\end{array}\\)<\/div>\r\n<\/div>\r\n<p id=\"fs-id1169148867608\">The graph of a quadratic function <em data-effect=\"italics\">f<\/em>(<em data-effect=\"italics\">x<\/em>) = <em data-effect=\"italics\">ax<\/em><sup>2<\/sup> + <em data-effect=\"italics\">bx<\/em> + <em data-effect=\"italics\">c<\/em> = 0 is a parabola. When we ask when is <em data-effect=\"italics\">ax<\/em><sup>2<\/sup> + <em data-effect=\"italics\">bx<\/em> + <em data-effect=\"italics\">c<\/em> &lt; 0, we are asking when is f(<em data-effect=\"italics\">x<\/em>) &lt; 0. We want to know when the parabola is below the <em data-effect=\"italics\">x<\/em>-axis.<\/p>\r\n<p id=\"fs-id1169149221517\">When we ask when is <em data-effect=\"italics\">ax<\/em><sup>2<\/sup> + <em data-effect=\"italics\">bx<\/em> + <em data-effect=\"italics\">c<\/em> &gt; 0, we are asking when is <em data-effect=\"italics\">f<\/em>(<em data-effect=\"italics\">x<\/em>) &gt; 0. We want to know when the parabola is above the <em data-effect=\"italics\">y<\/em>-axis.<\/p>\r\n<span data-type=\"media\" id=\"fs-id1169148828123\" data-alt=\"The first graph is an upward facing parabola, f of x, on an x y-coordinate plane. To the left of the function, f of x is greater than 0. Between the x-intercepts, f of x is less than 0. To the right of the function, f of x is greater than 0. The second graph is a downward-facing parabola, f of x, on an x y coordinate plane. To the left of the function, f of x is less than 0. Between the x-intercepts, f of x is greater than 0. To the right of the function, f of x is less than 0.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_001_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The first graph is an upward facing parabola, f of x, on an x y-coordinate plane. To the left of the function, f of x is greater than 0. Between the x-intercepts, f of x is less than 0. To the right of the function, f of x is greater than 0. The second graph is a downward-facing parabola, f of x, on an x y coordinate plane. To the left of the function, f of x is less than 0. Between the x-intercepts, f of x is greater than 0. To the right of the function, f of x is less than 0.\" \/><\/span>\r\n<div data-type=\"example\" id=\"fs-id1169147029625\" class=\"textbox textbox--examples\">\r\n<div data-type=\"title\">How to Solve a Quadratic Inequality Graphically<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149095397\">\r\n<div data-type=\"problem\" id=\"fs-id1169149335491\">\r\n<p id=\"fs-id1169146607663\">Solve \\({x}^{2}-6x+8&lt;0\\) graphically. Write the solution in interval notation.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169149156869\"><span data-type=\"media\" data-alt=\"The figure is a table with 3 columns. The first column is Step 1: Write the quadratic inequality in standard form. The second column says the inequality is in standard form. The third column says x squared minus 6 times x plus 8 less than 0.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_002a_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure is a table with 3 columns. The first column is Step 1: Write the quadratic inequality in standard form. The second column says the inequality is in standard form. The third column says x squared minus 6 times x plus 8 less than 0.\" \/><\/span><span data-type=\"media\" id=\"fs-id1169149144291\" data-alt=\"The figure is a table with 3 columns. The first column says Step 2-Graph the function f of x equals a times x squared plus b times x plus c using properties or transformations. The second column gives instructions and the third column shows the work for step 3 as follows. We will graph using properties. The function is f of x equals x squared minus 6 times x plus 8 where a equals 1, b equals negative 6, and c equals 8. Look at a in the function f of x equals x squared minus 6 times x plus 8. Since a is positive, the parabola opens upward. The equation of the axis of symmetry is the line x equals negative b divided by 2 times a, so x equals negative negative 6 divided by 2 times 1. X equals 3. The axis of symmetry is the line x equals 3. The vertex is on the axis of symmetry. Substitute x equals 3 into the function, so f of 3 equals 3 squared minus 6 times 3 plus 8. F of 3 equals negative 1, so the vertex is (3, negative 1). We find f of 0 in order to find the y-intercept, so f of 0 equals 0 squared minus 6 times 0 plus 8. F of 0 equals 8, so the y intercept is (0, 8). We use the axis of symmetry to find a point symmetric to the y-intercept. The y-intercept is 3 units left of the axis of symmetry, x equals 3. A point 3 units to the right of the axis of symmetry has x equals 6. Point symmetric to y-intercept is (6, 8). We solve f of x equals 0 in order to find the x-intercepts. We can solve this quadratic equation by factoring. 0 equals x squared minus 6 times x plus 8, 0 equals the quantity x minus 2 times the quantity x minus 4, x equals 2 or x equals 4. The x-intercepts are (2, 0) and (4, 0). We graph the vertex, intercepts, and the point symmetric to the y-intercept. We connect these 5 points to sketch the parabola shown that is upward-facing with the points found through this process.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_002b_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure is a table with 3 columns. The first column says Step 2-Graph the function f of x equals a times x squared plus b times x plus c using properties or transformations. The second column gives instructions and the third column shows the work for step 3 as follows. We will graph using properties. The function is f of x equals x squared minus 6 times x plus 8 where a equals 1, b equals negative 6, and c equals 8. Look at a in the function f of x equals x squared minus 6 times x plus 8. Since a is positive, the parabola opens upward. The equation of the axis of symmetry is the line x equals negative b divided by 2 times a, so x equals negative negative 6 divided by 2 times 1. X equals 3. The axis of symmetry is the line x equals 3. The vertex is on the axis of symmetry. Substitute x equals 3 into the function, so f of 3 equals 3 squared minus 6 times 3 plus 8. F of 3 equals negative 1, so the vertex is (3, negative 1). We find f of 0 in order to find the y-intercept, so f of 0 equals 0 squared minus 6 times 0 plus 8. F of 0 equals 8, so the y intercept is (0, 8). We use the axis of symmetry to find a point symmetric to the y-intercept. The y-intercept is 3 units left of the axis of symmetry, x equals 3. A point 3 units to the right of the axis of symmetry has x equals 6. Point symmetric to y-intercept is (6, 8). We solve f of x equals 0 in order to find the x-intercepts. We can solve this quadratic equation by factoring. 0 equals x squared minus 6 times x plus 8, 0 equals the quantity x minus 2 times the quantity x minus 4, x equals 2 or x equals 4. The x-intercepts are (2, 0) and (4, 0). We graph the vertex, intercepts, and the point symmetric to the y-intercept. We connect these 5 points to sketch the parabola shown that is upward-facing with the points found through this process.\" \/><\/span><span data-type=\"media\" id=\"fs-id1169148944404\" data-alt=\"The figure is a table with 3 columns. The first column says Step 3- Determine the solution from the graph. The second column gives instructions. X squared minus 6 x plus 8 less than 0. The inequality asks for the values of x which make the function less than 0. Which values of x make the parabola below the x-axis. We do not include the values 2, 4 as the inequality is strictly less than. The third column says The solution, in interval notation, is (2, 4).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_002c_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure is a table with 3 columns. The first column says Step 3- Determine the solution from the graph. The second column gives instructions. X squared minus 6 x plus 8 less than 0. The inequality asks for the values of x which make the function less than 0. Which values of x make the parabola below the x-axis. We do not include the values 2, 4 as the inequality is strictly less than. The third column says The solution, in interval notation, is (2, 4).\" \/><\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1169149087550\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1169146622094\">\r\n<div data-type=\"problem\" id=\"fs-id1169148951471\">\r\n<p id=\"fs-id1169146633988\"><span class=\"token\">\u24d0<\/span> Solve \\({x}^{2}+2x-8&lt;0\\) graphically and <span class=\"token\">\u24d1<\/span> write the solution in interval notation.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169149123010\">\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d0<\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1169146643216\" data-alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 2, negative 9), y-intercept of (0, 8), and axis of symmetry shown at x equals negative 2.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_301_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 2, negative 9), y-intercept of (0, 8), and axis of symmetry shown at x equals negative 2.\" \/><\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\\(\\left(-4,-2\\right)\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1169149307519\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1169149346432\">\r\n<div data-type=\"problem\" id=\"fs-id1169146637308\">\r\n<p id=\"fs-id1169149312399\"><span class=\"token\">\u24d0<\/span> Solve \\({x}^{2}-8x+12\\ge 0\\) graphically and <span class=\"token\">\u24d1<\/span> write the solution in interval notation.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169149293258\">\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d0<\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" data-alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (4, negative 4) and x-intercepts of (2, 0) and (6, 0).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_302_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (4, negative 4) and x-intercepts of (2, 0) and (6, 0).\" \/><\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\\(\\left(\\text{\u2212}\\infty ,2\\right]\\cup \\left[6,\\infty \\right)\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1169149030281\">We list the steps to take to solve a quadratic inequality graphically.<\/p>\r\n\r\n<div data-type=\"note\" id=\"fs-id1169149350941\" class=\"howto\">\r\n<div data-type=\"title\">Solve a quadratic inequality graphically.<\/div>\r\n<ol id=\"fs-id1169148988653\" class=\"stepwise\" type=\"1\">\r\n \t<li>Write the quadratic inequality in standard form.<\/li>\r\n \t<li>Graph the function \\(f\\left(x\\right)=a{x}^{2}+bx+c.\\)<\/li>\r\n \t<li>Determine the solution from the graph.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<p id=\"fs-id1169146633876\">In the last example, the parabola opened upward and in the next example, it opens downward. In both cases, we are looking for the part of the parabola that is below the <em data-effect=\"italics\">x<\/em>-axis but note how the position of the parabola affects the solution.<\/p>\r\n\r\n<div data-type=\"example\" id=\"fs-id1169149121645\" class=\"textbox textbox--examples\">\r\n<div data-type=\"exercise\" id=\"fs-id1169144381671\">\r\n<div data-type=\"problem\" id=\"fs-id1169146661918\">\r\n<p id=\"fs-id1169144379524\">Solve \\(\\text{\u2212}{x}^{2}-8x-12\\le 0\\) graphically. Write the solution in interval notation.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169148899359\">\r\n<table class=\"unnumbered unstyled can-break\" summary=\"This figure is step-by-step instructions on how to solve an inequality graphically. The quadratic inequality in standard form is negative x squared minus 8 times x minus 12 less than or equal to 0. Graph the function f of x equals negative x squared minus 8 times x minus 12 to find that the parabola opens upward. Find the line of symmetry by using the equation x equals negative b divided by 2 times a. Substitute in to get x equals negative negative 8 divided by 2 times negative 1 to find x equals negative 4. Find the vertex of f of x equals negative x squared minus 8 times x minus 12 by finding that f of negative 4 equals negative negative 4 squared minus 8 times negative 4 minus 12. That gives you f of negative 4 equals negative 16 minus 32 minus 12, which then reduces to f of negative 4 equals 4. The vertex is (negative 4, 4). Find the x-intercepts. Let f of x equal 0. Take the original function, f of x equals negative x squared minus 8 times x minus 12, then make it 0 equals negative x squared minus 8 times x minus 12. Factor to get 0 equals negative 1 times the quantity x plus 6 times the quantity x plus 2. Use the Zero Product Property to get x equals negative 6 and x equals negative 2. The x-intercepts are (negative 6, 0) and (negative 2, 0). The graph shown is the curve formed when plotting all the points just found. Then, determine the solution from the graph, (negative infinity, negative 6] in union with [negative 2, infinity). We include the x-intercepts as the inequality is \u201cless than or equal to.\u201d\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td data-valign=\"top\" data-align=\"left\">The quadratic inequality in standard form.<\/td>\r\n<td><\/td>\r\n<td data-valign=\"top\" data-align=\"center\">\\(-{x}^{2}-8x-12\\le 0\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-valign=\"top\" data-align=\"left\">Graph the function \\(f\\left(x\\right)=\\text{\u2212}{x}^{2}-8x-12\\).<\/td>\r\n<td><\/td>\r\n<td data-valign=\"top\" data-align=\"left\">The parabola opens downward.\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1169148939580\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_003b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-valign=\"top\" data-align=\"left\">Find the line of symmetry.<\/td>\r\n<td><\/td>\r\n<td data-valign=\"top\" data-align=\"left\">\\(\\phantom{\\rule{1.8em}{0ex}}x=-\\frac{b}{2a}\\)\r\n<div data-type=\"newline\"><\/div>\r\n\\(\\phantom{\\rule{1.8em}{0ex}}x=-\\frac{-8}{2\\left(-1\\right)}\\)\r\n<div data-type=\"newline\"><\/div>\r\n\\(\\phantom{\\rule{1.8em}{0ex}}x=-4\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-valign=\"top\" data-align=\"left\">Find the vertex.<\/td>\r\n<td><\/td>\r\n<td data-valign=\"top\" data-align=\"left\">\\(\\phantom{\\rule{0.65em}{0ex}}f\\left(x\\right)=\\text{\u2212}{x}^{2}-8x-12\\)\r\n<div data-type=\"newline\"><\/div>\r\n\\(f\\left(-4\\right)=\\text{\u2212}{\\left(-4\\right)}^{2}-8\\left(-4\\right)-12\\)\r\n<div data-type=\"newline\"><\/div>\r\n\\(f\\left(-4\\right)=-16+32-12\\)\r\n<div data-type=\"newline\"><\/div>\r\n\\(f\\left(-4\\right)=4\\)\r\n<div data-type=\"newline\"><\/div>\r\nVertex \\(\\left(-4,4\\right)\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-valign=\"top\" data-align=\"left\">Find the <em data-effect=\"italics\">x<\/em>-intercepts. Let \\(f\\left(x\\right)=0\\).<\/td>\r\n<td><\/td>\r\n<td data-valign=\"top\" data-align=\"left\">\\(\\phantom{\\rule{0.75em}{0ex}}f\\left(x\\right)=\\text{\u2212}{x}^{2}-8x-12\\)\r\n<div data-type=\"newline\"><\/div>\r\n\\(\\phantom{\\rule{1.95em}{0ex}}0=\\text{\u2212}{x}^{2}-8x-12\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-valign=\"top\" data-align=\"left\">Factor.\r\n<div data-type=\"newline\"><\/div>\r\nUse the Zero Product Property.<\/td>\r\n<td><\/td>\r\n<td data-valign=\"top\" data-align=\"left\">\\(\\phantom{\\rule{1.95em}{0ex}}0=-1\\left(x+6\\right)\\left(x+2\\right)\\)\r\n<div data-type=\"newline\"><\/div>\r\n\\(\\phantom{\\rule{2em}{0ex}}x=-6\\phantom{\\rule{1.5em}{0ex}}x=-2\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-valign=\"top\" data-align=\"left\">Graph the parabola.<\/td>\r\n<td><\/td>\r\n<td data-valign=\"top\" data-align=\"center\"><em data-effect=\"italics\">x<\/em>-intercepts \\(\\left(-6,0\\right),\\left(-2,0\\right)\\)\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1169148889714\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_003n_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-valign=\"top\" data-align=\"left\">Determine the solution from the graph.\r\n<div data-type=\"newline\"><\/div>\r\nWe include the <em data-effect=\"italics\">x<\/em>-intercepts as the inequality\r\n<div data-type=\"newline\"><\/div>\r\nis \u201cless than or equal to.\u201d<\/td>\r\n<td><\/td>\r\n<td data-valign=\"top\" data-align=\"center\">\\(\\left(\\text{\u2212}\\infty ,\\phantom{\\rule{0.2em}{0ex}}\\text{\u2212}6\\right]\\cup \\left[\\text{\u2212}2,\\phantom{\\rule{0.2em}{0ex}}\\infty \\right)\\)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1169148834153\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1169148862433\">\r\n<div data-type=\"problem\" id=\"fs-id1169148859315\">\r\n<p id=\"fs-id1169148845708\"><span class=\"token\">\u24d0<\/span> Solve \\(\\text{\u2212}{x}^{2}-6x-5&gt;0\\) graphically and <span class=\"token\">\u24d1<\/span> write the solution in interval notation.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\">\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d0<\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1169148824319\" data-alt=\"A downward-facing parabola on the x y-coordinate plane. It has a vertex of (negative 3, 4), a y-intercept at (0, negative 5), and an axis of symmetry shown at x equals negative 3.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_303_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"A downward-facing parabola on the x y-coordinate plane. It has a vertex of (negative 3, 4), a y-intercept at (0, negative 5), and an axis of symmetry shown at x equals negative 3.\" \/><\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\\(\\left(-1,5\\right)\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1169148948253\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1169148821286\">\r\n<div data-type=\"problem\" id=\"fs-id1169146633494\">\r\n<p id=\"fs-id1169149155077\"><span class=\"token\">\u24d0<\/span> Solve \\(\\text{\u2212}{x}^{2}+10x-16\\le 0\\) graphically and <span class=\"token\">\u24d1<\/span> write the solution in interval notation.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169146627483\">\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d0<\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1169148880103\" data-alt=\"A downward-facing parabola on the x y-coordinate plane. It has a vertex of (5, 9), a y-intercept at (0, negative 16), and an axis of symmetry of x equals 5.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_304_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"A downward-facing parabola on the x y-coordinate plane. It has a vertex of (5, 9), a y-intercept at (0, negative 16), and an axis of symmetry of x equals 5.\" \/><\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\\(\\left(\\text{\u2212}\\infty ,2\\right]\\cup \\left[8,\\infty \\right)\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1169149157937\">\r\n<h3 data-type=\"title\">Solve Quadratic Inequalities Algebraically<\/h3>\r\n<p id=\"fs-id1169149308017\">The algebraic method we will use is very similar to the method we used to solve rational inequalities. We will find the critical points for the inequality, which will be the solutions to the related quadratic equation. Remember a polynomial expression can change signs only where the expression is zero.<\/p>\r\n<p id=\"fs-id1169148867090\">We will use the <span data-type=\"term\" class=\"no-emphasis\">critical points<\/span> to divide the number line into intervals and then determine whether the quadratic expression willl be postive or negative in the interval. We then determine the solution for the inequality.<\/p>\r\n\r\n<div data-type=\"example\" id=\"fs-id1169149024679\" class=\"textbox textbox--examples\">\r\n<div data-type=\"title\">How To Solve Quadratic Inequalities Algebraically<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169148998010\">\r\n<div data-type=\"problem\" id=\"fs-id1169148846107\">\r\n<p id=\"fs-id1169148866808\">Solve \\({x}^{2}-x-12\\ge 0\\) algebraically. Write the solution in interval notation.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169149116677\"><span data-type=\"media\" id=\"fs-id1169149295000\" data-alt=\"This figure is a table giving the instructions for solving x squared minus x minus 12 greater than or equal to 0 algebraically. It consists of 3 columns where the instructions are given in the first column, the explanation in the second, and the work in the third. Step 1 is to write the quadratic inequality in standard form. The quadratic inequality in already in standard form, so x squared minus x minus 12 greater than or equal to 0.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_004a_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure is a table giving the instructions for solving x squared minus x minus 12 greater than or equal to 0 algebraically. It consists of 3 columns where the instructions are given in the first column, the explanation in the second, and the work in the third. Step 1 is to write the quadratic inequality in standard form. The quadratic inequality in already in standard form, so x squared minus x minus 12 greater than or equal to 0.\" \/><\/span><span data-type=\"media\" id=\"fs-id1169146640088\" data-alt=\"Step 2 is to determine the critical points -- the solutions to the related quadratic equation. To do this, change the inequality sign to an equal sign and then solve the equation. x squared minus x minus 12 equals 0 factors to the quantity x plus 3 times the quantity x minus 4 equals 0. Then, x plus 3 equals 0 and x minus 4 equals 0 to give x equals negative 3 and x equals 4.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_004b_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2 is to determine the critical points -- the solutions to the related quadratic equation. To do this, change the inequality sign to an equal sign and then solve the equation. x squared minus x minus 12 equals 0 factors to the quantity x plus 3 times the quantity x minus 4 equals 0. Then, x plus 3 equals 0 and x minus 4 equals 0 to give x equals negative 3 and x equals 4.\" \/><\/span><span data-type=\"media\" id=\"fs-id1169149066368\" data-alt=\"Step 3 is to use the critical points to divide the number line into intervals. Use negative 3 and 4 to divide the number line into intervals. A number line is shown that includes from left to right the values of negative 3, 0, and 4, with dotted lines on negative 3 and 4.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_004c_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3 is to use the critical points to divide the number line into intervals. Use negative 3 and 4 to divide the number line into intervals. A number line is shown that includes from left to right the values of negative 3, 0, and 4, with dotted lines on negative 3 and 4.\" \/><\/span><span data-type=\"media\" id=\"fs-id1169148969887\" data-alt=\"Step 4 says above the number line show the sign of each quadratic expression using test points from each interval substituted into the original inequality. X equals negative 5, x equals 0, and x equals 5 are chosen to test. The expression negative x squared minus x minus 12 is given with negative 5 squared minus negative 5 minus 12 underneath, which gives 18. The expression negative x squared minus x minus 12 is given with 0 squared minus 0 minus 12 underneath, which gives 12. The expression negative x squared minus x minus 12 is given with 5 squared minus 5 minus 12 underneath, which gives 8.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_004d_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 4 says above the number line show the sign of each quadratic expression using test points from each interval substituted into the original inequality. X equals negative 5, x equals 0, and x equals 5 are chosen to test. The expression negative x squared minus x minus 12 is given with negative 5 squared minus negative 5 minus 12 underneath, which gives 18. The expression negative x squared minus x minus 12 is given with 0 squared minus 0 minus 12 underneath, which gives 12. The expression negative x squared minus x minus 12 is given with 5 squared minus 5 minus 12 underneath, which gives 8.\" \/><\/span><span data-type=\"media\" id=\"fs-id1169149305655\" data-alt=\"For Step 5, determine the intervals where the inequality is correct. Write the solution in interval notation. x squared minus x minus 12 greater than or equal to 0 is shown. The inequality is positive in the first and last intervals and equals 0 at the points negative 4, 3 . The solution, in interval notation, is (negative infinity, negative 3] in union with [4, infinity).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_004e_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"For Step 5, determine the intervals where the inequality is correct. Write the solution in interval notation. x squared minus x minus 12 greater than or equal to 0 is shown. The inequality is positive in the first and last intervals and equals 0 at the points negative 4, 3 . The solution, in interval notation, is (negative infinity, negative 3] in union with [4, infinity).\" \/><\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1169148861231\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1169148878663\">\r\n<div data-type=\"problem\" id=\"fs-id1169148924858\">\r\n\r\nSolve \\({x}^{2}+2x-8\\ge 0\\) algebraically. Write the solution in interval notation.\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169148818997\">\r\n<p id=\"fs-id1169146654389\">\\(\\left(\\text{\u2212}\\infty ,-4\\right]\\cup \\left[2,\\infty \\right)\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1169148999680\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1169148898510\">\r\n<div data-type=\"problem\" id=\"fs-id1169149362258\">\r\n<p id=\"fs-id1169148888294\">Solve \\({x}^{2}-2x-15\\le 0\\) algebraically. Write the solution in interval notation.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169149292178\">\r\n<p id=\"fs-id1169148938679\">\\(\\left[-3,5\\right]\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1169149291093\">In this example, since the expression \\({x}^{2}-x-12\\) factors nicely, we can also find the sign in each interval much like we did when we solved rational inequalities. We find the sign of each of the factors, and then the sign of the product. Our number line would like this:<\/p>\r\n<span data-type=\"media\" id=\"fs-id1169148932873\" data-alt=\"The figure shows the expression x squared minus x minus 12 factored to the quantity of x plus 3 times the quantity of x minus 4. The image shows a number line showing dotted lines on negative 3 and 4. It shows the signs of the quantity x plus 3 to be negative, positive, positive, and the signs of the quantity x minus 4 to be negative, negative, positive. Under the number line, it shows the quantity x plus 3 times the quantity x minus 4 with the signs positive, negative, positive.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_005_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the expression x squared minus x minus 12 factored to the quantity of x plus 3 times the quantity of x minus 4. The image shows a number line showing dotted lines on negative 3 and 4. It shows the signs of the quantity x plus 3 to be negative, positive, positive, and the signs of the quantity x minus 4 to be negative, negative, positive. Under the number line, it shows the quantity x plus 3 times the quantity x minus 4 with the signs positive, negative, positive.\" \/><\/span>\r\n<p id=\"fs-id1169146665788\">The result is the same as we found using the other method.<\/p>\r\n<p id=\"fs-id1169149342624\">We summarize the steps here.<\/p>\r\n\r\n<div data-type=\"note\" id=\"fs-id1169149307510\" class=\"howto\">\r\n<div data-type=\"title\">Solve a quadratic inequality algebraically.<\/div>\r\n<ol id=\"fs-id1169149015056\" class=\"stepwise\" type=\"1\">\r\n \t<li>Write the quadratic inequality in standard form.<\/li>\r\n \t<li>Determine the critical points\u2014the solutions to the related quadratic equation.<\/li>\r\n \t<li>Use the critical points to divide the number line into intervals.<\/li>\r\n \t<li>Above the number line show the sign of each quadratic expression using test points from each interval substituted into the original inequality.<\/li>\r\n \t<li>Determine the intervals where the inequality is correct. Write the solution in interval notation.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div data-type=\"example\" id=\"fs-id1169144383249\" class=\"textbox textbox--examples\">\r\n<div data-type=\"exercise\" id=\"fs-id1169146627378\">\r\n<div data-type=\"problem\" id=\"fs-id1169146644117\">\r\n<p id=\"fs-id1169149002129\">Solve \\({x}^{2}+6x-7\\ge 0\\) algebraically. Write the solution in interval notation.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169148879146\">\r\n<table id=\"fs-id1169149003372\" class=\"unnumbered unstyled can-break\" summary=\"The figure is gives step-by-step instructions on how to solve negative x squared plus 6 times x minus 7 greater than or equal to 0 algebraically. Write the inequality in standard form. negative x squared plus 6 times x minus 7 greater than or equal to 0 is already in standard form. Multiply both sides of the inequality by negative 1, remember to reverse the inequality sign, to get x squared minus 6 times x plus 7 less than or equal to 0. Determine the critical points by solving the related quadratic equation, x squared minus 6 times x plus 7 equals 0. Write the quadratic formula, x equals negative b plus or minus the square root of b squared minus 4 times a times c all divided by 2 times a. Then substitute in the values of a, b, c, to get x equals negative negative 6 plus or minus the square root of negative 6 squared minus 4 times 1 times 7 all divided by 2 times 1. Simplify to get x equals 6 plus or minus the square root of 8 divided by 2. Remove the common factor of 2, x equals 2 times the quantity 3 plus or minus square root of 2 divided by 2 which gives x equals 3 plus or minus square root of 2. If x equals 3 plus square root of 2, x is approximately 1 and 6 tenths. If x equals 3 minus square root of 2, x is approximately 4 and 4 tenths. Use the critical points to divide the number line into intervals. A number line is shown with 1 and 6 tenths and 4 and 4 tenths. Test the numbers from each interval in the original inequality. On the number line, negative x squared plus 6 times x minus 7 is shown with the signs negative, positive, and negative. Determine the intervals where the inequality is correct. Write the solution in interval notation. negative x squared plus 6 times x minus 7 is greater than or equals to 0 in the middle interval, so the final answer is [3 minus square root of 2, 3 plus square root of 2\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td data-valign=\"top\" data-align=\"left\">Write the quadratic inequality in standard form.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">\\(-{x}^{2}+6x-7\\ge 0\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-valign=\"top\" data-align=\"left\">Multiply both sides of the inequality by \\(-1\\).\r\n<div data-type=\"newline\"><\/div>\r\nRemember to reverse the inequality sign.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">\\(\\phantom{\\rule{0.7em}{0ex}}{x}^{2}-6x+7\\le 0\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-valign=\"top\" data-align=\"left\">Determine the critical points by solving\r\n<div data-type=\"newline\"><\/div>\r\nthe related quadratic equation.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">\\(\\phantom{\\rule{0.7em}{0ex}}{x}^{2}-6x+7=0\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-valign=\"top\" data-align=\"left\">Write the Quadratic Formula.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">\\(\\phantom{\\rule{4.9em}{0ex}}x=\\frac{-b\u00b1\\sqrt{{b}^{2}-4ac}}{2a}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-valign=\"top\" data-align=\"left\">Then substitute in the values of \\(a,b,c\\).<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">\\(\\phantom{\\rule{4.9em}{0ex}}x=\\frac{-\\left(-6\\right)\u00b1\\sqrt{{\\left(-6\\right)}^{2}-4\\cdot 1\\cdot \\left(7\\right)}}{2\\cdot 1}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">\\(\\phantom{\\rule{4.9em}{0ex}}x=\\frac{6\u00b1\\sqrt{8}}{2}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-valign=\"top\" data-align=\"left\">Simplify the radical.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">\\(\\phantom{\\rule{4.9em}{0ex}}x=\\frac{6\u00b12\\sqrt{2}}{2}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-valign=\"top\" data-align=\"left\">Remove the common factor, 2.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">\\(\\phantom{\\rule{4.9em}{0ex}}x=\\frac{2\\left(3\u00b1\\sqrt{2}\\right)}{2}\\)\r\n<div data-type=\"newline\"><\/div>\r\n\\(\\phantom{\\rule{4.9em}{0ex}}x=3\u00b1\\sqrt{2}\\)\r\n<div data-type=\"newline\"><\/div>\r\n\\(\\phantom{\\rule{4.9em}{0ex}}x=3+\\sqrt{2}\\phantom{\\rule{2em}{0ex}}x=3-\\sqrt{2}\\)\r\n<div data-type=\"newline\"><\/div>\r\n\\(\\phantom{\\rule{4.9em}{0ex}}x\\approx 1.6\\phantom{\\rule{3.35em}{0ex}}x\\approx 4.4\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-valign=\"top\" data-align=\"left\">Use the critical points to divide the\r\n<div data-type=\"newline\"><\/div>\r\nnumber line into intervals.\r\n<div data-type=\"newline\"><\/div>\r\nTest numbers from each interval\r\n<div data-type=\"newline\"><\/div>\r\nin the original inequality.<\/td>\r\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1169149087038\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_006l_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-valign=\"top\" data-align=\"left\">Determine the intervals where the\r\n<div data-type=\"newline\"><\/div>\r\ninequality is correct. Write the solution\r\n<div data-type=\"newline\"><\/div>\r\nin interval notation.<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">\\(\\text{\u2212}{x}^{2}+6x-7\\ge 0\\) in the middle interval\r\n<div data-type=\"newline\"><\/div>\r\n\\(\\left[3-\\sqrt{2},\\phantom{\\rule{0.5em}{0ex}}3+\\sqrt{2}\\right]\\)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1169148974557\">\r\n<div data-type=\"problem\">\r\n<p id=\"fs-id1169149087343\">Solve \\(\\text{\u2212}{x}^{2}+2x+1\\ge 0\\) algebraically. Write the solution in interval notation.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\">\r\n<p id=\"fs-id1169149000975\">\\(\\left[-1-\\sqrt{2},-1+\\sqrt{2}\\right]\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1169149037957\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1169149122051\">\r\n<div data-type=\"problem\" id=\"fs-id1169149135306\">\r\n<p id=\"fs-id1169148994901\">Solve \\(\\text{\u2212}{x}^{2}+8x-14&lt;0\\) algebraically. Write the solution in interval notation.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169149088969\">\r\n\r\n\\(\\left(\\text{\u2212}\\infty ,4-\\sqrt{2}\\right)\\cup \\left(4+\\sqrt{2},\\infty \\right)\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1169149309531\">The solutions of the quadratic inequalities in each of the previous examples, were either an interval or the union of two intervals. This resulted from the fact that, in each case we found two solutions to the corresponding quadratic equation <em data-effect=\"italics\">ax<\/em><sup>2<\/sup> + <em data-effect=\"italics\">bx<\/em> + <em data-effect=\"italics\">c<\/em> = 0. These two solutions then gave us either the two <em data-effect=\"italics\">x-<\/em>intercepts for the graph or the two critical points to divide the number line into intervals.<\/p>\r\n<p id=\"fs-id1169148999242\">This correlates to our previous discussion of the number and type of solutions to a quadratic equation using the discriminant.<\/p>\r\n<p id=\"fs-id1169149135716\">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>\r\n<span data-type=\"media\" data-alt=\"The figure is a table with 3 columns. Column 1 is labeled discriminant, column 2 is Number\/Type of solution, and column 3 is Typical Graph. Reading across the columns, if b squared minus 4 times a times c is greater than 0, there will be 2 real solutions because there are 2 x-intercepts on the graph. The image of a typical graph an upward or downward parabola with 2 x-intercepts. If the discriminant b squared minus 4 times a times c is equals to 0, then there is 1 real solution because there is 1 x-intercept on the graph. The image of the typical graph is an upward- or downward-facing parabola that has a vertex on the x-axis instead of crossing through it. If the discriminant b squared minus 4 times a times c is less than 0, there are 2 complex solutions because there is no x-intercept. The image of the typical graph shows an upward- or downward-facing parabola that does not cross the x-axis.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_007_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure is a table with 3 columns. Column 1 is labeled discriminant, column 2 is Number\/Type of solution, and column 3 is Typical Graph. Reading across the columns, if b squared minus 4 times a times c is greater than 0, there will be 2 real solutions because there are 2 x-intercepts on the graph. The image of a typical graph an upward or downward parabola with 2 x-intercepts. If the discriminant b squared minus 4 times a times c is equals to 0, then there is 1 real solution because there is 1 x-intercept on the graph. The image of the typical graph is an upward- or downward-facing parabola that has a vertex on the x-axis instead of crossing through it. If the discriminant b squared minus 4 times a times c is less than 0, there are 2 complex solutions because there is no x-intercept. The image of the typical graph shows an upward- or downward-facing parabola that does not cross the x-axis.\" \/><\/span>\r\n<p id=\"fs-id1169148888286\">The last row of the table shows us when the parabolas never intersect the <em data-effect=\"italics\">x<\/em>-axis. Using the Quadratic Formula to solve the quadratic equation, the radicand is a negative. We get two complex solutions.<\/p>\r\n<p id=\"fs-id1169148894484\">In the next example, the quadratic inequality solutions will result from the solution of the quadratic equation being complex.<\/p>\r\n\r\n<div data-type=\"example\" id=\"fs-id1169148951346\" class=\"textbox textbox--examples\">\r\n<div data-type=\"exercise\" id=\"fs-id1169149308650\">\r\n<div data-type=\"problem\">\r\n\r\nSolve, writing any solution in interval notation:\r\n\r\n<span class=\"token\">\u24d0<\/span>\\({x}^{2}-3x+4&gt;0\\)<span class=\"token\">\u24d1<\/span>\\({x}^{2}-3x+4\\le 0\\)\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169148973208\">\r\n<p id=\"fs-id1169148970174\"><span class=\"token\">\u24d0<\/span><\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<table id=\"fs-id1169149220879\" class=\"unnumbered unstyled can-break\" summary=\"The figure is gives step-by-step instructions on how to solve x squared minus 3 times x plus 4 greater than 0 algebraically. Write the inequality in standard form. x squared minus 3 times x plus 4 greater than 0 is already in standard form. Determine the critical points by solving the related quadratic equation, x squared minus 3 times x plus 4 equals 0. Write the quadratic formula, x equals negative b plus or minus the square root of b squared minus 4 times a times c all divided by 2 times a. Then substitute in the values of a, b, c, to get x equals negative negative 3 plus or minus the square root of negative 3 squared minus 4 times 1 times 4 all divided by 2 times 1. Simplify to get x equals 3 plus or minus the square root of negative 7 divided by 2. Simplify the radicand to get x equals 3 plus or minus the square root of 7 times i divided by 2. The complex solutions tell us that parabola does not intercept the x-axis. Also, the parabola opens upward. This tells us that the paprabola is completely above the x-axis, as the image of an upward-facing parabola that does not cross the x-axis shows.\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td data-valign=\"top\" data-align=\"left\">Write the quadratic inequality in standard form.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">\\(\\phantom{\\rule{0.8em}{0ex}}-{x}^{2}-3x+4&gt;0\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-valign=\"top\" data-align=\"left\">Determine the critical points by solving\r\n<div data-type=\"newline\"><\/div>\r\nthe related quadratic equation.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">\\(\\phantom{\\rule{1.5em}{0ex}}{x}^{2}-3x+4=0\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-valign=\"top\" data-align=\"left\">Write the Quadratic Formula.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">\\(x=\\frac{-b\u00b1\\sqrt{{b}^{2}-4ac}}{2a}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-valign=\"top\" data-align=\"left\">Then substitute in the values of \\(a,b,c\\).<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">\\(x=\\frac{-\\left(\\text{\u2212}3\\right)\u00b1\\sqrt{{\\left(\\text{\u2212}3\\right)}^{2}-4\\cdot 1\\cdot \\left(4\\right)}}{2\\cdot 1}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">\\(x=\\frac{3\u00b1\\sqrt{-7}}{2}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-valign=\"top\" data-align=\"left\">Simplify the radicand.<\/td>\r\n<td data-valign=\"top\" data-align=\"left\">\\(x=\\frac{3\u00b1\\sqrt{7}i}{2}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-valign=\"top\" data-align=\"left\">The complex solutions tell us the\r\n<div data-type=\"newline\"><\/div>\r\nparabola does not intercept the <em data-effect=\"italics\">x<\/em>-axis.\r\n<div data-type=\"newline\"><\/div>\r\nAlso, the parabola opens upward. This\r\n<div data-type=\"newline\"><\/div>\r\ntells us that the parabola is completely above the <em data-effect=\"italics\">x<\/em>-axis.<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">Complex solutions\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1169149219736\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_008g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1169148989687\">We are to find the solution to \\({x}^{2}-3x+4&gt;0.\\) Since for all values of \\(x\\) the graph is above the <em data-effect=\"italics\">x<\/em>-axis, all values of <em data-effect=\"italics\">x<\/em> make the inequality true. In interval notation we write \\(\\left(\\text{\u2212}\\infty ,\\infty \\right).\\)<\/p>\r\n<p id=\"fs-id1169149123625\"><span class=\"token\">\u24d1<\/span><\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n\\(\\begin{array}{cccc}\\text{Write the quadratic inequality in standard form.}\\hfill &amp; &amp; &amp; {x}^{2}-3x+4\\le 0\\hfill \\\\ \\begin{array}{c}\\text{Determine the critical points by solving}\\hfill \\\\ \\text{the related quadratic equation}\\hfill \\end{array}\\hfill &amp; &amp; &amp; {x}^{2}-3x+4=0\\hfill \\end{array}\\)\r\n<p id=\"fs-id1169148938454\">Since the corresponding quadratic equation is the same as in part (a), the parabola will be the same. The parabola opens upward and is completely above the <em data-effect=\"italics\">x<\/em>-axis\u2014no part of it is below the <em data-effect=\"italics\">x<\/em>-axis.<\/p>\r\n<p id=\"fs-id1169148948396\">We are to find the solution to \\({x}^{2}-3x+4\\le 0.\\) Since for all values of <em data-effect=\"italics\">x<\/em> the graph is never below the <em data-effect=\"italics\">x<\/em>-axis, no values of <em data-effect=\"italics\">x<\/em> make the inequality true. There is no solution to the inequality.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1169144562776\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1169148891264\">\r\n<div data-type=\"problem\" id=\"fs-id1169148881288\">\r\n<p id=\"fs-id1169148930301\">Solve and write any solution in interval notation:<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d0<\/span> \\(\\text{\u2212}{x}^{2}+2x-4\\le 0\\) <span class=\"token\">\u24d1<\/span> \\(\\text{\u2212}{x}^{2}+2x-4\\ge 0\\)\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169148826303\">\r\n<p id=\"fs-id1169149196732\"><span class=\"token\">\u24d0<\/span>\\(\\left(\\text{\u2212}\\infty ,\\infty \\right)\\)<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span> no solution\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"note\" id=\"fs-id1169148910806\" class=\"try\">\r\n<div data-type=\"exercise\" id=\"fs-id1169148964170\">\r\n<div data-type=\"problem\" id=\"fs-id1169149319585\">\r\n<p id=\"fs-id1169148884129\">Solve and write any solution in interval notation:<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d0<\/span> \\({x}^{2}+3x+3&lt;0\\) <span class=\"token\">\u24d1<\/span> \\({x}^{2}+3x+3&gt;0\\)\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169148843341\">\r\n<p id=\"fs-id1169149293529\"><span class=\"token\">\u24d0<\/span> no solution<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span> \\(\\left(\\text{\u2212}\\infty ,\\infty \\right)\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1169148894040\">\r\n<h3 data-type=\"title\">Key Concepts<\/h3>\r\n<ul id=\"fs-id1169148898632\" data-bullet-style=\"bullet\">\r\n \t<li>Solve a Quadratic Inequality Graphically\r\n<ol id=\"fs-id1169149094904\" class=\"stepwise\" type=\"1\">\r\n \t<li>Write the quadratic inequality in standard form.<\/li>\r\n \t<li>Graph the function \\(f\\left(x\\right)=a{x}^{2}+bx+c\\) using properties or transformations.<\/li>\r\n \t<li>Determine the solution from the graph.<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>How to Solve a Quadratic Inequality Algebraically\r\n<ol id=\"fs-id1169148967529\" class=\"stepwise\" type=\"1\">\r\n \t<li>Write the quadratic inequality in standard form.<\/li>\r\n \t<li>Determine the critical points -- the solutions to the related quadratic equation.<\/li>\r\n \t<li>Use the critical points to divide the number line into intervals.<\/li>\r\n \t<li>Above the number line show the sign of each quadratic expression using test points from each interval substituted into the original inequality.<\/li>\r\n \t<li>Determine the intervals where the inequality is correct. Write the solution in interval notation.<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1169148867354\">\r\n<h3 data-type=\"title\">Section Exercises<\/h3>\r\n<div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1169148894275\">\r\n<h4 data-type=\"title\">Practice Makes Perfect<\/h4>\r\n<p id=\"fs-id1169148951165\"><strong data-effect=\"bold\">Solve Quadratic Inequalities Graphically<\/strong><\/p>\r\n<p id=\"fs-id1169146743578\">In the following exercises, <span class=\"token\">\u24d0<\/span> solve graphically and <span class=\"token\">\u24d1<\/span> write the solution in interval notation.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1169146731344\">\r\n<div data-type=\"problem\" id=\"fs-id1169148926529\">\r\n<p id=\"fs-id1169148964346\">\\({x}^{2}+6x+5&gt;0\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169149026035\">\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d0<\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1169146745490\" data-alt=\"The graph shown is an upward-facing parabola with vertex (negative 3, negative 4) and y-intercept (0,5).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_305_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph shown is an upward-facing parabola with vertex (negative 3, negative 4) and y-intercept (0,5).\" \/><\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\\(\\left(\\text{\u2212}\\infty ,-5\\right)\\cup \\left(-1,\\infty \\right)\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149375310\">\r\n<div data-type=\"problem\" id=\"fs-id1169148915246\">\r\n<p id=\"fs-id1169149113062\">\\({x}^{2}+4x-12&lt;0\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169148851196\">\r\n<div data-type=\"problem\" id=\"fs-id1169146731454\">\r\n<p id=\"fs-id1169146617305\">\\({x}^{2}+4x+3\\le 0\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169146611390\">\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d0<\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1169149376574\" data-alt=\"The graph shown is an upward facing parabola with vertex (negative 2, negative 1) and y-intercept (0,3).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_307_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph shown is an upward facing parabola with vertex (negative 2, negative 1) and y-intercept (0,3).\" \/><\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\\(\\left[-3,-1\\right]\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149222448\">\r\n<div data-type=\"problem\" id=\"fs-id1169149008120\">\r\n<p id=\"fs-id1169148996581\">\\({x}^{2}-6x+8\\ge 0\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169146642948\">\r\n<div data-type=\"problem\" id=\"fs-id1169149343778\">\r\n<p id=\"fs-id1169146741072\">\\(\\text{\u2212}{x}^{2}-3x+18\\le 0\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169148992135\">\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d0<\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1169149009812\" data-alt=\"The graph shown is a downward-facing parabola with vertex (negative 1 and 5 tenths, 20) and y-intercept (0, 18).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_309_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph shown is a downward-facing parabola with vertex (negative 1 and 5 tenths, 20) and y-intercept (0, 18).\" \/><\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\\(\\left(\\text{\u2212}\\infty ,-6\\right]\\cup \\left[3,\\infty \\right)\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149029983\">\r\n<div data-type=\"problem\" id=\"fs-id1169149109867\">\r\n<p id=\"fs-id1169149037612\">\\(\\text{\u2212}{x}^{2}+2x+24&lt;0\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169148880298\">\r\n<div data-type=\"problem\" id=\"fs-id1169149361788\">\r\n<p id=\"fs-id1169149295280\">\\(\\text{\u2212}{x}^{2}+x+12\\ge 0\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169146618321\">\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d0<\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1169149017568\" data-alt=\"The graph shown is a downward facing parabola with a y-intercept of (0, 12) and x-intercepts (negative 3, 0) and (4, 0).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_311_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph shown is a downward facing parabola with a y-intercept of (0, 12) and x-intercepts (negative 3, 0) and (4, 0).\" \/><\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\\(\\left[-3,4\\right]\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169147029171\">\r\n<div data-type=\"problem\" id=\"fs-id1169144565964\">\r\n<p id=\"fs-id1169149223052\">\\(\\text{\u2212}{x}^{2}+2x+15&gt;0\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1169148933030\">In the following exercises, solve each inequality algebraically and write any solution in interval notation.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1169148992200\">\r\n<div data-type=\"problem\" id=\"fs-id1169148992202\">\r\n<p id=\"fs-id1169149012746\">\\({x}^{2}+3x-4\\ge 0\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169149016572\">\r\n<p id=\"fs-id1169149113119\">\\(\\left(\\text{\u2212}\\infty ,-4\\right]\\cup \\left[1,\\infty \\right)\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149113889\">\r\n<div data-type=\"problem\" id=\"fs-id1169146719158\">\r\n<p id=\"fs-id1169146719160\">\\({x}^{2}+x-6\\le 0\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149370800\">\r\n<div data-type=\"problem\" id=\"fs-id1169149034315\">\r\n<p id=\"fs-id1169148967931\">\\({x}^{2}-7x+10&lt;0\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169149305222\">\r\n<p id=\"fs-id1169149305224\">\\(\\left(2,5\\right)\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169148878510\">\r\n<div data-type=\"problem\" id=\"fs-id1169146612846\">\r\n<p id=\"fs-id1169148974075\">\\({x}^{2}-4x+3&gt;0\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149303922\">\r\n<div data-type=\"problem\" id=\"fs-id1169149303924\">\r\n<p id=\"fs-id1169149369970\">\\({x}^{2}+8x&gt;-15\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169146648690\">\r\n<p id=\"fs-id1169148985706\">\\(\\left(\\text{\u2212}\\infty ,-5\\right)\\cup \\left(-3,\\infty \\right)\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169148956652\">\r\n<div data-type=\"problem\" id=\"fs-id1169148956654\">\r\n<p id=\"fs-id1169144382480\">\\({x}^{2}+8x&lt;-12\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149100890\">\r\n<div data-type=\"problem\" id=\"fs-id1169149066384\">\r\n<p id=\"fs-id1169149066386\">\\({x}^{2}-4x+2\\le 0\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169149346346\">\r\n<p id=\"fs-id1169149346190\">\\(\\left[2-\\sqrt{2},2+\\sqrt{2}\\right]\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169146741668\">\r\n<div data-type=\"problem\" id=\"fs-id1169146741670\">\r\n<p id=\"fs-id1169149341634\">\\(\\text{\u2212}{x}^{2}+8x-11&lt;0\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169148983875\">\r\n<div data-type=\"problem\" id=\"fs-id1169148983877\">\r\n<p id=\"fs-id1169148889146\">\\({x}^{2}-10x&gt;-19\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169149117548\">\r\n<p id=\"fs-id1169148952146\">\\(\\left(\\text{\u2212}\\infty ,5-\\sqrt{6}\\right)\\cup \\left(5+\\sqrt{6},\\infty \\right)\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169148885396\">\r\n<div data-type=\"problem\" id=\"fs-id1169148885398\">\r\n<p id=\"fs-id1169149223155\">\\({x}^{2}+6x&lt;-3\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149123398\">\r\n<div data-type=\"problem\" id=\"fs-id1169149123401\">\r\n<p id=\"fs-id1169149219766\">\\(-6{x}^{2}+19x-10\\ge 0\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169148837495\">\r\n<p id=\"fs-id1169148885412\">\\(\\left(\\text{\u2212}\\infty ,-\\frac{5}{2}\\right]\\cup \\left[-\\frac{2}{3},\\infty \\right)\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169148989556\">\r\n<div data-type=\"problem\" id=\"fs-id1169148989558\">\r\n<p id=\"fs-id1169148967548\">\\(-3{x}^{2}-4x+4\\le 0\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149093113\">\r\n<div data-type=\"problem\" id=\"fs-id1169149093116\">\r\n<p id=\"fs-id1169149103082\">\\(-2{x}^{2}+7x+4\\ge 0\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169148871567\">\r\n<p id=\"fs-id1169148925172\">\\(\\left[\\text{\u2212}\\frac{1}{2},4\\right]\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169148996604\">\r\n<div data-type=\"problem\" id=\"fs-id1169148996606\">\r\n<p id=\"fs-id1169148983820\">\\(2{x}^{2}+5x-12&gt;0\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149011049\">\r\n<div data-type=\"problem\" id=\"fs-id1169149011051\">\r\n<p id=\"fs-id1169149013353\">\\({x}^{2}+3x+5&gt;0\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169149009744\">\r\n<p id=\"fs-id1169149009747\">\\(\\left(\\text{\u2212}\\infty ,\\infty \\right).\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169148862364\">\r\n<div data-type=\"problem\" id=\"fs-id1169148862366\">\r\n<p id=\"fs-id1169149029184\">\\({x}^{2}-3x+6\\le 0\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169148988479\">\r\n<div data-type=\"problem\" id=\"fs-id1169148988481\">\r\n<p id=\"fs-id1169149009558\">\\(\\text{\u2212}{x}^{2}+x-7&gt;0\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169148947767\">\r\n<p id=\"fs-id1169148947769\">no solution<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169147107055\">\r\n<div data-type=\"problem\" id=\"fs-id1169149287002\">\r\n<p id=\"fs-id1169149287004\">\\(\\text{\u2212}{x}^{2}-4x-5&lt;0\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169144564505\">\r\n<div data-type=\"problem\" id=\"fs-id1169144564507\">\r\n<p id=\"fs-id1169148965224\">\\(-2{x}^{2}+8x-10&lt;0\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169144365729\">\r\n<p id=\"fs-id1169148912386\">\\(\\left(\\text{\u2212}\\infty ,\\infty \\right).\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169146655785\">\r\n<div data-type=\"problem\" id=\"fs-id1169148930404\">\r\n<p id=\"fs-id1169148930406\">\\(\\text{\u2212}{x}^{2}+2x-7\\ge 0\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"writing\" data-depth=\"2\" id=\"fs-id1169149330036\">\r\n<h4 data-type=\"title\">Writing Exercises<\/h4>\r\n<div data-type=\"exercise\" id=\"fs-id1169149095709\">\r\n<div data-type=\"problem\" id=\"fs-id1169149008282\">\r\n<p id=\"fs-id1169149008284\">Explain critical points and how they are used to solve quadratic inequalities algebraically.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169144365711\">\r\n<p id=\"fs-id1169148969298\">Answers will vary.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169144563561\">\r\n<div data-type=\"problem\" id=\"fs-id1169149009433\">\r\n<p id=\"fs-id1169149009435\">Solve \\({x}^{2}+2x\\ge 8\\) both graphically and algebraically. Which method do you prefer, and why?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169148957614\">\r\n<div data-type=\"problem\" id=\"fs-id1169149096351\">\r\n<p id=\"fs-id1169149096353\">Describe the steps needed to solve a quadratic inequality graphically.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169148974155\">\r\n<p id=\"fs-id1169149112730\">Answers will vary.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169148987788\">\r\n<div data-type=\"problem\" id=\"fs-id1169148959814\">\r\n<p id=\"fs-id1169148959817\">Describe the steps needed to solve a quadratic inequality algebraically.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1169148871792\">\r\n<h4 data-type=\"title\">Self Check<\/h4>\r\n<p id=\"fs-id1169146731597\"><span class=\"token\">\u24d0<\/span> After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.<\/p>\r\n<span data-type=\"media\" id=\"fs-id1169148974036\" data-alt=\"This figure is a list to assess your understanding of the concepts presented in this section. It has 4 columns labeled I can\u2026, Confidently, With some help, and No-I don\u2019t get it! Below I can\u2026, there is solve quadratic inequalities graphically and solve quadratic inequalities algebraically. The other columns are left blank for you to check you understanding.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_201_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure is a list to assess your understanding of the concepts presented in this section. It has 4 columns labeled I can\u2026, Confidently, With some help, and No-I don\u2019t get it! Below I can\u2026, there is solve quadratic inequalities graphically and solve quadratic inequalities algebraically. The other columns are left blank for you to check you understanding.\" \/><\/span>\r\n<p id=\"fs-id1169148970119\"><span class=\"token\">\u24d1<\/span> On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"review-exercises\" data-depth=\"1\" id=\"fs-id1169148997490\">\r\n<h3 data-type=\"title\">Chapter Review Exercises<\/h3>\r\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1169148993623\">\r\n<h4 data-type=\"title\"><a href=\"\/contents\/b9659e42-3afa-4449-81d9-a017c35de140\" class=\"target-chapter\">Solve Quadratic Equations Using the Square Root Property<\/a><\/h4>\r\n<p id=\"fs-id1169148935132\"><strong data-effect=\"bold\">Solve Quadratic Equations of the form <em data-effect=\"italics\">ax<\/em><sup>2<\/sup> = <em data-effect=\"italics\">k<\/em> Using the Square Root Property<\/strong><\/p>\r\n<p id=\"fs-id1169149007258\">In the following exercises, solve using the Square Root Property.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1169149007261\">\r\n<div data-type=\"problem\" id=\"fs-id1169149305649\">\r\n<p id=\"fs-id1169149305651\">\\({y}^{2}=144\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169149028652\">\r\n<p id=\"fs-id1169149028654\">\\(y=\u00b112\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169148996016\">\r\n<div data-type=\"problem\" id=\"fs-id1169148996018\">\r\n<p id=\"fs-id1169149005194\">\\({n}^{2}-80=0\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149014399\">\r\n<div data-type=\"problem\" id=\"fs-id1169149014401\">\r\n<p id=\"fs-id1169148959976\">\\(4{a}^{2}=100\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169148970019\">\r\n<p id=\"fs-id1169149005881\">\\(a=\u00b15\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149043513\">\r\n<div data-type=\"problem\" id=\"fs-id1169148956183\">\r\n<p id=\"fs-id1169148956186\">\\(2{b}^{2}=72\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149143870\">\r\n<div data-type=\"problem\" id=\"fs-id1169149031270\">\r\n<p id=\"fs-id1169149031272\">\\({r}^{2}+32=0\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169149094648\">\r\n<p id=\"fs-id1169148828209\">\\(r=\u00b14\\sqrt{2}i\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149118485\">\r\n<div data-type=\"problem\" id=\"fs-id1169149118488\">\r\n<p id=\"fs-id1169149294753\">\\({t}^{2}+18=0\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149095532\">\r\n<div data-type=\"problem\" id=\"fs-id1169149095534\">\r\n<p id=\"fs-id1169149028835\">\\(\\frac{2}{3}{w}^{2}-20=30\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169149000087\">\r\n<p id=\"fs-id1169148993617\">\\(w=\u00b15\\sqrt{3}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169148870393\">\r\n<div data-type=\"problem\" id=\"fs-id1169149287688\">\r\n<p id=\"fs-id1169149287690\">11. \\(5{c}^{2}+3=19\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1169146617977\"><strong data-effect=\"bold\">Solve Quadratic Equations of the Form \\(a{\\left(x-h\\right)}^{2}=k\\) Using the Square Root Property<\/strong><\/p>\r\nIn the following exercises, solve using the Square Root Property.\r\n<div data-type=\"exercise\" id=\"fs-id1169146744166\">\r\n<div data-type=\"problem\" id=\"fs-id1169146744168\">\r\n<p id=\"fs-id1169149344983\">\\({\\left(p-5\\right)}^{2}+3=19\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169148957520\">\r\n<p id=\"fs-id1169148957522\">\\(p=-1,9\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169146731705\">\r\n<div data-type=\"problem\" id=\"fs-id1169146647727\">\r\n<p id=\"fs-id1169146647729\">\\({\\left(u+1\\right)}^{2}=45\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169148970873\">\r\n<div data-type=\"problem\" id=\"fs-id1169148970875\">\r\n<p id=\"fs-id1169149158618\">\\({\\left(x-\\frac{1}{4}\\right)}^{2}=\\frac{3}{16}\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169147086922\">\r\n<p id=\"fs-id1169147086924\">\\(x=\\frac{1}{4}\u00b1\\frac{\\sqrt{3}}{4}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169148890807\">\r\n<div data-type=\"problem\" id=\"fs-id1169146657246\">\r\n<p id=\"fs-id1169146657248\">\\({\\left(y-\\frac{2}{3}\\right)}^{2}=\\frac{2}{9}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169146742770\">\r\n<div data-type=\"problem\" id=\"fs-id1169146644219\">\r\n<p id=\"fs-id1169146644221\">\\({\\left(n-4\\right)}^{2}-50=150\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169149123657\">\r\n<p id=\"fs-id1169149123659\">\\(n=4\u00b110\\sqrt{2}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169148926362\">\r\n<div data-type=\"problem\" id=\"fs-id1169149319656\">\r\n<p id=\"fs-id1169149319659\">\\({\\left(4c-1\\right)}^{2}=-18\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169146627605\">\r\n<div data-type=\"problem\" id=\"fs-id1169146627607\">\r\n<p id=\"fs-id1169146627609\">\\({n}^{2}+10n+25=12\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169144377735\">\r\n<p id=\"fs-id1169149141082\">\\(n=-5\u00b12\\sqrt{3}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169148963466\">\r\n<div data-type=\"problem\" id=\"fs-id1169148963469\">\r\n<p id=\"fs-id1169148963471\">\\(64{a}^{2}+48a+9=81\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1169149329012\">\r\n<h4 data-type=\"title\"><a href=\"\/contents\/3e7f365d-4885-4b7a-bb2a-4358cbc00d2e\" class=\"target-chapter\">Solve Quadratic Equations by Completing the Square<\/a><\/h4>\r\n<p id=\"fs-id1169148968740\"><strong data-effect=\"bold\">Solve Quadratic Equations Using Completing the Square<\/strong><\/p>\r\n<p id=\"fs-id1169148968745\">In the following exercises, complete the square to make a perfect square trinomial. Then write the result as a binomial squared.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1169148984228\">\r\n<div data-type=\"problem\" id=\"fs-id1169148984230\">\r\n<p id=\"fs-id1169149112748\">\\({x}^{2}+22x\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169149100741\">\r\n<p id=\"fs-id1169148843264\">\\({\\left(x+11\\right)}^{2}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169144564417\">\r\n<div data-type=\"problem\" id=\"fs-id1169148966854\">\r\n<p id=\"fs-id1169148966856\">\\({m}^{2}-8m\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149011925\">\r\n<div data-type=\"problem\" id=\"fs-id1169149011928\">\r\n<p id=\"fs-id1169149011930\">\\({a}^{2}-3a\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169146662472\">\r\n<p id=\"fs-id1169146662475\">\\({\\left(a-\\frac{3}{2}\\right)}^{2}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169146660855\">\r\n<div data-type=\"problem\" id=\"fs-id1169146660857\">\r\n<p id=\"fs-id1169146660859\">\\({b}^{2}+13b\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1169149087768\">In the following exercises, solve by completing the square.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1169149087771\">\r\n<div data-type=\"problem\" id=\"fs-id1169149087773\">\r\n<p id=\"fs-id1169149219038\">\\({d}^{2}+14d=-13\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169148926998\">\r\n<p id=\"fs-id1169148927000\">\\(d=-13,-1\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169148948193\">\r\n<div data-type=\"problem\" id=\"fs-id1169148948196\">\r\n<p id=\"fs-id1169148970671\">\\({y}^{2}-6y=36\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169148951536\">\r\n<div data-type=\"problem\" id=\"fs-id1169148951538\">\r\n<p id=\"fs-id1169148951540\">\\({m}^{2}+6m=-109\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169149143890\">\r\n<p id=\"fs-id1169149143892\">\\(m=-3\u00b110i\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169148889866\">\r\n<div data-type=\"problem\" id=\"fs-id1169148889868\">\r\n<p id=\"fs-id1169148995968\">\\({t}^{2}-12t=-40\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169147107888\">\r\n<div data-type=\"problem\" id=\"fs-id1169146653503\">\r\n<p id=\"fs-id1169146653505\">\\({v}^{2}-14v=-31\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169149361822\">\r\n<p id=\"fs-id1169146607654\">\\(v=7\u00b13\\sqrt{2}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149018178\">\r\n<div data-type=\"problem\" id=\"fs-id1169149018181\">\r\n<p id=\"fs-id1169149018183\">\\({w}^{2}-20w=100\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169146662389\">\r\n<div data-type=\"problem\" id=\"fs-id1169148989123\">\r\n<p id=\"fs-id1169148989125\">\\({m}^{2}+10m-4=-13\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169146662548\">\r\n<p id=\"fs-id1169146662550\">\\(m=-9,-1\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169148884917\">\r\n<div data-type=\"problem\" id=\"fs-id1169148884919\">\r\n<p id=\"fs-id1169149027712\">\\({n}^{2}-6n+11=34\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169146648896\">\r\n<div data-type=\"problem\" id=\"fs-id1169146648898\">\r\n<p id=\"fs-id1169146653908\">\\({a}^{2}=3a+8\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169149215120\">\r\n<p id=\"fs-id1169149215122\">\\(a=\\frac{3}{2}\u00b1\\frac{\\sqrt{41}}{2}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149309533\">\r\n<div data-type=\"problem\" id=\"fs-id1169149309535\">\r\n<p id=\"fs-id1169149309537\">\\({b}^{2}=11b-5\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169146744234\">\r\n<div data-type=\"problem\" id=\"fs-id1169149007007\">\r\n<p id=\"fs-id1169149007010\">\\(\\left(u+8\\right)\\left(u+4\\right)=14\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169148998766\">\r\n<p id=\"fs-id1169146594818\">\\(u=-6\u00b12\\sqrt{2}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149112785\">\r\n<div data-type=\"problem\" id=\"fs-id1169149112787\">\r\n<p id=\"fs-id1169149007107\">\\(\\left(z-10\\right)\\left(z+2\\right)=28\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1169149091903\"><strong data-effect=\"bold\">Solve Quadratic Equations of the form <em data-effect=\"italics\">ax<\/em><sup>2<\/sup> + <em data-effect=\"italics\">bx<\/em> + <em data-effect=\"italics\">c<\/em> = 0 by Completing the Square<\/strong><\/p>\r\n<p id=\"fs-id1169148927233\">In the following exercises, solve by completing the square.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1169144566586\">\r\n<div data-type=\"problem\" id=\"fs-id1169144566588\">\r\n<p id=\"fs-id1169144566590\">\\(3{p}^{2}-18p+15=15\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169146637275\">\r\n<p id=\"fs-id1169146637277\">\\(p=0,6\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149373207\">\r\n<div data-type=\"problem\" id=\"fs-id1169149373210\">\r\n<p id=\"fs-id1169149373212\">\\(5{q}^{2}+70q+20=0\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149376741\">\r\n<div data-type=\"problem\" id=\"fs-id1169148988547\">\r\n<p id=\"fs-id1169148988549\">\\(4{y}^{2}-6y=4\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169149007395\">\r\n<p id=\"fs-id1169149007398\">\\(y=-\\frac{1}{2},2\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169148925211\">\r\n<div data-type=\"problem\" id=\"fs-id1169149335772\">\r\n<p id=\"fs-id1169149335774\">\\(2{x}^{2}+2x=4\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169148879311\">\r\n<div data-type=\"problem\" id=\"fs-id1169148879313\">\r\n<p id=\"fs-id1169148879315\">\\(3{c}^{2}+2c=9\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169148908999\">\r\n<p id=\"fs-id1169148909001\">\\(c=-\\frac{1}{3}\u00b1\\frac{2\\sqrt{7}}{3}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169144564314\">\r\n<div data-type=\"problem\" id=\"fs-id1169144564317\">\r\n<p id=\"fs-id1169149121477\">\\(4{d}^{2}-2d=8\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149003467\">\r\n<div data-type=\"problem\" id=\"fs-id1169149003469\">\r\n<p id=\"fs-id1169144379514\">\\(2{x}^{2}+6x=-5\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169148995457\">\r\n<p id=\"fs-id1169149124152\">\\(x=\\frac{3}{2}\u00b1\\frac{1}{2}i\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169148871025\">\r\n<div data-type=\"problem\" id=\"fs-id1169148871027\">\r\n<p id=\"fs-id1169148871029\">\\(2{x}^{2}+4x=-5\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1169149005906\">\r\n<h4 data-type=\"title\"><a href=\"\/contents\/f045a37e-bf7c-4818-95a1-e29172da48b4\" class=\"target-chapter\">Solve Quadratic Equations Using the Quadratic Formula<\/a><\/h4>\r\n<p id=\"fs-id1169148983922\">In the following exercises, solve by using the Quadratic Formula.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1169148983925\">\r\n<div data-type=\"problem\" id=\"fs-id1169148983927\">\r\n<p id=\"fs-id1169149037735\">\\(4{x}^{2}-5x+1=0\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169148891295\">\r\n<p id=\"fs-id1169148891297\">\\(x=\\frac{1}{4},1\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169148984100\">\r\n<div data-type=\"problem\" id=\"fs-id1169148984102\">\r\n<p id=\"fs-id1169148984104\">\\(7{y}^{2}+4y-3=0\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169146740292\">\r\n<div data-type=\"problem\" id=\"fs-id1169146740294\">\r\n<p id=\"fs-id1169144545760\">\\({r}^{2}-r-42=0\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169149121257\">\r\n<p id=\"fs-id1169149014927\">\\(r=-6,7\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149013093\">\r\n<div data-type=\"problem\" id=\"fs-id1169149121079\">\r\n<p id=\"fs-id1169149121081\">\\({t}^{2}+13t+22=0\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169148968116\">\r\n<div data-type=\"problem\" id=\"fs-id1169148968118\">\r\n<p id=\"fs-id1169148867792\">\\(4{v}^{2}+v-5=0\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169148947416\">\r\n<p id=\"fs-id1169148947418\">\\(v=\\frac{-1\u00b1\\sqrt{21}}{8}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169146651115\">\r\n<div data-type=\"problem\" id=\"fs-id1169146651117\">\r\n<p id=\"fs-id1169146651119\">\\(2{w}^{2}+9w+2=0\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169148938507\">\r\n<div data-type=\"problem\" id=\"fs-id1169148938509\">\r\n<p id=\"fs-id1169148938511\">\\(3{m}^{2}+8m+2=0\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169146654033\">\r\n<p id=\"fs-id1169146654035\">\\(m=\\frac{-4\u00b1\\sqrt{10}}{3}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169148912005\">\r\n<div data-type=\"problem\" id=\"fs-id1169148912007\">\r\n<p id=\"fs-id1169148912009\">\\(5{n}^{2}+2n-1=0\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149114464\">\r\n<div data-type=\"problem\" id=\"fs-id1169149114466\">\r\n<p id=\"fs-id1169149114468\">\\(6{a}^{2}-5a+2=0\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169149156972\">\r\n<p id=\"fs-id1169149156974\">\\(a=\\frac{5}{12}\u00b1\\frac{\\sqrt{23}}{12}i\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149116251\">\r\n<div data-type=\"problem\" id=\"fs-id1169149108716\">\r\n<p id=\"fs-id1169149108718\">\\(4{b}^{2}-b+8=0\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149313134\">\r\n<div data-type=\"problem\" id=\"fs-id1169149220735\">\r\n<p id=\"fs-id1169149220737\">\\(u\\left(u-10\\right)+3=0\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169148926228\">\r\n<p id=\"fs-id1169148926230\">\\(u=5\u00b1\\sqrt{21}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149293737\">\r\n<div data-type=\"problem\" id=\"fs-id1169149293739\">\r\n<p id=\"fs-id1169149293741\">\\(5z\\left(z-2\\right)=3\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169148971298\">\r\n<div data-type=\"problem\" id=\"fs-id1169148971300\">\r\n<p id=\"fs-id1169148971302\">\\(\\frac{1}{8}{p}^{2}-\\frac{1}{5}p=-\\frac{1}{20}\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169149025180\">\r\n<p id=\"fs-id1169149025182\">\\(p=\\frac{4\u00b1\\sqrt{5}}{5}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149308978\">\r\n<div data-type=\"problem\" id=\"fs-id1169149065758\">\r\n<p id=\"fs-id1169149065761\">\\(\\frac{2}{5}{q}^{2}+\\frac{3}{10}q=\\frac{1}{10}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169148870452\">\r\n<div data-type=\"problem\" id=\"fs-id1169148870454\">\r\n<p id=\"fs-id1169148870456\">\\(4{c}^{2}+4c+1=0\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169149154819\">\r\n<p id=\"fs-id1169149154821\">\\(c=-\\frac{1}{2}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169144603954\">\r\n<div data-type=\"problem\" id=\"fs-id1169144603956\">\r\n<p id=\"fs-id1169144603958\">\\(9{d}^{2}-12d=-4\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1169149285662\"><strong data-effect=\"bold\">Use the Discriminant to Predict the Number of Solutions of a Quadratic Equation<\/strong><\/p>\r\n<p id=\"fs-id1169149285668\">In the following exercises, determine the number of solutions for each quadratic equation.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1169146651299\">\r\n<div data-type=\"problem\" id=\"fs-id1169146651301\">\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d0<\/span>\\(9{x}^{2}-6x+1=0\\)\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\\(3{y}^{2}-8y+1=0\\)\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d2<\/span>\\(7{m}^{2}+12m+4=0\\)\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d3<\/span>\\(5{n}^{2}-n+1=0\\)\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169149024795\">\r\n<p id=\"fs-id1169149024797\"><span class=\"token\">\u24d0<\/span> 1 <span class=\"token\">\u24d1<\/span> 2 <span class=\"token\">\u24d2<\/span> 2 <span class=\"token\">\u24d3<\/span> 2<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169146606787\">\r\n<div data-type=\"problem\" id=\"fs-id1169146606789\">\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d0<\/span>\\(5{x}^{2}-7x-8=0\\)\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\\(7{x}^{2}-10x+5=0\\)\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d2<\/span>\\(25{x}^{2}-90x+81=0\\)\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d3<\/span>\\(15{x}^{2}-8x+4=0\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1169149349068\"><strong data-effect=\"bold\">Identify the Most Appropriate Method to Use to Solve a Quadratic Equation<\/strong><\/p>\r\n<p id=\"fs-id1169149109241\">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>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1169149109246\">\r\n<div data-type=\"problem\" id=\"fs-id1169144365602\">\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d0<\/span>\\(16{r}^{2}-8r+1=0\\)\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\\(5{t}^{2}-8t+3=9\\)\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d2<\/span>\\(3{\\left(c+2\\right)}^{2}=15\\)\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169149027651\">\r\n<p id=\"fs-id1169149027654\"><span class=\"token\">\u24d0<\/span> factor <span class=\"token\">\u24d1<\/span> Quadratic Formula <span class=\"token\">\u24d2<\/span> square root<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149302439\">\r\n<div data-type=\"problem\" id=\"fs-id1169149302441\">\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d0<\/span>\\(4{d}^{2}+10d-5=21\\)\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\\(25{x}^{2}-60x+36=0\\)\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d2<\/span>\\(6{\\left(5v-7\\right)}^{2}=150\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1169149109187\">\r\n<h4 data-type=\"title\"><a href=\"\/contents\/48820a47-b89e-428f-9534-a71207245a16\" class=\"target-chapter\">Solve Equations in Quadratic Form<\/a><\/h4>\r\n<p id=\"fs-id1169149003863\"><strong data-effect=\"bold\">Solve Equations in Quadratic Form<\/strong><\/p>\r\n<p id=\"fs-id1169149370495\">In the following exercises, solve.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1169149370498\">\r\n<div data-type=\"problem\" id=\"fs-id1169149370500\">\r\n<p id=\"fs-id1169149370290\">\\({x}^{4}-14{x}^{2}+24=0\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169149373720\">\r\n<p id=\"fs-id1169149373722\">\\(x=\u00b1\\sqrt{2},\\phantom{\\rule{0.2em}{0ex}}\\text{}\\phantom{\\rule{0.2em}{0ex}}x=\u00b12\\sqrt{3}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169146661162\">\r\n<div data-type=\"problem\" id=\"fs-id1169146661722\">\r\n<p id=\"fs-id1169146661724\">\\({x}^{4}+4{x}^{2}-32=0\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169146656250\">\r\n<div data-type=\"problem\" id=\"fs-id1169146656252\">\r\n<p id=\"fs-id1169146656254\">\\(4{x}^{4}-5{x}^{2}+1=0\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169146657125\">\r\n<p id=\"fs-id1169146657127\">\\(x=\u00b11,\\phantom{\\rule{0.2em}{0ex}}\\text{}\\phantom{\\rule{0.2em}{0ex}}x=\u00b1\\frac{1}{2}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149376033\">\r\n<div data-type=\"problem\" id=\"fs-id1169149376035\">\r\n<p id=\"fs-id1169149375166\">\\({\\left(2y+3\\right)}^{2}+3\\left(2y+3\\right)-28=0\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169146637673\">\r\n<div data-type=\"problem\" id=\"fs-id1169146637675\">\r\n<p id=\"fs-id1169146637677\">\\(x+3\\sqrt{x}-28=0\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169146637854\">\r\n<p id=\"fs-id1169146637856\">\\(x=16\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169146643865\">\r\n<div data-type=\"problem\" id=\"fs-id1169146643867\">\r\n<p id=\"fs-id1169146643869\">\\(6x+5\\sqrt{x}-6=0\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169146644998\">\r\n<div data-type=\"problem\" id=\"fs-id1169146645048\">\r\n<p id=\"fs-id1169146645050\">\\({x}^{\\frac{2}{3}}-10{x}^{\\frac{1}{3}}+24=0\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169148934926\">\r\n<p id=\"fs-id1169146652899\">\\(x=64,x=216\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149367711\">\r\n<div data-type=\"problem\" id=\"fs-id1169149367713\">\r\n<p id=\"fs-id1169149367688\">\\(x+7{x}^{\\frac{1}{2}}+6=0\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149376147\">\r\n<div data-type=\"problem\" id=\"fs-id1169149376149\">\r\n<p id=\"fs-id1169149376151\">\\(8{x}^{-2}-2{x}^{-1}-3=0\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169149003331\">\r\n<p id=\"fs-id1169149330215\">\\(x=-2,x=\\frac{4}{3}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1169149009036\">\r\n<h4 data-type=\"title\"><a href=\"\/contents\/b6bba63f-4f97-49ab-ac8a-01c3927118a7\" class=\"target-chapter\">Solve Applications of Quadratic Equations<\/a><\/h4>\r\n<p id=\"fs-id1169146731708\"><strong data-effect=\"bold\">Solve Applications Modeled by Quadratic Equations<\/strong><\/p>\r\n<p id=\"fs-id1169146731714\">In the following exercises, solve by using the method of factoring, the square root principle, or the Quadratic Formula. Round your answers to the nearest tenth, if needed.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1169149349822\">\r\n<div data-type=\"problem\" id=\"fs-id1169149349824\">\r\n<p id=\"fs-id1169149349826\">Find two consecutive odd numbers whose product is 323.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149335626\">\r\n<div data-type=\"problem\" id=\"fs-id1169149335628\">\r\n<p id=\"fs-id1169149335630\">Find two consecutive even numbers whose product is 624.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169149156873\">\r\n<p id=\"fs-id1169149156875\">Two consecutive even numbers whose product is 624 are 24 and 26, and \u221224 and \u221226.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149094500\">\r\n<div data-type=\"problem\" id=\"fs-id1169149094502\">\r\n<p id=\"fs-id1169149094504\">A triangular banner has an area of 351 square centimeters. The length of the base is two centimeters longer than four times the height. Find the height and length of the base.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149123561\">\r\n<div data-type=\"problem\" id=\"fs-id1169149123563\">\r\n<p id=\"fs-id1169149123565\">Julius built a triangular display case for his coin collection. The height of the display case is six inches less than twice the width of the base. The area of the of the back of the case is 70 square inches. Find the height and width of the case.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169148895521\">\r\n<p id=\"fs-id1169148895523\">The height is 14 inches and the width is 10 inches.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169148989338\">\r\n<div data-type=\"problem\" id=\"fs-id1169148989340\">\r\n<p id=\"fs-id1169148989342\">A tile mosaic in the shape of a right triangle is used as the corner of a rectangular pathway. The hypotenuse of the mosaic is 5 feet. One side of the mosaic is twice as long as the other side. What are the lengths of the sides? Round to the nearest tenth.<\/p>\r\n<span data-type=\"media\" id=\"fs-id1169147029064\" data-alt=\"A rectangle is shown is a right triangle in the corner. The hypotenuse of the triangle is 5 feet, the longer leg is 2 times s and the shorter leg is s.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_202_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"A rectangle is shown is a right triangle in the corner. The hypotenuse of the triangle is 5 feet, the longer leg is 2 times s and the shorter leg is s.\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149370246\">\r\n<div data-type=\"problem\" id=\"fs-id1169149370248\">\r\n<p id=\"fs-id1169148962548\">A rectangular piece of plywood has a diagonal which measures two feet more than the width. The length of the plywood is twice the width. What is the length of the plywood\u2019s diagonal? Round to the nearest tenth.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169149152558\">\r\n<p id=\"fs-id1169149152561\">The length of the diagonal is 3.6 feet.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149152566\">\r\n<div data-type=\"problem\" id=\"fs-id1169149116008\">\r\n<p id=\"fs-id1169149116010\">The front walk from the street to Pam\u2019s house has an area of 250 square feet. Its length is two less than four times its width. Find the length and width of the sidewalk. Round to the nearest tenth.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149338846\">\r\n<div data-type=\"problem\" id=\"fs-id1169149338848\">\r\n<p id=\"fs-id1169149338850\">For Sophia\u2019s graduation party, several tables of the same width will be arranged end to end to give serving table with a total area of 75 square feet. The total length of the tables will be two more than three times the width. Find the length and width of the serving table so Sophia can purchase the correct size tablecloth . Round answer to the nearest tenth.<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169149087629\">\r\n<p id=\"fs-id1169149087631\">The width of the serving table is 4.7 feet and the length is 16.1 feet.<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1169149087636\" data-alt=\"Four tables arranged end-to-end are shown. Together, they have an area of 75 feet. The short side measures w and the long side measures 3 times w plus 2.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_203_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Four tables arranged end-to-end are shown. Together, they have an area of 75 feet. The short side measures w and the long side measures 3 times w plus 2.\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149017955\">\r\n<div data-type=\"problem\" id=\"fs-id1169149017957\">\r\n<p id=\"fs-id1169149017959\">A ball is thrown vertically in the air with a velocity of 160 ft\/sec. Use the formula <em data-effect=\"italics\">h<\/em> = \u221216<em data-effect=\"italics\">t<\/em><sup>2<\/sup> + <em data-effect=\"italics\">v<\/em><sub>0<\/sub><em data-effect=\"italics\">t<\/em> to determine when the ball will be 384 feet from the ground. Round to the nearest tenth.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169148927292\">\r\n<div data-type=\"problem\" id=\"fs-id1169148927295\">\r\n<p id=\"fs-id1169148927297\">The couple took a small airplane for a quick flight up to the wine country for a romantic dinner and then returned home. The plane flew a total of 5 hours and each way the trip was 360 miles. If the plane was flying at 150 mph, what was the speed of the wind that affected the plane?<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169149011762\">\r\n<p id=\"fs-id1169149011764\">The speed of the wind was 30 mph.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149004012\">\r\n<div data-type=\"problem\" id=\"fs-id1169149004014\">\r\n<p id=\"fs-id1169149004016\">Ezra kayaked up the river and then back in a total time of 6 hours. The trip was 4 miles each way and the current was difficult. If Roy kayaked at a speed of 5 mph, what was the speed of the current?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149033290\">\r\n<div data-type=\"problem\" id=\"fs-id1169149033292\">\r\n<p id=\"fs-id1169149033294\">Two handymen can do a home repair in 2 hours if they work together. One of the men takes 3 hours more than the other man to finish the job by himself. How long does it take for each handyman to do the home repair individually?<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169144382246\">\r\n\r\nOne man takes 3 hours and the other man 6 hours to finish the repair alone.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1169148969732\">\r\n<h4 data-type=\"title\"><a href=\"\/contents\/3de1ae34-6225-4751-be75-a17b3e0e665b\" class=\"target-chapter\">Graph Quadratic Functions Using Properties<\/a><\/h4>\r\n<p id=\"fs-id1169148929691\"><strong data-effect=\"bold\">Recognize the Graph of a Quadratic Function<\/strong><\/p>\r\n<p id=\"fs-id1169148929696\">In the following exercises, graph by plotting point.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1169144551404\">\r\n<div data-type=\"problem\" id=\"fs-id1169144551406\">\r\n<p id=\"fs-id1169144551408\">Graph \\(y={x}^{2}-2\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169148991591\">\r\n<div data-type=\"problem\" id=\"fs-id1169148991593\">\r\n<p id=\"fs-id1169148991595\">Graph \\(y=\\text{\u2212}{x}^{2}+3\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169149089320\">\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1169149089323\" data-alt=\"This figure shows a downward-opening parabola on the x y-coordinate plane. It has a vertex of (3, 0) and other points of (negative 2, negative 1) and (2, negative 1).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_314_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a downward-opening parabola on the x y-coordinate plane. It has a vertex of (3, 0) and other points of (negative 2, negative 1) and (2, negative 1).\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1169149065516\">In the following exercises, determine if the following parabolas open up or down.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1169149065519\">\r\n<div data-type=\"problem\" id=\"fs-id1169146655769\">\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d0<\/span>\\(y=-3{x}^{2}+3x-1\\)\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\\(y=5{x}^{2}+6x+3\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149230086\">\r\n<div data-type=\"problem\" id=\"fs-id1169149230089\">\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d0<\/span>\\(y={x}^{2}+8x-1\\)\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\\(y=-4{x}^{2}-7x+1\\)\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169149095454\">\r\n<p id=\"fs-id1169149095457\"><span class=\"token\">\u24d0<\/span> up <span class=\"token\">\u24d1<\/span> down<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1169149095361\"><strong data-effect=\"bold\">Find the Axis of Symmetry and Vertex of a Parabola<\/strong><\/p>\r\n<p id=\"fs-id1169148994045\">In the following exercises, find <span class=\"token\">\u24d0<\/span> the equation of the axis of symmetry and <span class=\"token\">\u24d1<\/span> the vertex.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1169148955071\">\r\n<div data-type=\"problem\" id=\"fs-id1169148955073\">\r\n<p id=\"fs-id1169148955075\">\\(y=\\text{\u2212}{x}^{2}+6x+8\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149285597\">\r\n<div data-type=\"problem\" id=\"fs-id1169149369843\">\r\n<p id=\"fs-id1169149369845\">\\(y=2{x}^{2}-8x+1\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169149211429\">\r\n<p id=\"fs-id1169148939472\">\\(x=2;\\left(2,-7\\right)\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1169146633836\"><strong data-effect=\"bold\">Find the Intercepts of a Parabola<\/strong><\/p>\r\n<p id=\"fs-id1169144603974\">In the following exercises, find the <em data-effect=\"italics\">x<\/em>- and <em data-effect=\"italics\">y<\/em>-intercepts.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1169149007215\">\r\n<div data-type=\"problem\" id=\"fs-id1169149007217\">\r\n<p id=\"fs-id1169149007219\">\\(y={x}^{2}-4x+5\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169148869070\">\r\n<div data-type=\"problem\" id=\"fs-id1169148869072\">\r\n<p id=\"fs-id1169148869075\">\\(y={x}^{2}-8x+15\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169149341584\">\r\n<p id=\"fs-id1169146617675\">\\(\\begin{array}{c}y:\\left(0,15\\right)\\hfill \\\\ x:\\left(3,0\\right),\\left(5,0\\right)\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169148959755\">\r\n<div data-type=\"problem\" id=\"fs-id1169148959757\">\r\n<p id=\"fs-id1169148984708\">\\(y={x}^{2}-4x+10\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149038744\">\r\n<div data-type=\"problem\" id=\"fs-id1169149038746\">\r\n<p id=\"fs-id1169149038748\">\\(y=-5{x}^{2}-30x-46\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169149025866\">\r\n<p id=\"fs-id1169149025868\">\\(\\begin{array}{c}y:\\left(0,-46\\right)\\hfill \\\\ x:\\text{none}\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169148963942\">\r\n<div data-type=\"problem\" id=\"fs-id1169148963944\">\r\n<p id=\"fs-id1169148963946\">\\(y=16{x}^{2}-8x+1\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149285285\">\r\n<div data-type=\"problem\" id=\"fs-id1169149285288\">\r\n<p id=\"fs-id1169149285290\">\\(y={x}^{2}+16x+64\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169149109297\">\r\n<p id=\"fs-id1169149109299\">\\(\\begin{array}{c}y:\\left(0,-64\\right)\\hfill \\\\ x:\\left(-8,0\\right)\\hfill \\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1169148963400\"><strong data-effect=\"bold\">Graph Quadratic Functions Using Properties<\/strong><\/p>\r\n<p id=\"fs-id1169148924491\">In the following exercises, graph by using its properties.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1169148924494\">\r\n<div data-type=\"problem\" id=\"fs-id1169148924496\">\r\n<p id=\"fs-id1169148924498\">\\(y={x}^{2}+8x+15\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149370164\">\r\n<div data-type=\"problem\" id=\"fs-id1169149370167\">\r\n<p id=\"fs-id1169149370169\">\\(y={x}^{2}-2x-3\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169146657085\">\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1169146657090\" data-alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (1, negative 4) and a y-intercept of (0, negative 3).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_316_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (1, negative 4) and a y-intercept of (0, negative 3).\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149024631\">\r\n<div data-type=\"problem\" id=\"fs-id1169149024633\">\r\n<p id=\"fs-id1169144564487\">\\(y=\\text{\u2212}{x}^{2}+8x-16\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149309999\">\r\n<div data-type=\"problem\" id=\"fs-id1169149310001\">\r\n<p id=\"fs-id1169149310003\">\\(y=4{x}^{2}-4x+1\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169148995767\">\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1169148995770\" data-alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (one-half, 0) and a y-intercept of (0, 1).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_318_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (one-half, 0) and a y-intercept of (0, 1).\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149349320\">\r\n<div data-type=\"problem\" id=\"fs-id1169149349322\">\r\n<p id=\"fs-id1169149349324\">\\(y={x}^{2}+6x+13\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169148996221\">\r\n<div data-type=\"problem\" id=\"fs-id1169148996223\">\r\n<p id=\"fs-id1169148996225\">\\(y=-2{x}^{2}-8x-12\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169146655944\">\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1169146655947\" data-alt=\"This figure shows a downward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 2, negative 4) and a y-intercept of (0, negative 12).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_320_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a downward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 2, negative 4) and a y-intercept of (0, negative 12).\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1169149291277\"><strong data-effect=\"bold\">Solve Maximum and Minimum Applications<\/strong><\/p>\r\n<p id=\"fs-id1169149039976\">In the following exercises, find the minimum or maximum value.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1169149039979\">\r\n<div data-type=\"problem\" id=\"fs-id1169149039981\">\r\n<p id=\"fs-id1169149039983\">\\(y=7{x}^{2}+14x+6\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149029750\">\r\n<div data-type=\"problem\" id=\"fs-id1169149029752\">\r\n<p id=\"fs-id1169146660811\">\\(y=-3{x}^{2}+12x-10\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169147028313\">\r\n<p id=\"fs-id1169147028316\">The maximum value is 2 when <em data-effect=\"italics\">x<\/em> = 2.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1169148957096\">In the following exercises, solve. Rounding answers to the nearest tenth.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1169149336292\">\r\n<div data-type=\"problem\" id=\"fs-id1169149336294\">\r\n<p id=\"fs-id1169149336296\">A ball is thrown upward from the ground with an initial velocity of 112 ft\/sec. Use the quadratic equation <em data-effect=\"italics\">h<\/em> = \u221216<em data-effect=\"italics\">t<\/em><sup>2<\/sup> + 112<em data-effect=\"italics\">t<\/em> to find how long it will take the ball to reach maximum height, and then find the maximum height.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169148972750\">\r\n<div data-type=\"problem\" id=\"fs-id1169148972752\">\r\n<p id=\"fs-id1169148969278\">A daycare facility is enclosing a rectangular area along the side of their building for the children to play outdoors. They need to maximize the area using 180 feet of fencing on three sides of the yard. The quadratic equation <em data-effect=\"italics\">A<\/em> = \u22122<em data-effect=\"italics\">x<\/em><sup>2<\/sup> + 180<em data-effect=\"italics\">x<\/em> gives the area, <em data-effect=\"italics\">A<\/em>, of the yard for the length, <em data-effect=\"italics\">x<\/em>, of the building that will border the yard. Find the length of the building that should border the yard to maximize the area, and then find the maximum area.<\/p>\r\n<span data-type=\"media\" id=\"fs-id1169149292285\" data-alt=\"An odd-shaped figure is given. 3 sides of a rectangle are attached to the right side of the figure.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_204_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"An odd-shaped figure is given. 3 sides of a rectangle are attached to the right side of the figure.\" \/><\/span>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169149114447\">\r\n<p id=\"fs-id1169149114449\">The length adjacent to the building is 90 feet giving a maximum area of 4,050 square feet.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1169144376789\">\r\n<h4 data-type=\"title\"><a href=\"\/contents\/9054900c-f191-40ed-8ceb-e4f051bbbf2b\" class=\"target-chapter\">Graph Quadratic Functions Using Transformations<\/a><\/h4>\r\n<p id=\"fs-id1169148909034\"><strong data-effect=\"bold\">Graph Quadratic Functions of the form \\(f\\left(x\\right)={x}^{2}+k\\)<\/strong><\/p>\r\n<p id=\"fs-id1169149030734\">In the following exercises, graph each function using a vertical shift.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1169149030738\">\r\n<div data-type=\"problem\" id=\"fs-id1169149030740\">\r\n<p id=\"fs-id1169149214960\">\\(g\\left(x\\right)={x}^{2}+4\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169146740423\">\r\n<div data-type=\"problem\" id=\"fs-id1169146744005\">\r\n<p id=\"fs-id1169146744007\">\\(h\\left(x\\right)={x}^{2}-3\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169148939925\">\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1169148939929\" data-alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 3, 0) and other points of (negative 1, negative 2) and (1, negative 2).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_322_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 3, 0) and other points of (negative 1, negative 2) and (1, negative 2).\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1169146744134\">In the following exercises, graph each function using a horizontal shift.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1169146744137\">\r\n<div data-type=\"problem\" id=\"fs-id1169146744139\">\r\n<p id=\"fs-id1169146744141\">\\(f\\left(x\\right)={\\left(x+1\\right)}^{2}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169147085150\">\r\n<div data-type=\"problem\" id=\"fs-id1169147085152\">\r\n<p id=\"fs-id1169147085155\">\\(g\\left(x\\right)={\\left(x-3\\right)}^{2}\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169149230329\">\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1169148923644\" data-alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (3, 0) and other points of (2, 1) and (4,1).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_324_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (3, 0) and other points of (2, 1) and (4,1).\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1169149345843\">In the following exercises, graph each function using transformations.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1169149345846\">\r\n<div data-type=\"problem\" id=\"fs-id1169149345848\">\r\n<p id=\"fs-id1169149345850\">\\(f\\left(x\\right)={\\left(x+2\\right)}^{2}+3\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149114367\">\r\n<div data-type=\"problem\" id=\"fs-id1169149114369\">\r\n<p id=\"fs-id1169149114372\">\\(f\\left(x\\right)={\\left(x+3\\right)}^{2}-2\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169149220778\">\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1169146740834\" data-alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 3, negative 2) and other points of (negative 5, 2) and (negative 1, 2).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_326_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 3, negative 2) and other points of (negative 5, 2) and (negative 1, 2).\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169148843323\">\r\n<div data-type=\"problem\" id=\"fs-id1169148843325\">\r\n<p id=\"fs-id1169148843327\">\\(f\\left(x\\right)={\\left(x-1\\right)}^{2}+4\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149043690\">\r\n<div data-type=\"problem\" id=\"fs-id1169149043692\">\r\n<p id=\"fs-id1169149043694\">\\(f\\left(x\\right)={\\left(x-4\\right)}^{2}-3\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169148937884\">\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1169149222724\" data-alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (4, negative 3) and other points of (3, negative 2) and (5, negative 2).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_328_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (4, negative 3) and other points of (3, negative 2) and (5, negative 2).\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1169144376807\"><strong data-effect=\"bold\">Graph Quadratic Functions of the form \\(f\\left(x\\right)=a{x}^{2}\\)<\/strong><\/p>\r\n<p id=\"fs-id1169148972924\">In the following exercises, graph each function.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1169148972927\">\r\n<div data-type=\"problem\" id=\"fs-id1169149032750\">\r\n<p id=\"fs-id1169149032752\">\\(f\\left(x\\right)=2{x}^{2}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169147107658\">\r\n<div data-type=\"problem\" id=\"fs-id1169148963022\">\r\n<p id=\"fs-id1169148963024\">\\(f\\left(x\\right)=\\text{\u2212}{x}^{2}\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169147028990\">\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1169147028995\" data-alt=\"This figure shows a downward-opening parabola on the x y-coordinate plane. It has a vertex of (0, 0) and other points of (negative 1, negative 1) and (1, negative 1).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_330_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a downward-opening parabola on the x y-coordinate plane. It has a vertex of (0, 0) and other points of (negative 1, negative 1) and (1, negative 1).\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169146744334\">\r\n<div data-type=\"problem\" id=\"fs-id1169147028998\">\r\n<p id=\"fs-id1169149304791\">\\(f\\left(x\\right)=\\frac{1}{2}{x}^{2}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1169149287025\"><strong data-effect=\"bold\">Graph Quadratic Functions Using Transformations<\/strong><\/p>\r\n<p id=\"fs-id1169149172194\">In the following exercises, rewrite each function in the \\(f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k\\) form by completing the square.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1169148969375\">\r\n<div data-type=\"problem\" id=\"fs-id1169148969377\">\r\n<p id=\"fs-id1169148875162\">\\(f\\left(x\\right)=2{x}^{2}-4x-4\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169149038778\">\r\n<p id=\"fs-id1169149038780\">\\(f\\left(x\\right)=2{\\left(x-1\\right)}^{2}-6\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169148990578\">\r\n<div data-type=\"problem\" id=\"fs-id1169148990580\">\r\n<p id=\"fs-id1169148990582\">\\(f\\left(x\\right)=3{x}^{2}+12x+8\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1169149110351\">In the following exercises, <span class=\"token\">\u24d0<\/span> rewrite each function in \\(f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k\\) form and <span class=\"token\">\u24d1<\/span> graph it by using transformations.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1169148994595\">\r\n<div data-type=\"problem\" id=\"fs-id1169148994597\">\r\n<p id=\"fs-id1169146815184\">\\(f\\left(x\\right)=3{x}^{2}-6x-1\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169149155501\">\r\n<p id=\"fs-id1169146631384\"><span class=\"token\">\u24d0<\/span>\\(f\\left(x\\right)=3{\\left(x-1\\right)}^{2}-4\\)<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1169148965135\" data-alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (1, negative 4) and other points of (0, negative 1) and (2, negative 1).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_332_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (1, negative 4) and other points of (0, negative 1) and (2, negative 1).\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169148987024\">\r\n<div data-type=\"problem\" id=\"fs-id1169148987026\">\r\n<p id=\"fs-id1169148987028\">\\(f\\left(x\\right)=-2{x}^{2}-12x-5\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149346707\">\r\n<div data-type=\"problem\" id=\"fs-id1169149346709\">\r\n<p id=\"fs-id1169149346711\">\\(f\\left(x\\right)=2{x}^{2}+4x+6\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169149341018\">\r\n<p id=\"fs-id1169149341020\"><span class=\"token\">\u24d0<\/span>\\(f\\left(x\\right)=2{\\left(x+1\\right)}^{2}+4\\)<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1169149293605\" data-alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 1, 4) and other points of (negative 2, 6) and (0, 6).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_334_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 1, 4) and other points of (negative 2, 6) and (0, 6).\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169148947324\">\r\n<div data-type=\"problem\" id=\"fs-id1169148947326\">\r\n<p id=\"fs-id1169148947329\">\\(f\\left(x\\right)=3{x}^{2}-12x+7\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1169149195662\">In the following exercises, <span class=\"token\">\u24d0<\/span> rewrite each function in \\(f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k\\) form and <span class=\"token\">\u24d1<\/span> graph it using properties.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1169149220010\">\r\n<div data-type=\"problem\" id=\"fs-id1169149220012\">\r\n<p id=\"fs-id1169149220014\">\\(f\\left(x\\right)=-3{x}^{2}-12x-5\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169149012893\">\r\n<p id=\"fs-id1169149012896\"><span class=\"token\">\u24d0<\/span>\\(f\\left(x\\right)=-3{\\left(x+2\\right)}^{2}+7\\)<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1169149001060\" data-alt=\"This figure shows a downward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 2, 7) and other points of (negative 4, negative 5) and (0, negative 5).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_336_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a downward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 2, 7) and other points of (negative 4, negative 5) and (0, negative 5).\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149015898\">\r\n<div data-type=\"problem\" id=\"fs-id1169149015900\">\r\n<p id=\"fs-id1169149000933\">\\(f\\left(x\\right)=2{x}^{2}-12x+7\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1169149104832\"><strong data-effect=\"bold\">Find a Quadratic Function from its Graph<\/strong><\/p>\r\n<p id=\"fs-id1171791302843\">In the following exercises, write the quadratic function in \\(f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k\\) form.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1169148911498\">\r\n<div data-type=\"problem\" id=\"fs-id1169149007350\"><span data-type=\"media\" id=\"fs-id1169149007352\" data-alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 1, negative 1) and other points of (negative 2, negative 4) and (0, negative 4).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_205_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 1, negative 1) and other points of (negative 2, negative 4) and (0, negative 4).\" \/><\/span><\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169144566540\">\r\n<p id=\"fs-id1169144566542\">\\(f\\left(x\\right)={\\left(x+1\\right)}^{2}-5\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169148938157\">\r\n<div data-type=\"problem\" id=\"fs-id1169148938159\"><span data-type=\"media\" id=\"fs-id1169148938161\" data-alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (2, 4) and other points of (0, 8) and (4, 8).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_206_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (2, 4) and other points of (0, 8) and (4, 8).\" \/><\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1169148924238\">\r\n<h4 data-type=\"title\"><a href=\"\/contents\/4f54d544-fb3e-4336-9f8c-57e6b8489257\" class=\"target-chapter\">Solve Quadratic Inequalities<\/a><\/h4>\r\n<p id=\"fs-id1169144565221\"><strong data-effect=\"bold\">Solve Quadratic Inequalities Graphically<\/strong><\/p>\r\n<p id=\"fs-id1169144565227\">In the following exercises, solve graphically and write the solution in interval notation.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1169144565232\">\r\n<div data-type=\"problem\" id=\"fs-id1169149008556\">\r\n<p id=\"fs-id1169149008559\">\\({x}^{2}-x-6&gt;0\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169149107840\">\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d0<\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1169149107850\" data-alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (one-half, negative 6 and one-fourth) and other points of (0, negative 6) and (1, negative 6).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_338_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (one-half, negative 6 and one-fourth) and other points of (0, negative 6) and (1, negative 6).\" \/><\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d1<\/span>\\(\\left(-\\infty ,-2\\right)\\cup \\left(3,\\infty \\right)\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149115800\">\r\n<div data-type=\"problem\" id=\"fs-id1169149115802\">\r\n<p id=\"fs-id1169149115804\">\\({x}^{2}+4x+3\\le 0\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149100215\">\r\n<div data-type=\"problem\" id=\"fs-id1169149100218\">\r\n<p id=\"fs-id1169149100220\">\\(\\text{\u2212}{x}^{2}-x+2\\ge 0\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169148952525\">\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d0<\/span>\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1169149008950\" data-alt=\"This figure shows a downward-opening parabola on the x y-coordinate plane. It has a vertex of (negative one-half, 2 and one-fourth) and other points of (negative 2, 0) and (1, 0).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_340_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a downward-opening parabola on the x y-coordinate plane. It has a vertex of (negative one-half, 2 and one-fourth) and other points of (negative 2, 0) and (1, 0).\" \/><\/span>\r\n<div data-type=\"newline\"><\/div>\r\n\\(\\left[-2,1\\right]\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149003165\">\r\n<div data-type=\"problem\" id=\"fs-id1169149003167\">\r\n<p id=\"fs-id1169149006401\">\\(\\text{\u2212}{x}^{2}+2x+3&lt;0\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1169149042015\">In the following exercises, solve each inequality algebraically and write any solution in interval notation.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1169149042019\">\r\n<div data-type=\"problem\" id=\"fs-id1169149042022\">\r\n<p id=\"fs-id1169149042024\">\\({x}^{2}-6x+8&lt;0\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169146659084\">\r\n<p id=\"fs-id1169146669955\">\\(\\left(2,4\\right)\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149015035\">\r\n<div data-type=\"problem\" id=\"fs-id1169149015037\">\r\n<p id=\"fs-id1169149015040\">\\({x}^{2}+x&gt;12\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169148962536\">\r\n<div data-type=\"problem\" id=\"fs-id1169148962538\">\r\n<p id=\"fs-id1169148962540\">\\({x}^{2}-6x+4\\le 0\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169149096065\">\r\n<p id=\"fs-id1169149096067\">\\(\\left[3-\\sqrt{5},3+\\sqrt{5}\\right]\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149112768\">\r\n<div data-type=\"problem\" id=\"fs-id1169149112770\">\r\n<p id=\"fs-id1169149112772\">\\(2{x}^{2}+7x-4&gt;0\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149028228\">\r\n<div data-type=\"problem\" id=\"fs-id1169149033349\">\r\n<p id=\"fs-id1169149033351\">\\(\\text{\u2212}{x}^{2}+x-6&gt;0\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169149144392\">\r\n<p id=\"fs-id1169149144394\">no solution<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149144400\">\r\n<div data-type=\"problem\" id=\"fs-id1169148962968\">\r\n<p id=\"fs-id1169148962970\">\\({x}^{2}-2x+4\\ge 0\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"practice-test\" data-depth=\"1\" id=\"fs-id1169149012734\">\r\n<h3 data-type=\"title\">Practice Test<\/h3>\r\n<div data-type=\"exercise\" id=\"fs-id1169149012741\">\r\n<div data-type=\"problem\" id=\"fs-id1169146742654\">\r\n<p id=\"fs-id1169146742656\">Use the Square Root Property to solve the quadratic equation \\(3{\\left(w+5\\right)}^{2}=27.\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169149143992\">\r\n<p id=\"fs-id1169149143994\">\\(w=-2,w=-8\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169144565579\">\r\n<div data-type=\"problem\" id=\"fs-id1169144565582\">\r\n<p id=\"fs-id1169144565584\">Use Completing the Square to solve the quadratic equation \\({a}^{2}-8a+7=23.\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169148879638\">\r\n<div data-type=\"problem\" id=\"fs-id1169149094530\">\r\n<p id=\"fs-id1169149094532\">Use the Quadratic Formula to solve the quadratic equation \\(2{m}^{2}-5m+3=0.\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169146741118\">\r\n<p id=\"fs-id1169146741120\">\\(m=1,m=\\frac{3}{2}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1169149279941\">Solve the following quadratic equations. Use any method.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1169149295358\">\r\n<div data-type=\"problem\" id=\"fs-id1169149295360\">\r\n<p id=\"fs-id1169149295362\">\\(2x\\left(3x-2\\right)-1=0\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149121563\">\r\n<div data-type=\"problem\" id=\"fs-id1169149121566\">\r\n<p id=\"fs-id1169149121568\">\\(\\frac{9}{4}{y}^{2}-3y+1=0\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169149311826\">\r\n<p id=\"fs-id1169149156327\">\\(y=\\frac{2}{3}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1169149029934\">Use the discriminant to determine the number and type of solutions of each quadratic equation.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1169149029938\">\r\n<div data-type=\"problem\" id=\"fs-id1169149029940\">\r\n<p id=\"fs-id1169149122254\">\\(6{p}^{2}-13p+7=0\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149095062\">\r\n<div data-type=\"problem\" id=\"fs-id1169149095064\">\r\n<p id=\"fs-id1169149345203\">\\(3{q}^{2}-10q+12=0\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169148958290\">\r\n<p id=\"fs-id1169148958292\">2 complex<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1169148958297\">Solve each equation.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1169149295913\">\r\n<div data-type=\"problem\" id=\"fs-id1169149295915\">\r\n<p id=\"fs-id1169149295917\">\\(4{x}^{4}-17{x}^{2}+4=0\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169148879788\">\r\n<div data-type=\"problem\" id=\"fs-id1169148879790\">\r\n<p id=\"fs-id1169148879792\">\\({y}^{\\frac{2}{3}}+2{y}^{\\frac{1}{3}}-3=0\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169148870657\">\r\n<p id=\"fs-id1169148870659\">\\(y=1,y=-27\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1169148995102\">For each parabola, find <span class=\"token\">\u24d0<\/span> which direction it opens, <span class=\"token\">\u24d1<\/span> the equation of the axis of symmetry, <span class=\"token\">\u24d2<\/span> the vertex, <span class=\"token\">\u24d3<\/span> the <em data-effect=\"italics\">x-<\/em> and <em data-effect=\"italics\">y<\/em>-intercepts, and e) the maximum or minimum value.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1169146669108\">\r\n<div data-type=\"problem\" id=\"fs-id1169146669110\">\r\n<p id=\"fs-id1169146669112\">\\(y=3{x}^{2}+6x+8\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169148967539\">\r\n<div data-type=\"problem\" id=\"fs-id1169148967542\">\r\n<p id=\"fs-id1169148967544\">\\(y=\\text{\u2212}{x}^{2}-8x+16\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169148908757\">\r\n<p id=\"fs-id1169149042133\"><span class=\"token\">\u24d0<\/span> down <span class=\"token\">\u24d1<\/span> \\(x=-4\\)<\/p>\r\n\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d2<\/span> \\(\\left(-4,0\\right)\\) <span class=\"token\">\u24d3<\/span> \\(y\\text{:}\\phantom{\\rule{0.2em}{0ex}}\\left(0,16\\right)\\text{;}\\phantom{\\rule{0.2em}{0ex}}x\\text{:}\\phantom{\\rule{0.2em}{0ex}}\\left(-4,0\\right)\\)\r\n<div data-type=\"newline\"><\/div>\r\n<span class=\"token\">\u24d4<\/span> minimum value of \\(-4\\) when \\(x=0\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1169149140297\">Graph each quadratic function using intercepts, the vertex, and the equation of the axis of symmetry.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1169149140300\">\r\n<div data-type=\"problem\" id=\"fs-id1169149140302\">\r\n<p id=\"fs-id1169149140304\">\\(f\\left(x\\right)={x}^{2}+6x+9\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149115844\">\r\n<div data-type=\"problem\" id=\"fs-id1169149103441\">\r\n<p id=\"fs-id1169149103443\">\\(f\\left(x\\right)=-2{x}^{2}+8x+4\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169147029418\"><span data-type=\"media\" id=\"fs-id1169149033999\" data-alt=\"This figure shows a downward-opening parabola on the x y-coordinate plane. It has a vertex of (2, 12) and other points of (0, 4) and (4, 4).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_343_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a downward-opening parabola on the x y-coordinate plane. It has a vertex of (2, 12) and other points of (0, 4) and (4, 4).\" \/><\/span><\/div>\r\n<\/div>\r\n<p id=\"fs-id1169149013976\">In the following exercises, graph each function using transformations.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1169149013979\">\r\n<div data-type=\"problem\" id=\"fs-id1169149013982\">\r\n<p id=\"fs-id1169149013984\">\\(f\\left(x\\right)={\\left(x+3\\right)}^{2}+2\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149306709\">\r\n<div data-type=\"problem\" id=\"fs-id1169149306711\">\r\n<p id=\"fs-id1169149306714\">\\(f\\left(x\\right)={x}^{2}-4x-1\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169148951771\">\r\n<div data-type=\"newline\"><\/div>\r\n<span data-type=\"media\" id=\"fs-id1169149009270\" data-alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (2, negative 5) and other points of (0, negative 1) and (4, negative 1).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_345_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (2, negative 5) and other points of (0, negative 1) and (4, negative 1).\" \/><\/span>\r\n<div data-type=\"newline\"><\/div>\r\n\\(f\\left(x\\right)=2{\\left(x-1\\right)}^{2}-6\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1169148925023\">In the following exercises, solve each inequality algebraically and write any solution in interval notation.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1169148925027\">\r\n<div data-type=\"problem\" id=\"fs-id1169148925029\">\r\n<p id=\"fs-id1169149279506\">\\({x}^{2}-6x-8\\le 0\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149110365\">\r\n<div data-type=\"problem\" id=\"fs-id1169149110367\">\r\n<p id=\"fs-id1169146742867\">\\(2{x}^{2}+x-10&gt;0\\)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169148999276\">\r\n<p id=\"fs-id1169148999278\">\\(\\left(\\text{\u2212}\\infty ,-\\frac{5}{2}\\right)\\cup \\left(2,\\infty \\right)\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1169146629166\">Model the situation with a quadratic equation and solve by any method.<\/p>\r\n\r\n<div data-type=\"exercise\" id=\"fs-id1169146629169\">\r\n<div data-type=\"problem\" id=\"fs-id1169146629171\">\r\n<p id=\"fs-id1169146629173\">Find two consecutive even numbers whose product is 360.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149312273\">\r\n<div data-type=\"problem\" id=\"fs-id1169149312276\">\r\n<p id=\"fs-id1169149294719\">The length of a diagonal of a rectangle is three more than the width. The length of the rectangle is three times the width. Find the length of the diagonal. (Round to the nearest tenth.)<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"solution\" id=\"fs-id1169149294725\">\r\n<p id=\"fs-id1169149294727\">The diagonal is 3.8 units long.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"exercise\" id=\"fs-id1169149009528\">\r\n<div data-type=\"problem\" id=\"fs-id1169149009530\">\r\n<p id=\"fs-id1169149009532\">A water balloon is launched upward at the rate of 86 ft\/sec. Using the formula <em data-effect=\"italics\">h<\/em> = \u221216<em data-effect=\"italics\">t<\/em><sup>2<\/sup> + 86<em data-effect=\"italics\">t<\/em> find how long it will take the balloon to reach the maximum height, and then find the maximum height. Round to the nearest tenth.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"glossary\" class=\"textbox shaded\">\r\n<h3 data-type=\"glossary-title\">Glossary<\/h3>\r\n<dl id=\"fs-id1169148990888\">\r\n \t<dt>quadratic inequality<\/dt>\r\n \t<dd id=\"fs-id1169148990891\">A quadratic inequality is an inequality that contains a quadratic expression.<\/dd>\r\n<\/dl>\r\n<\/div>","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 inequalities graphically<\/li>\n<li>Solve quadratic inequalities algebraically<\/li>\n<\/ul>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169148924848\" class=\"be-prepared\">\n<p id=\"fs-id1169146648302\">Before you get started, take this readiness quiz.<\/p>\n<ol type=\"1\">\n<li>Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-21b0ccac9532b38cbfb0795124185ded_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#120;&#45;&#51;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"86\" style=\"vertical-align: 0px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p>If you missed this problem, review <a href=\"\/contents\/9f100e8f-6d15-4cae-bc22-c306e9d7d55c#fs-id1167836432956\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<li>Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-da97db4912cb5d3be7bcb08f33c3e849_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#121;&#61;&#49;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"98\" style=\"vertical-align: -4px;\" \/>.\n<div data-type=\"newline\"><\/div>\n<p>If you missed this problem, review <a href=\"\/contents\/da8478b4-93bc-4919-81a1-5e3267050e7e#fs-id1167836625705\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<li>Solve <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-05ce65d5124e319aa5376eac9942d600_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#120;&#45;&#56;&#125;&#62;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"92\" style=\"vertical-align: -9px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p>If you missed this problem, review <a href=\"\/contents\/a68b06f6-2833-4512-b24f-c0da889a8759#fs-id1167835534361\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<\/ol>\n<\/div>\n<p id=\"fs-id1169148924000\">We have learned how to solve linear inequalities and rational inequalities previously. Some of the techniques we used to solve them were the same and some were different.<\/p>\n<p id=\"fs-id1165926586447\">We will now learn to solve inequalities that have a quadratic expression. We will use some of the techniques from solving linear and rational inequalities as well as quadratic equations.<\/p>\n<p id=\"fs-id1169149008053\">We will solve quadratic inequalities two ways\u2014both graphically and algebraically.<\/p>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1169144377740\">\n<h3 data-type=\"title\">Solve Quadratic Inequalities Graphically<\/h3>\n<p id=\"fs-id1169149115761\">A <span data-type=\"term\" class=\"no-emphasis\">quadratic equation<\/span> is in standard form when written as <em data-effect=\"italics\">ax<\/em><sup>2<\/sup> + <em data-effect=\"italics\">bx<\/em> + <em data-effect=\"italics\">c<\/em> = 0. If we replace the equal sign with an inequality sign, we have a <span data-type=\"term\">quadratic inequality<\/span> in standard form.<\/p>\n<div data-type=\"note\" id=\"fs-id1169148991490\">\n<div data-type=\"title\">Quadratic Inequality<\/div>\n<p>A <strong data-effect=\"bold\">quadratic inequality<\/strong> is an inequality that contains a quadratic expression.<\/p>\n<p id=\"fs-id1169148916649\">The standard form of a quadratic inequality is written:<\/p>\n<div data-type=\"equation\" id=\"fs-id1169148962762\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0a0063d3bbdb1f7d3ddafe28ec98dacd_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;&#125;&#97;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#120;&#43;&#99;&#60;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#97;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#120;&#43;&#99;&#92;&#108;&#101;&#32;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#97;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#120;&#43;&#99;&#62;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#97;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#120;&#43;&#99;&#92;&#103;&#101;&#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=\"338\" style=\"vertical-align: -14px;\" \/><\/div>\n<\/div>\n<p id=\"fs-id1169148867608\">The graph of a quadratic function <em data-effect=\"italics\">f<\/em>(<em data-effect=\"italics\">x<\/em>) = <em data-effect=\"italics\">ax<\/em><sup>2<\/sup> + <em data-effect=\"italics\">bx<\/em> + <em data-effect=\"italics\">c<\/em> = 0 is a parabola. When we ask when is <em data-effect=\"italics\">ax<\/em><sup>2<\/sup> + <em data-effect=\"italics\">bx<\/em> + <em data-effect=\"italics\">c<\/em> &lt; 0, we are asking when is f(<em data-effect=\"italics\">x<\/em>) &lt; 0. We want to know when the parabola is below the <em data-effect=\"italics\">x<\/em>-axis.<\/p>\n<p id=\"fs-id1169149221517\">When we ask when is <em data-effect=\"italics\">ax<\/em><sup>2<\/sup> + <em data-effect=\"italics\">bx<\/em> + <em data-effect=\"italics\">c<\/em> &gt; 0, we are asking when is <em data-effect=\"italics\">f<\/em>(<em data-effect=\"italics\">x<\/em>) &gt; 0. We want to know when the parabola is above the <em data-effect=\"italics\">y<\/em>-axis.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1169148828123\" data-alt=\"The first graph is an upward facing parabola, f of x, on an x y-coordinate plane. To the left of the function, f of x is greater than 0. Between the x-intercepts, f of x is less than 0. To the right of the function, f of x is greater than 0. The second graph is a downward-facing parabola, f of x, on an x y coordinate plane. To the left of the function, f of x is less than 0. Between the x-intercepts, f of x is greater than 0. To the right of the function, f of x is less than 0.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_001_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The first graph is an upward facing parabola, f of x, on an x y-coordinate plane. To the left of the function, f of x is greater than 0. Between the x-intercepts, f of x is less than 0. To the right of the function, f of x is greater than 0. The second graph is a downward-facing parabola, f of x, on an x y coordinate plane. To the left of the function, f of x is less than 0. Between the x-intercepts, f of x is greater than 0. To the right of the function, f of x is less than 0.\" \/><\/span><\/p>\n<div data-type=\"example\" id=\"fs-id1169147029625\" class=\"textbox textbox--examples\">\n<div data-type=\"title\">How to Solve a Quadratic Inequality Graphically<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149095397\">\n<div data-type=\"problem\" id=\"fs-id1169149335491\">\n<p id=\"fs-id1169146607663\">Solve <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bf33ffa63a06dafad984244d3005dc16_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#120;&#43;&#56;&#60;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"122\" style=\"vertical-align: -2px;\" \/> graphically. Write the solution in interval notation.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149156869\"><span data-type=\"media\" data-alt=\"The figure is a table with 3 columns. The first column is Step 1: Write the quadratic inequality in standard form. The second column says the inequality is in standard form. The third column says x squared minus 6 times x plus 8 less than 0.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_002a_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure is a table with 3 columns. The first column is Step 1: Write the quadratic inequality in standard form. The second column says the inequality is in standard form. The third column says x squared minus 6 times x plus 8 less than 0.\" \/><\/span><span data-type=\"media\" id=\"fs-id1169149144291\" data-alt=\"The figure is a table with 3 columns. The first column says Step 2-Graph the function f of x equals a times x squared plus b times x plus c using properties or transformations. The second column gives instructions and the third column shows the work for step 3 as follows. We will graph using properties. The function is f of x equals x squared minus 6 times x plus 8 where a equals 1, b equals negative 6, and c equals 8. Look at a in the function f of x equals x squared minus 6 times x plus 8. Since a is positive, the parabola opens upward. The equation of the axis of symmetry is the line x equals negative b divided by 2 times a, so x equals negative negative 6 divided by 2 times 1. X equals 3. The axis of symmetry is the line x equals 3. The vertex is on the axis of symmetry. Substitute x equals 3 into the function, so f of 3 equals 3 squared minus 6 times 3 plus 8. F of 3 equals negative 1, so the vertex is (3, negative 1). We find f of 0 in order to find the y-intercept, so f of 0 equals 0 squared minus 6 times 0 plus 8. F of 0 equals 8, so the y intercept is (0, 8). We use the axis of symmetry to find a point symmetric to the y-intercept. The y-intercept is 3 units left of the axis of symmetry, x equals 3. A point 3 units to the right of the axis of symmetry has x equals 6. Point symmetric to y-intercept is (6, 8). We solve f of x equals 0 in order to find the x-intercepts. We can solve this quadratic equation by factoring. 0 equals x squared minus 6 times x plus 8, 0 equals the quantity x minus 2 times the quantity x minus 4, x equals 2 or x equals 4. The x-intercepts are (2, 0) and (4, 0). We graph the vertex, intercepts, and the point symmetric to the y-intercept. We connect these 5 points to sketch the parabola shown that is upward-facing with the points found through this process.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_002b_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure is a table with 3 columns. The first column says Step 2-Graph the function f of x equals a times x squared plus b times x plus c using properties or transformations. The second column gives instructions and the third column shows the work for step 3 as follows. We will graph using properties. The function is f of x equals x squared minus 6 times x plus 8 where a equals 1, b equals negative 6, and c equals 8. Look at a in the function f of x equals x squared minus 6 times x plus 8. Since a is positive, the parabola opens upward. The equation of the axis of symmetry is the line x equals negative b divided by 2 times a, so x equals negative negative 6 divided by 2 times 1. X equals 3. The axis of symmetry is the line x equals 3. The vertex is on the axis of symmetry. Substitute x equals 3 into the function, so f of 3 equals 3 squared minus 6 times 3 plus 8. F of 3 equals negative 1, so the vertex is (3, negative 1). We find f of 0 in order to find the y-intercept, so f of 0 equals 0 squared minus 6 times 0 plus 8. F of 0 equals 8, so the y intercept is (0, 8). We use the axis of symmetry to find a point symmetric to the y-intercept. The y-intercept is 3 units left of the axis of symmetry, x equals 3. A point 3 units to the right of the axis of symmetry has x equals 6. Point symmetric to y-intercept is (6, 8). We solve f of x equals 0 in order to find the x-intercepts. We can solve this quadratic equation by factoring. 0 equals x squared minus 6 times x plus 8, 0 equals the quantity x minus 2 times the quantity x minus 4, x equals 2 or x equals 4. The x-intercepts are (2, 0) and (4, 0). We graph the vertex, intercepts, and the point symmetric to the y-intercept. We connect these 5 points to sketch the parabola shown that is upward-facing with the points found through this process.\" \/><\/span><span data-type=\"media\" id=\"fs-id1169148944404\" data-alt=\"The figure is a table with 3 columns. The first column says Step 3- Determine the solution from the graph. The second column gives instructions. X squared minus 6 x plus 8 less than 0. The inequality asks for the values of x which make the function less than 0. Which values of x make the parabola below the x-axis. We do not include the values 2, 4 as the inequality is strictly less than. The third column says The solution, in interval notation, is (2, 4).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_002c_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure is a table with 3 columns. The first column says Step 3- Determine the solution from the graph. The second column gives instructions. X squared minus 6 x plus 8 less than 0. The inequality asks for the values of x which make the function less than 0. Which values of x make the parabola below the x-axis. We do not include the values 2, 4 as the inequality is strictly less than. The third column says The solution, in interval notation, is (2, 4).\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169149087550\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169146622094\">\n<div data-type=\"problem\" id=\"fs-id1169148951471\">\n<p id=\"fs-id1169146633988\"><span class=\"token\">\u24d0<\/span> Solve <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-05c9665055307689d39ee39b60e8f1c6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#120;&#45;&#56;&#60;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"122\" style=\"vertical-align: -2px;\" \/> graphically and <span class=\"token\">\u24d1<\/span> write the solution in interval notation.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149123010\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1169146643216\" data-alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 2, negative 9), y-intercept of (0, 8), and axis of symmetry shown at x equals negative 2.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_301_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 2, negative 9), y-intercept of (0, 8), and axis of symmetry shown at x equals negative 2.\" \/><\/span><\/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-4ce13ff56205290ac2f68e7ba2f28d73_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#44;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"65\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169149307519\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169149346432\">\n<div data-type=\"problem\" id=\"fs-id1169146637308\">\n<p id=\"fs-id1169149312399\"><span class=\"token\">\u24d0<\/span> Solve <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-adfa084126e3113c4481a313a4702d6c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#56;&#120;&#43;&#49;&#50;&#92;&#103;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"131\" style=\"vertical-align: -3px;\" \/> graphically and <span class=\"token\">\u24d1<\/span> write the solution in interval notation.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149293258\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" data-alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (4, negative 4) and x-intercepts of (2, 0) and (6, 0).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_302_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (4, negative 4) and x-intercepts of (2, 0) and (6, 0).\" \/><\/span><\/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-c0d1c7e8b3eab7380596dabfdc6ffb10_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#44;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#93;&#92;&#99;&#117;&#112;&#32;&#92;&#108;&#101;&#102;&#116;&#91;&#54;&#44;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"111\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1169149030281\">We list the steps to take to solve a quadratic inequality graphically.<\/p>\n<div data-type=\"note\" id=\"fs-id1169149350941\" class=\"howto\">\n<div data-type=\"title\">Solve a quadratic inequality graphically.<\/div>\n<ol id=\"fs-id1169148988653\" class=\"stepwise\" type=\"1\">\n<li>Write the quadratic inequality in standard form.<\/li>\n<li>Graph the function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5a0b6fa82f59c470088d6e34f484552d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#97;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#120;&#43;&#99;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"161\" style=\"vertical-align: -4px;\" \/><\/li>\n<li>Determine the solution from the graph.<\/li>\n<\/ol>\n<\/div>\n<p id=\"fs-id1169146633876\">In the last example, the parabola opened upward and in the next example, it opens downward. In both cases, we are looking for the part of the parabola that is below the <em data-effect=\"italics\">x<\/em>-axis but note how the position of the parabola affects the solution.<\/p>\n<div data-type=\"example\" id=\"fs-id1169149121645\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1169144381671\">\n<div data-type=\"problem\" id=\"fs-id1169146661918\">\n<p id=\"fs-id1169144379524\">Solve <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-719c3283f77483f12ba7cca88398eb49_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#56;&#120;&#45;&#49;&#50;&#92;&#108;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"131\" style=\"vertical-align: -3px;\" \/> graphically. Write the solution in interval notation.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148899359\">\n<table class=\"unnumbered unstyled can-break\" summary=\"This figure is step-by-step instructions on how to solve an inequality graphically. The quadratic inequality in standard form is negative x squared minus 8 times x minus 12 less than or equal to 0. Graph the function f of x equals negative x squared minus 8 times x minus 12 to find that the parabola opens upward. Find the line of symmetry by using the equation x equals negative b divided by 2 times a. Substitute in to get x equals negative negative 8 divided by 2 times negative 1 to find x equals negative 4. Find the vertex of f of x equals negative x squared minus 8 times x minus 12 by finding that f of negative 4 equals negative negative 4 squared minus 8 times negative 4 minus 12. That gives you f of negative 4 equals negative 16 minus 32 minus 12, which then reduces to f of negative 4 equals 4. The vertex is (negative 4, 4). Find the x-intercepts. Let f of x equal 0. Take the original function, f of x equals negative x squared minus 8 times x minus 12, then make it 0 equals negative x squared minus 8 times x minus 12. Factor to get 0 equals negative 1 times the quantity x plus 6 times the quantity x plus 2. Use the Zero Product Property to get x equals negative 6 and x equals negative 2. The x-intercepts are (negative 6, 0) and (negative 2, 0). The graph shown is the curve formed when plotting all the points just found. Then, determine the solution from the graph, (negative infinity, negative 6] in union with [negative 2, infinity). We include the x-intercepts as the inequality is \u201cless than or equal to.\u201d\" data-label=\"\">\n<tbody>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">The quadratic inequality in standard form.<\/td>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b3a1421d6ddfca77b30cc2cc132d54d7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#56;&#120;&#45;&#49;&#50;&#92;&#108;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"143\" style=\"vertical-align: -3px;\" \/><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Graph the function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-291bfb1ebee9e8cb32d66780a56c39a0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#56;&#120;&#45;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"158\" style=\"vertical-align: -4px;\" \/>.<\/td>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\">The parabola opens downward.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1169148939580\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_003b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Find the line of symmetry.<\/td>\n<td><\/td>\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-3ca56fe254ea7893db96cc9c75027551_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#56;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#98;&#125;&#123;&#50;&#97;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"64\" style=\"vertical-align: -6px;\" \/><\/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-3fc7042bd8ca821f2ad0a605d587b5e3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#56;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#56;&#125;&#123;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"85\" style=\"vertical-align: -9px;\" \/><\/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-6d8ca9d84eff3e8f1c9ca889664ec93d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#56;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#61;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"57\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Find the vertex.<\/td>\n<td><\/td>\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-d683e31ce67f5f4932bf8701ba2956f9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#54;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#56;&#120;&#45;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"158\" style=\"vertical-align: -4px;\" \/><\/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-2b51db6ba02fd7c3975dc55714b35b0f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#45;&#56;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"226\" style=\"vertical-align: -4px;\" \/><\/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-4e656c965cdda846670d6e4f656e68ac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#49;&#54;&#43;&#51;&#50;&#45;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"183\" style=\"vertical-align: -4px;\" \/><\/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-6a71d7e5033ae878c4e48e1dd58e85e7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"83\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p>Vertex <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6bd03bb09a34e49dd9f3ebf548e3dab1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Find the <em data-effect=\"italics\">x<\/em>-intercepts. Let <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-506e7f62456359347409a646ee8199fa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"70\" style=\"vertical-align: -4px;\" \/>.<\/td>\n<td><\/td>\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-856bbd1b9613ad12c243b61485afc239_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#55;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#56;&#120;&#45;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"159\" style=\"vertical-align: -4px;\" \/><\/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-fe533174e4203cd3b2b3264b4ccdc30b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#57;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#48;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#56;&#120;&#45;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"129\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Factor.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>Use the Zero Product Property.<\/td>\n<td><\/td>\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-f53ece02587ebae5cb5fc7f6fc941acf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#57;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#48;&#61;&#45;&#49;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"169\" style=\"vertical-align: -4px;\" \/><\/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-3ae040eb8d57444926f44fff53f3234d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#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;&#120;&#61;&#45;&#54;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#61;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"139\" style=\"vertical-align: 0px;\" \/><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Graph the parabola.<\/td>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"center\"><em data-effect=\"italics\">x<\/em>-intercepts <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a417a7897a86a734a1427dcf32690b6f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#54;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"115\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1169148889714\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_003n_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Determine the solution from the graph.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>We include the <em data-effect=\"italics\">x<\/em>-intercepts as the inequality<\/p>\n<div data-type=\"newline\"><\/div>\n<p>is \u201cless than or equal to.\u201d<\/td>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0132bc62c84059a070aacb35460483e0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#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;&#8722;&#125;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#93;&#92;&#99;&#117;&#112;&#32;&#92;&#108;&#101;&#102;&#116;&#91;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#50;&#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;&#105;&#110;&#102;&#116;&#121;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"118\" style=\"vertical-align: -5px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169148834153\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169148862433\">\n<div data-type=\"problem\" id=\"fs-id1169148859315\">\n<p id=\"fs-id1169148845708\"><span class=\"token\">\u24d0<\/span> Solve <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a1d020b2fdbc9bbd503488ced8881ead_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#120;&#45;&#53;&#62;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"122\" style=\"vertical-align: 0px;\" \/> graphically and <span class=\"token\">\u24d1<\/span> write the solution in interval notation.<\/p>\n<\/div>\n<div data-type=\"solution\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1169148824319\" data-alt=\"A downward-facing parabola on the x y-coordinate plane. It has a vertex of (negative 3, 4), a y-intercept at (0, negative 5), and an axis of symmetry shown at x equals negative 3.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_303_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"A downward-facing parabola on the x y-coordinate plane. It has a vertex of (negative 3, 4), a y-intercept at (0, negative 5), and an axis of symmetry shown at x equals negative 3.\" \/><\/span><\/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-1c6cf6f67551105173d9fb3cab5966cf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#44;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169148948253\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169148821286\">\n<div data-type=\"problem\" id=\"fs-id1169146633494\">\n<p id=\"fs-id1169149155077\"><span class=\"token\">\u24d0<\/span> Solve <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dcd8d2f68296094cc06daf62562b65b0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#48;&#120;&#45;&#49;&#54;&#92;&#108;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"139\" style=\"vertical-align: -3px;\" \/> graphically and <span class=\"token\">\u24d1<\/span> write the solution in interval notation.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169146627483\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1169148880103\" data-alt=\"A downward-facing parabola on the x y-coordinate plane. It has a vertex of (5, 9), a y-intercept at (0, negative 16), and an axis of symmetry of x equals 5.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_304_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"A downward-facing parabola on the x y-coordinate plane. It has a vertex of (5, 9), a y-intercept at (0, negative 16), and an axis of symmetry of x equals 5.\" \/><\/span><\/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-95b36c38ae0151bdcfb2590cb6a597c3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#44;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#93;&#92;&#99;&#117;&#112;&#32;&#92;&#108;&#101;&#102;&#116;&#91;&#56;&#44;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"111\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1169149157937\">\n<h3 data-type=\"title\">Solve Quadratic Inequalities Algebraically<\/h3>\n<p id=\"fs-id1169149308017\">The algebraic method we will use is very similar to the method we used to solve rational inequalities. We will find the critical points for the inequality, which will be the solutions to the related quadratic equation. Remember a polynomial expression can change signs only where the expression is zero.<\/p>\n<p id=\"fs-id1169148867090\">We will use the <span data-type=\"term\" class=\"no-emphasis\">critical points<\/span> to divide the number line into intervals and then determine whether the quadratic expression willl be postive or negative in the interval. We then determine the solution for the inequality.<\/p>\n<div data-type=\"example\" id=\"fs-id1169149024679\" class=\"textbox textbox--examples\">\n<div data-type=\"title\">How To Solve Quadratic Inequalities Algebraically<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148998010\">\n<div data-type=\"problem\" id=\"fs-id1169148846107\">\n<p id=\"fs-id1169148866808\">Solve <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ed81b082789edc9dea471891fb006ffa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#120;&#45;&#49;&#50;&#92;&#103;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"122\" style=\"vertical-align: -3px;\" \/> algebraically. Write the solution in interval notation.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149116677\"><span data-type=\"media\" id=\"fs-id1169149295000\" data-alt=\"This figure is a table giving the instructions for solving x squared minus x minus 12 greater than or equal to 0 algebraically. It consists of 3 columns where the instructions are given in the first column, the explanation in the second, and the work in the third. Step 1 is to write the quadratic inequality in standard form. The quadratic inequality in already in standard form, so x squared minus x minus 12 greater than or equal to 0.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_004a_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure is a table giving the instructions for solving x squared minus x minus 12 greater than or equal to 0 algebraically. It consists of 3 columns where the instructions are given in the first column, the explanation in the second, and the work in the third. Step 1 is to write the quadratic inequality in standard form. The quadratic inequality in already in standard form, so x squared minus x minus 12 greater than or equal to 0.\" \/><\/span><span data-type=\"media\" id=\"fs-id1169146640088\" data-alt=\"Step 2 is to determine the critical points -- the solutions to the related quadratic equation. To do this, change the inequality sign to an equal sign and then solve the equation. x squared minus x minus 12 equals 0 factors to the quantity x plus 3 times the quantity x minus 4 equals 0. Then, x plus 3 equals 0 and x minus 4 equals 0 to give x equals negative 3 and x equals 4.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_004b_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2 is to determine the critical points -- the solutions to the related quadratic equation. To do this, change the inequality sign to an equal sign and then solve the equation. x squared minus x minus 12 equals 0 factors to the quantity x plus 3 times the quantity x minus 4 equals 0. Then, x plus 3 equals 0 and x minus 4 equals 0 to give x equals negative 3 and x equals 4.\" \/><\/span><span data-type=\"media\" id=\"fs-id1169149066368\" data-alt=\"Step 3 is to use the critical points to divide the number line into intervals. Use negative 3 and 4 to divide the number line into intervals. A number line is shown that includes from left to right the values of negative 3, 0, and 4, with dotted lines on negative 3 and 4.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_004c_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3 is to use the critical points to divide the number line into intervals. Use negative 3 and 4 to divide the number line into intervals. A number line is shown that includes from left to right the values of negative 3, 0, and 4, with dotted lines on negative 3 and 4.\" \/><\/span><span data-type=\"media\" id=\"fs-id1169148969887\" data-alt=\"Step 4 says above the number line show the sign of each quadratic expression using test points from each interval substituted into the original inequality. X equals negative 5, x equals 0, and x equals 5 are chosen to test. The expression negative x squared minus x minus 12 is given with negative 5 squared minus negative 5 minus 12 underneath, which gives 18. The expression negative x squared minus x minus 12 is given with 0 squared minus 0 minus 12 underneath, which gives 12. The expression negative x squared minus x minus 12 is given with 5 squared minus 5 minus 12 underneath, which gives 8.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_004d_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 4 says above the number line show the sign of each quadratic expression using test points from each interval substituted into the original inequality. X equals negative 5, x equals 0, and x equals 5 are chosen to test. The expression negative x squared minus x minus 12 is given with negative 5 squared minus negative 5 minus 12 underneath, which gives 18. The expression negative x squared minus x minus 12 is given with 0 squared minus 0 minus 12 underneath, which gives 12. The expression negative x squared minus x minus 12 is given with 5 squared minus 5 minus 12 underneath, which gives 8.\" \/><\/span><span data-type=\"media\" id=\"fs-id1169149305655\" data-alt=\"For Step 5, determine the intervals where the inequality is correct. Write the solution in interval notation. x squared minus x minus 12 greater than or equal to 0 is shown. The inequality is positive in the first and last intervals and equals 0 at the points negative 4, 3 . The solution, in interval notation, is (negative infinity, negative 3] in union with [4, infinity).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_004e_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"For Step 5, determine the intervals where the inequality is correct. Write the solution in interval notation. x squared minus x minus 12 greater than or equal to 0 is shown. The inequality is positive in the first and last intervals and equals 0 at the points negative 4, 3 . The solution, in interval notation, is (negative infinity, negative 3] in union with [4, infinity).\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169148861231\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169148878663\">\n<div data-type=\"problem\" id=\"fs-id1169148924858\">\n<p>Solve <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-82926b00159f2f1623580c4a1474d8a3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#120;&#45;&#56;&#92;&#103;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"122\" style=\"vertical-align: -3px;\" \/> algebraically. Write the solution in interval notation.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148818997\">\n<p id=\"fs-id1169146654389\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e4e497c930e01b9a2110b8bd390d2a79_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#44;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#93;&#92;&#99;&#117;&#112;&#32;&#92;&#108;&#101;&#102;&#116;&#91;&#50;&#44;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"124\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169148999680\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169148898510\">\n<div data-type=\"problem\" id=\"fs-id1169149362258\">\n<p id=\"fs-id1169148888294\">Solve <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e5a34b6994da18ffd0ba7f09c8bc18f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#120;&#45;&#49;&#53;&#92;&#108;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"131\" style=\"vertical-align: -3px;\" \/> algebraically. Write the solution in interval notation.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149292178\">\n<p id=\"fs-id1169148938679\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e0c4b6fce9885126fae07de9aa2a2504_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#45;&#51;&#44;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#93;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"45\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1169149291093\">In this example, since the expression <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-703f7c4f04a6f58a1ee1fe54a96deb93_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#120;&#45;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"88\" style=\"vertical-align: -1px;\" \/> factors nicely, we can also find the sign in each interval much like we did when we solved rational inequalities. We find the sign of each of the factors, and then the sign of the product. Our number line would like this:<\/p>\n<p><span data-type=\"media\" id=\"fs-id1169148932873\" data-alt=\"The figure shows the expression x squared minus x minus 12 factored to the quantity of x plus 3 times the quantity of x minus 4. The image shows a number line showing dotted lines on negative 3 and 4. It shows the signs of the quantity x plus 3 to be negative, positive, positive, and the signs of the quantity x minus 4 to be negative, negative, positive. Under the number line, it shows the quantity x plus 3 times the quantity x minus 4 with the signs positive, negative, positive.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_005_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the expression x squared minus x minus 12 factored to the quantity of x plus 3 times the quantity of x minus 4. The image shows a number line showing dotted lines on negative 3 and 4. It shows the signs of the quantity x plus 3 to be negative, positive, positive, and the signs of the quantity x minus 4 to be negative, negative, positive. Under the number line, it shows the quantity x plus 3 times the quantity x minus 4 with the signs positive, negative, positive.\" \/><\/span><\/p>\n<p id=\"fs-id1169146665788\">The result is the same as we found using the other method.<\/p>\n<p id=\"fs-id1169149342624\">We summarize the steps here.<\/p>\n<div data-type=\"note\" id=\"fs-id1169149307510\" class=\"howto\">\n<div data-type=\"title\">Solve a quadratic inequality algebraically.<\/div>\n<ol id=\"fs-id1169149015056\" class=\"stepwise\" type=\"1\">\n<li>Write the quadratic inequality in standard form.<\/li>\n<li>Determine the critical points\u2014the solutions to the related quadratic equation.<\/li>\n<li>Use the critical points to divide the number line into intervals.<\/li>\n<li>Above the number line show the sign of each quadratic expression using test points from each interval substituted into the original inequality.<\/li>\n<li>Determine the intervals where the inequality is correct. Write the solution in interval notation.<\/li>\n<\/ol>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1169144383249\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1169146627378\">\n<div data-type=\"problem\" id=\"fs-id1169146644117\">\n<p id=\"fs-id1169149002129\">Solve <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3a54b19eef82a2e73959f8ae6e550763_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#120;&#45;&#55;&#92;&#103;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"122\" style=\"vertical-align: -3px;\" \/> algebraically. Write the solution in interval notation.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148879146\">\n<table id=\"fs-id1169149003372\" class=\"unnumbered unstyled can-break\" summary=\"The figure is gives step-by-step instructions on how to solve negative x squared plus 6 times x minus 7 greater than or equal to 0 algebraically. Write the inequality in standard form. negative x squared plus 6 times x minus 7 greater than or equal to 0 is already in standard form. Multiply both sides of the inequality by negative 1, remember to reverse the inequality sign, to get x squared minus 6 times x plus 7 less than or equal to 0. Determine the critical points by solving the related quadratic equation, x squared minus 6 times x plus 7 equals 0. Write the quadratic formula, x equals negative b plus or minus the square root of b squared minus 4 times a times c all divided by 2 times a. Then substitute in the values of a, b, c, to get x equals negative negative 6 plus or minus the square root of negative 6 squared minus 4 times 1 times 7 all divided by 2 times 1. Simplify to get x equals 6 plus or minus the square root of 8 divided by 2. Remove the common factor of 2, x equals 2 times the quantity 3 plus or minus square root of 2 divided by 2 which gives x equals 3 plus or minus square root of 2. If x equals 3 plus square root of 2, x is approximately 1 and 6 tenths. If x equals 3 minus square root of 2, x is approximately 4 and 4 tenths. Use the critical points to divide the number line into intervals. A number line is shown with 1 and 6 tenths and 4 and 4 tenths. Test the numbers from each interval in the original inequality. On the number line, negative x squared plus 6 times x minus 7 is shown with the signs negative, positive, and negative. Determine the intervals where the inequality is correct. Write the solution in interval notation. negative x squared plus 6 times x minus 7 is greater than or equals to 0 in the middle interval, so the final answer is [3 minus square root of 2, 3 plus square root of 2\" data-label=\"\">\n<tbody>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Write the quadratic inequality in standard form.<\/td>\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-1764006037ec7bba047f0511405e71c6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#120;&#45;&#55;&#92;&#103;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"134\" style=\"vertical-align: -3px;\" \/><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Multiply both sides of the inequality by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7b34c01098c83fa602de54e9d74d63a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"21\" style=\"vertical-align: -1px;\" \/>.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>Remember to reverse the inequality sign.<\/td>\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-dc124a07c8c3a829050bba415c32c09d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#55;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#120;&#43;&#55;&#92;&#108;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"122\" style=\"vertical-align: -3px;\" \/><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Determine the critical points by solving<\/p>\n<div data-type=\"newline\"><\/div>\n<p>the related quadratic equation.<\/td>\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-76e2419c4179771698ac7f2f74c4350c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#55;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#120;&#43;&#55;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"122\" style=\"vertical-align: -2px;\" \/><\/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\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0f9a04bbeda04ba40e3a6f0e4eb3c8aa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#46;&#57;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#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=\"108\" style=\"vertical-align: -6px;\" \/><\/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=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-faafdb98e50582f0267df19f645cc167_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#46;&#57;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&plusmn;&#92;&#115;&#113;&#114;&#116;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#92;&#99;&#100;&#111;&#116;&#32;&#49;&#92;&#99;&#100;&#111;&#116;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#125;&#123;&#50;&#92;&#99;&#100;&#111;&#116;&#32;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"31\" width=\"175\" style=\"vertical-align: -7px;\" \/><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\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-076b58ab10ce9757da563f86fe9d4eba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#46;&#57;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#54;&plusmn;&#92;&#115;&#113;&#114;&#116;&#123;&#56;&#125;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"62\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Simplify the radical.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-45854234514fa65e5063fe875c2e1d84_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#46;&#57;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#54;&plusmn;&#50;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#125;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"69\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Remove the common factor, 2.<\/td>\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-95e0793e7a9f9d3a1a0a24ccfec58cad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#46;&#57;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&plusmn;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"30\" width=\"82\" style=\"vertical-align: -6px;\" \/><\/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-ff302517151f0216547cc0d37b7ab29a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#46;&#57;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#61;&#51;&plusmn;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"66\" style=\"vertical-align: -2px;\" \/><\/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-9e12b830db14b204358af0d01ab2a65d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#46;&#57;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#61;&#51;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#125;&#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;&#120;&#61;&#51;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"211\" style=\"vertical-align: -2px;\" \/><\/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-e7f8a523e1d7f2f3d46ebfb427d30e49_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#46;&#57;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#92;&#97;&#112;&#112;&#114;&#111;&#120;&#32;&#49;&#46;&#54;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#46;&#51;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#92;&#97;&#112;&#112;&#114;&#111;&#120;&#32;&#52;&#46;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"172\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Use the critical points to divide the<\/p>\n<div data-type=\"newline\"><\/div>\n<p>number line into intervals.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>Test numbers from each interval<\/p>\n<div data-type=\"newline\"><\/div>\n<p>in the original inequality.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1169149087038\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_006l_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Determine the intervals where the<\/p>\n<div data-type=\"newline\"><\/div>\n<p>inequality is correct. Write the solution<\/p>\n<div data-type=\"newline\"><\/div>\n<p>in interval notation.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8025da24693d56a405052eb261462d27_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#120;&#45;&#55;&#92;&#103;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"122\" style=\"vertical-align: -3px;\" \/> in the middle interval<\/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-0fd104fd64ba9e9b6225607799ea0053_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#51;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#125;&#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;&#51;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#93;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"133\" style=\"vertical-align: -7px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169148974557\">\n<div data-type=\"problem\">\n<p id=\"fs-id1169149087343\">Solve <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c923adaeb66b28b09cd476d2e3391c51_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#120;&#43;&#49;&#92;&#103;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"122\" style=\"vertical-align: -3px;\" \/> algebraically. Write the solution in interval notation.<\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1169149000975\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3853756d4fd43fe30bb66d9157de305a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#45;&#49;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#125;&#44;&#45;&#49;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#93;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"152\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169149037957\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169149122051\">\n<div data-type=\"problem\" id=\"fs-id1169149135306\">\n<p id=\"fs-id1169148994901\">Solve <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-53f9f9f3cdbd5188385e09fc9b1b1379_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#56;&#120;&#45;&#49;&#52;&#60;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"131\" style=\"vertical-align: -2px;\" \/> algebraically. Write the solution in interval notation.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149088969\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ca39805f272dd1fbd886eec7d05930d5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#44;&#52;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#99;&#117;&#112;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#125;&#44;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"208\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1169149309531\">The solutions of the quadratic inequalities in each of the previous examples, were either an interval or the union of two intervals. This resulted from the fact that, in each case we found two solutions to the corresponding quadratic equation <em data-effect=\"italics\">ax<\/em><sup>2<\/sup> + <em data-effect=\"italics\">bx<\/em> + <em data-effect=\"italics\">c<\/em> = 0. These two solutions then gave us either the two <em data-effect=\"italics\">x-<\/em>intercepts for the graph or the two critical points to divide the number line into intervals.<\/p>\n<p id=\"fs-id1169148999242\">This correlates to our previous discussion of the number and type of solutions to a quadratic equation using the discriminant.<\/p>\n<p id=\"fs-id1169149135716\">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-ce78e2da43dbf8e758d3c5c14d7f44ac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#92;&#110;&#101;&#32;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"46\" style=\"vertical-align: -4px;\" \/><\/p>\n<p><span data-type=\"media\" data-alt=\"The figure is a table with 3 columns. Column 1 is labeled discriminant, column 2 is Number\/Type of solution, and column 3 is Typical Graph. Reading across the columns, if b squared minus 4 times a times c is greater than 0, there will be 2 real solutions because there are 2 x-intercepts on the graph. The image of a typical graph an upward or downward parabola with 2 x-intercepts. If the discriminant b squared minus 4 times a times c is equals to 0, then there is 1 real solution because there is 1 x-intercept on the graph. The image of the typical graph is an upward- or downward-facing parabola that has a vertex on the x-axis instead of crossing through it. If the discriminant b squared minus 4 times a times c is less than 0, there are 2 complex solutions because there is no x-intercept. The image of the typical graph shows an upward- or downward-facing parabola that does not cross the x-axis.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_007_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure is a table with 3 columns. Column 1 is labeled discriminant, column 2 is Number\/Type of solution, and column 3 is Typical Graph. Reading across the columns, if b squared minus 4 times a times c is greater than 0, there will be 2 real solutions because there are 2 x-intercepts on the graph. The image of a typical graph an upward or downward parabola with 2 x-intercepts. If the discriminant b squared minus 4 times a times c is equals to 0, then there is 1 real solution because there is 1 x-intercept on the graph. The image of the typical graph is an upward- or downward-facing parabola that has a vertex on the x-axis instead of crossing through it. If the discriminant b squared minus 4 times a times c is less than 0, there are 2 complex solutions because there is no x-intercept. The image of the typical graph shows an upward- or downward-facing parabola that does not cross the x-axis.\" \/><\/span><\/p>\n<p id=\"fs-id1169148888286\">The last row of the table shows us when the parabolas never intersect the <em data-effect=\"italics\">x<\/em>-axis. Using the Quadratic Formula to solve the quadratic equation, the radicand is a negative. We get two complex solutions.<\/p>\n<p id=\"fs-id1169148894484\">In the next example, the quadratic inequality solutions will result from the solution of the quadratic equation being complex.<\/p>\n<div data-type=\"example\" id=\"fs-id1169148951346\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1169149308650\">\n<div data-type=\"problem\">\n<p>Solve, writing any solution in interval notation:<\/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-b3967ad9ca1873e5d54fd40d93991e0b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#120;&#43;&#52;&#62;&#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-63bb89eb3df03c2a69e9f52b2d0cd63b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#120;&#43;&#52;&#92;&#108;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"122\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148973208\">\n<p id=\"fs-id1169148970174\"><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1169149220879\" class=\"unnumbered unstyled can-break\" summary=\"The figure is gives step-by-step instructions on how to solve x squared minus 3 times x plus 4 greater than 0 algebraically. Write the inequality in standard form. x squared minus 3 times x plus 4 greater than 0 is already in standard form. Determine the critical points by solving the related quadratic equation, x squared minus 3 times x plus 4 equals 0. Write the quadratic formula, x equals negative b plus or minus the square root of b squared minus 4 times a times c all divided by 2 times a. Then substitute in the values of a, b, c, to get x equals negative negative 3 plus or minus the square root of negative 3 squared minus 4 times 1 times 4 all divided by 2 times 1. Simplify to get x equals 3 plus or minus the square root of negative 7 divided by 2. Simplify the radicand to get x equals 3 plus or minus the square root of 7 times i divided by 2. The complex solutions tell us that parabola does not intercept the x-axis. Also, the parabola opens upward. This tells us that the paprabola is completely above the x-axis, as the image of an upward-facing parabola that does not cross the x-axis shows.\" data-label=\"\">\n<tbody>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Write the quadratic inequality in standard form.<\/td>\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-46cc75451b11d9af89735e9bcee43bb8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#56;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#120;&#43;&#52;&#62;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"139\" style=\"vertical-align: -2px;\" \/><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Determine the critical points by solving<\/p>\n<div data-type=\"newline\"><\/div>\n<p>the related quadratic equation.<\/td>\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-9385e884e77b06ce7c894373956e6d7a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#120;&#43;&#52;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"121\" style=\"vertical-align: -2px;\" \/><\/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\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-63785482d5553cf0cf6a2d1a8b49ce0e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#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=\"108\" style=\"vertical-align: -6px;\" \/><\/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=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dd78496e52a14bc405ca2c674a11821e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&plusmn;&#92;&#115;&#113;&#114;&#116;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#92;&#99;&#100;&#111;&#116;&#32;&#49;&#92;&#99;&#100;&#111;&#116;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#125;&#123;&#50;&#92;&#99;&#100;&#111;&#116;&#32;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"31\" width=\"154\" style=\"vertical-align: -7px;\" \/><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\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-d6399c5b850a4bf88cb0d5ca390eed75_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&plusmn;&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#55;&#125;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"72\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Simplify the radicand.<\/td>\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-8e5d2a0a4afb6afa824b5ab2dbfe9036_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&plusmn;&#92;&#115;&#113;&#114;&#116;&#123;&#55;&#125;&#105;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"66\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">The complex solutions tell us the<\/p>\n<div data-type=\"newline\"><\/div>\n<p>parabola does not intercept the <em data-effect=\"italics\">x<\/em>-axis.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>Also, the parabola opens upward. This<\/p>\n<div data-type=\"newline\"><\/div>\n<p>tells us that the parabola is completely above the <em data-effect=\"italics\">x<\/em>-axis.<\/td>\n<td data-valign=\"top\" data-align=\"center\">Complex solutions<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1169149219736\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_008g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1169148989687\">We are to find the solution to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cd06dadde0c2dc30e463efcf8dbfb365_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#120;&#43;&#52;&#62;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"126\" style=\"vertical-align: -2px;\" \/> Since for all values of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/> the graph is above the <em data-effect=\"italics\">x<\/em>-axis, all values of <em data-effect=\"italics\">x<\/em> make the inequality true. In interval notation we write <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5df92a2cc0a08c383e86c9bcbca032bf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#44;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"63\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"fs-id1169149123625\"><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-f024ecfc915ebcae16aee45957d74677_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#87;&#114;&#105;&#116;&#101;&#32;&#116;&#104;&#101;&#32;&#113;&#117;&#97;&#100;&#114;&#97;&#116;&#105;&#99;&#32;&#105;&#110;&#101;&#113;&#117;&#97;&#108;&#105;&#116;&#121;&#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;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#120;&#43;&#52;&#92;&#108;&#101;&#32;&#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;&#68;&#101;&#116;&#101;&#114;&#109;&#105;&#110;&#101;&#32;&#116;&#104;&#101;&#32;&#99;&#114;&#105;&#116;&#105;&#99;&#97;&#108;&#32;&#112;&#111;&#105;&#110;&#116;&#115;&#32;&#98;&#121;&#32;&#115;&#111;&#108;&#118;&#105;&#110;&#103;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#32;&#114;&#101;&#108;&#97;&#116;&#101;&#100;&#32;&#113;&#117;&#97;&#100;&#114;&#97;&#116;&#105;&#99;&#32;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#125;&#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;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#120;&#43;&#52;&#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=\"62\" width=\"547\" style=\"vertical-align: -25px;\" \/><\/p>\n<p id=\"fs-id1169148938454\">Since the corresponding quadratic equation is the same as in part (a), the parabola will be the same. The parabola opens upward and is completely above the <em data-effect=\"italics\">x<\/em>-axis\u2014no part of it is below the <em data-effect=\"italics\">x<\/em>-axis.<\/p>\n<p id=\"fs-id1169148948396\">We are to find the solution to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1ca5dc858d1ca3dc4295c35cbef712cd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#120;&#43;&#52;&#92;&#108;&#101;&#32;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"126\" style=\"vertical-align: -3px;\" \/> Since for all values of <em data-effect=\"italics\">x<\/em> the graph is never below the <em data-effect=\"italics\">x<\/em>-axis, no values of <em data-effect=\"italics\">x<\/em> make the inequality true. There is no solution to the inequality.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169144562776\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169148891264\">\n<div data-type=\"problem\" id=\"fs-id1169148881288\">\n<p id=\"fs-id1169148930301\">Solve and write any solution in interval notation:<\/p>\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-ebdaada13dadee0edd65bdf556105779_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#120;&#45;&#52;&#92;&#108;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"122\" style=\"vertical-align: -3px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f9747f88c9be932a2c9e8f06d6a9450c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#120;&#45;&#52;&#92;&#103;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"122\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148826303\">\n<p id=\"fs-id1169149196732\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-90eaa2f0f55290692476556ec134a082_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#44;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"56\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> no solution<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169148910806\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169148964170\">\n<div data-type=\"problem\" id=\"fs-id1169149319585\">\n<p id=\"fs-id1169148884129\">Solve and write any solution in interval notation:<\/p>\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-f0ad8ff3789081b25d6e434b66d28555_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#120;&#43;&#51;&#60;&#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-396c79c2f09d441acb990ae214f00a63_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#120;&#43;&#51;&#62;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"122\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148843341\">\n<p id=\"fs-id1169149293529\"><span class=\"token\">\u24d0<\/span> no solution<\/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-90eaa2f0f55290692476556ec134a082_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#44;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"56\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1169148894040\">\n<h3 data-type=\"title\">Key Concepts<\/h3>\n<ul id=\"fs-id1169148898632\" data-bullet-style=\"bullet\">\n<li>Solve a Quadratic Inequality Graphically\n<ol id=\"fs-id1169149094904\" class=\"stepwise\" type=\"1\">\n<li>Write the quadratic inequality in standard form.<\/li>\n<li>Graph the function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7ff39a37a230408f7c9a6410a33dbe1c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#97;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#120;&#43;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"157\" style=\"vertical-align: -4px;\" \/> using properties or transformations.<\/li>\n<li>Determine the solution from the graph.<\/li>\n<\/ol>\n<\/li>\n<li>How to Solve a Quadratic Inequality Algebraically\n<ol id=\"fs-id1169148967529\" class=\"stepwise\" type=\"1\">\n<li>Write the quadratic inequality in standard form.<\/li>\n<li>Determine the critical points &#8212; the solutions to the related quadratic equation.<\/li>\n<li>Use the critical points to divide the number line into intervals.<\/li>\n<li>Above the number line show the sign of each quadratic expression using test points from each interval substituted into the original inequality.<\/li>\n<li>Determine the intervals where the inequality is correct. Write the solution in interval notation.<\/li>\n<\/ol>\n<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1169148867354\">\n<h3 data-type=\"title\">Section Exercises<\/h3>\n<div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1169148894275\">\n<h4 data-type=\"title\">Practice Makes Perfect<\/h4>\n<p id=\"fs-id1169148951165\"><strong data-effect=\"bold\">Solve Quadratic Inequalities Graphically<\/strong><\/p>\n<p id=\"fs-id1169146743578\">In the following exercises, <span class=\"token\">\u24d0<\/span> solve graphically and <span class=\"token\">\u24d1<\/span> write the solution in interval notation.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169146731344\">\n<div data-type=\"problem\" id=\"fs-id1169148926529\">\n<p id=\"fs-id1169148964346\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f939b0b161a4a06c702dad34c6826b76_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#120;&#43;&#53;&#62;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"122\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149026035\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1169146745490\" data-alt=\"The graph shown is an upward-facing parabola with vertex (negative 3, negative 4) and y-intercept (0,5).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_305_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph shown is an upward-facing parabola with vertex (negative 3, negative 4) and y-intercept (0,5).\" \/><\/span><\/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-3a27478a4214dfefa8a4af21cb5ee207_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#44;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#99;&#117;&#112;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#44;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" 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-id1169149375310\">\n<div data-type=\"problem\" id=\"fs-id1169148915246\">\n<p id=\"fs-id1169149113062\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a44b54f9bd1c8c6d1fe1809dea583455_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#120;&#45;&#49;&#50;&#60;&#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-id1169148851196\">\n<div data-type=\"problem\" id=\"fs-id1169146731454\">\n<p id=\"fs-id1169146617305\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7211043c9125c152733b46138983814e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#120;&#43;&#51;&#92;&#108;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"122\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169146611390\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1169149376574\" data-alt=\"The graph shown is an upward facing parabola with vertex (negative 2, negative 1) and y-intercept (0,3).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_307_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph shown is an upward facing parabola with vertex (negative 2, negative 1) and y-intercept (0,3).\" \/><\/span><\/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-e32da4e784cc207e4118f4050da12516_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#45;&#51;&#44;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#93;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"59\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149222448\">\n<div data-type=\"problem\" id=\"fs-id1169149008120\">\n<p id=\"fs-id1169148996581\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8857a231db16cb7212ef3288433b7cf2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#120;&#43;&#56;&#92;&#103;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"122\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169146642948\">\n<div data-type=\"problem\" id=\"fs-id1169149343778\">\n<p id=\"fs-id1169146741072\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-685519a2375c3ec7bcb8a1f79c5c6b2f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#120;&#43;&#49;&#56;&#92;&#108;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"131\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148992135\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1169149009812\" data-alt=\"The graph shown is a downward-facing parabola with vertex (negative 1 and 5 tenths, 20) and y-intercept (0, 18).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_309_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph shown is a downward-facing parabola with vertex (negative 1 and 5 tenths, 20) and y-intercept (0, 18).\" \/><\/span><\/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-2c727129524f5cc6e0f328328b5f8576_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#44;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#93;&#92;&#99;&#117;&#112;&#32;&#92;&#108;&#101;&#102;&#116;&#91;&#51;&#44;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"124\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149029983\">\n<div data-type=\"problem\" id=\"fs-id1169149109867\">\n<p id=\"fs-id1169149037612\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-05c182159484013a5ec56a1d83d31333_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#120;&#43;&#50;&#52;&#60;&#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-id1169148880298\">\n<div data-type=\"problem\" id=\"fs-id1169149361788\">\n<p id=\"fs-id1169149295280\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ed0301f36bf3a9584402c55915d709fa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#120;&#43;&#49;&#50;&#92;&#103;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"122\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169146618321\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1169149017568\" data-alt=\"The graph shown is a downward facing parabola with a y-intercept of (0, 12) and x-intercepts (negative 3, 0) and (4, 0).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_311_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph shown is a downward facing parabola with a y-intercept of (0, 12) and x-intercepts (negative 3, 0) and (4, 0).\" \/><\/span><\/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-d95e622d274b2bf6035b71ca11229908_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#45;&#51;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#93;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"45\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147029171\">\n<div data-type=\"problem\" id=\"fs-id1169144565964\">\n<p id=\"fs-id1169149223052\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-605b6a9ca08e7d8e98ab3f6d67fdec5f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#120;&#43;&#49;&#53;&#62;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"131\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169148933030\">In the following exercises, solve each inequality algebraically and write any solution in interval notation.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169148992200\">\n<div data-type=\"problem\" id=\"fs-id1169148992202\">\n<p id=\"fs-id1169149012746\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7f77b2512b0884d591cdd4b08b518d4f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#120;&#45;&#52;&#92;&#103;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"122\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149016572\">\n<p id=\"fs-id1169149113119\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ea2a64264defe66863949ccc7498c509_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#44;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#93;&#92;&#99;&#117;&#112;&#32;&#92;&#108;&#101;&#102;&#116;&#91;&#49;&#44;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"124\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149113889\">\n<div data-type=\"problem\" id=\"fs-id1169146719158\">\n<p id=\"fs-id1169146719160\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fbbcebdb4fd84906535d2b3f741f9734_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#120;&#45;&#54;&#92;&#108;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"113\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149370800\">\n<div data-type=\"problem\" id=\"fs-id1169149034315\">\n<p id=\"fs-id1169148967931\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-32165f88f5242cb359d73275d4e9ccfc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#55;&#120;&#43;&#49;&#48;&#60;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"131\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149305222\">\n<p id=\"fs-id1169149305224\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2da1eb750fc283f55cb9396d5536b47a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148878510\">\n<div data-type=\"problem\" id=\"fs-id1169146612846\">\n<p id=\"fs-id1169148974075\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ae31bb06ba08d4e15ce3ebdd6aaf4da6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#120;&#43;&#51;&#62;&#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-id1169149303922\">\n<div data-type=\"problem\" id=\"fs-id1169149303924\">\n<p id=\"fs-id1169149369970\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5a59bec57dc988890112e8a0551d766a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#56;&#120;&#62;&#45;&#49;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"113\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169146648690\">\n<p id=\"fs-id1169148985706\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2ccd14758eb0ab443cc12f81e5b06901_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#44;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#99;&#117;&#112;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#44;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" 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-id1169148956652\">\n<div data-type=\"problem\" id=\"fs-id1169148956654\">\n<p id=\"fs-id1169144382480\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-29eaeb27dcccc873710046b173e86443_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#56;&#120;&#60;&#45;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"113\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149100890\">\n<div data-type=\"problem\" id=\"fs-id1169149066384\">\n<p id=\"fs-id1169149066386\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2e2bb0fe7a81bf8df6a6a7b39f987490_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#120;&#43;&#50;&#92;&#108;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"122\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149346346\">\n<p id=\"fs-id1169149346190\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fd12d342a88f6e080cd5d2031d2367c1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#50;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#125;&#44;&#50;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#93;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"125\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169146741668\">\n<div data-type=\"problem\" id=\"fs-id1169146741670\">\n<p id=\"fs-id1169149341634\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9fa62ec3be37243a841eb17d809f06fe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#56;&#120;&#45;&#49;&#49;&#60;&#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-id1169148983875\">\n<div data-type=\"problem\" id=\"fs-id1169148983877\">\n<p id=\"fs-id1169148889146\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ed2b14f9667eea39ebd7edb075b1fecf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#48;&#120;&#62;&#45;&#49;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"123\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149117548\">\n<p id=\"fs-id1169148952146\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-00b04b60db3f85d5f421d8d4d24bb58b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#44;&#53;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#54;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#99;&#117;&#112;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#54;&#125;&#44;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"208\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148885396\">\n<div data-type=\"problem\" id=\"fs-id1169148885398\">\n<p id=\"fs-id1169149223155\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-93c279a803e612307cd8813dec1031ab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#120;&#60;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"105\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149123398\">\n<div data-type=\"problem\" id=\"fs-id1169149123401\">\n<p id=\"fs-id1169149219766\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-76b4a40e9eaf6aa83f892660784e3e25_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#57;&#120;&#45;&#49;&#48;&#92;&#103;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"161\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148837495\">\n<p id=\"fs-id1169148885412\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d1a97ee263646501e76f125b5f4218a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#44;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#93;&#92;&#99;&#117;&#112;&#32;&#92;&#108;&#101;&#102;&#116;&#91;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#44;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"147\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148989556\">\n<div data-type=\"problem\" id=\"fs-id1169148989558\">\n<p id=\"fs-id1169148967548\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-55b646e1765bae882038b5fe353d1b0b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#120;&#43;&#52;&#92;&#108;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"143\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149093113\">\n<div data-type=\"problem\" id=\"fs-id1169149093116\">\n<p id=\"fs-id1169149103082\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e3a6cf0e12a5d010917a2f3ee0561f88_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#55;&#120;&#43;&#52;&#92;&#103;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"143\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148871567\">\n<p id=\"fs-id1169148925172\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-da7bf76d0b29c5ac2804895dbf47643c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#93;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"36\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148996604\">\n<div data-type=\"problem\" id=\"fs-id1169148996606\">\n<p id=\"fs-id1169148983820\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0385a1c0150ad28fb90f380caacaa714_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#53;&#120;&#45;&#49;&#50;&#62;&#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-id1169149011049\">\n<div data-type=\"problem\" id=\"fs-id1169149011051\">\n<p id=\"fs-id1169149013353\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-eb372392cc86241bd891465c9ffc3f61_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#120;&#43;&#53;&#62;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"122\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149009744\">\n<p id=\"fs-id1169149009747\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5df92a2cc0a08c383e86c9bcbca032bf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#44;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"63\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148862364\">\n<div data-type=\"problem\" id=\"fs-id1169148862366\">\n<p id=\"fs-id1169149029184\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8e04a2830771444b41f4ac4bf9d477a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#120;&#43;&#54;&#92;&#108;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"122\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148988479\">\n<div data-type=\"problem\" id=\"fs-id1169148988481\">\n<p id=\"fs-id1169149009558\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-410aa078ea205d593eedbbc0f4b40e4d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#120;&#45;&#55;&#62;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"113\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148947767\">\n<p id=\"fs-id1169148947769\">no solution<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147107055\">\n<div data-type=\"problem\" id=\"fs-id1169149287002\">\n<p id=\"fs-id1169149287004\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5cb9705bf8919794bf0976aa45f7e086_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#120;&#45;&#53;&#60;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"122\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169144564505\">\n<div data-type=\"problem\" id=\"fs-id1169144564507\">\n<p id=\"fs-id1169148965224\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7e4fd7478b2414332a6544c547f41b3e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#56;&#120;&#45;&#49;&#48;&#60;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"152\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169144365729\">\n<p id=\"fs-id1169148912386\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5df92a2cc0a08c383e86c9bcbca032bf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#44;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"63\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169146655785\">\n<div data-type=\"problem\" id=\"fs-id1169148930404\">\n<p id=\"fs-id1169148930406\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ef43456861f42ca3f98f1fde04fb36a6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#120;&#45;&#55;&#92;&#103;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"122\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"writing\" data-depth=\"2\" id=\"fs-id1169149330036\">\n<h4 data-type=\"title\">Writing Exercises<\/h4>\n<div data-type=\"exercise\" id=\"fs-id1169149095709\">\n<div data-type=\"problem\" id=\"fs-id1169149008282\">\n<p id=\"fs-id1169149008284\">Explain critical points and how they are used to solve quadratic inequalities algebraically.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169144365711\">\n<p id=\"fs-id1169148969298\">Answers will vary.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169144563561\">\n<div data-type=\"problem\" id=\"fs-id1169149009433\">\n<p id=\"fs-id1169149009435\">Solve <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-62cb1d2a23a8eb91722378b3d089c090_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#120;&#92;&#103;&#101;&#32;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"91\" style=\"vertical-align: -3px;\" \/> both graphically and algebraically. Which method do you prefer, and why?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148957614\">\n<div data-type=\"problem\" id=\"fs-id1169149096351\">\n<p id=\"fs-id1169149096353\">Describe the steps needed to solve a quadratic inequality graphically.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148974155\">\n<p id=\"fs-id1169149112730\">Answers will vary.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148987788\">\n<div data-type=\"problem\" id=\"fs-id1169148959814\">\n<p id=\"fs-id1169148959817\">Describe the steps needed to solve a quadratic inequality algebraically.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1169148871792\">\n<h4 data-type=\"title\">Self Check<\/h4>\n<p id=\"fs-id1169146731597\"><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-id1169148974036\" data-alt=\"This figure is a list to assess your understanding of the concepts presented in this section. It has 4 columns labeled I can\u2026, Confidently, With some help, and No-I don\u2019t get it! Below I can\u2026, there is solve quadratic inequalities graphically and solve quadratic inequalities algebraically. The other columns are left blank for you to check you understanding.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_201_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure is a list to assess your understanding of the concepts presented in this section. It has 4 columns labeled I can\u2026, Confidently, With some help, and No-I don\u2019t get it! Below I can\u2026, there is solve quadratic inequalities graphically and solve quadratic inequalities algebraically. The other columns are left blank for you to check you understanding.\" \/><\/span><\/p>\n<p id=\"fs-id1169148970119\"><span class=\"token\">\u24d1<\/span> On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?<\/p>\n<\/div>\n<\/div>\n<div class=\"review-exercises\" data-depth=\"1\" id=\"fs-id1169148997490\">\n<h3 data-type=\"title\">Chapter Review Exercises<\/h3>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1169148993623\">\n<h4 data-type=\"title\"><a href=\"\/contents\/b9659e42-3afa-4449-81d9-a017c35de140\" class=\"target-chapter\">Solve Quadratic Equations Using the Square Root Property<\/a><\/h4>\n<p id=\"fs-id1169148935132\"><strong data-effect=\"bold\">Solve Quadratic Equations of the form <em data-effect=\"italics\">ax<\/em><sup>2<\/sup> = <em data-effect=\"italics\">k<\/em> Using the Square Root Property<\/strong><\/p>\n<p id=\"fs-id1169149007258\">In the following exercises, solve using the Square Root Property.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169149007261\">\n<div data-type=\"problem\" id=\"fs-id1169149305649\">\n<p id=\"fs-id1169149305651\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0aed8eea4223cca133c2d8ce9fee49e9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#52;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"68\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149028652\">\n<p id=\"fs-id1169149028654\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9d92b43d505ccd2e3ed17e3a1e77442e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&plusmn;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"50\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148996016\">\n<div data-type=\"problem\" id=\"fs-id1169148996018\">\n<p id=\"fs-id1169149005194\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fe2cf52b5d33aaa47ca716101be09c9f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#45;&#56;&#48;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"90\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149014399\">\n<div data-type=\"problem\" id=\"fs-id1169149014401\">\n<p id=\"fs-id1169148959976\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d65df6734cdbd1ea49c153b981be36cd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#48;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"77\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148970019\">\n<p id=\"fs-id1169149005881\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-784a5a3ab70d9fa10e32944f96f1ea0c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&plusmn;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"41\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149043513\">\n<div data-type=\"problem\" id=\"fs-id1169148956183\">\n<p id=\"fs-id1169148956186\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-415bddc218c2389eb1ef841dfcfcbed8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#61;&#55;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"65\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149143870\">\n<div data-type=\"problem\" id=\"fs-id1169149031270\">\n<p id=\"fs-id1169149031272\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bae1ef1b0d3d1b18ca460e8d5589cbc1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#114;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#50;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"88\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149094648\">\n<p id=\"fs-id1169148828209\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3eee4b053f5d02333fa3f083c2705dc5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#114;&#61;&plusmn;&#52;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#125;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"71\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149118485\">\n<div data-type=\"problem\" id=\"fs-id1169149118488\">\n<p id=\"fs-id1169149294753\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4cef4c62f948106bb7843b593b57d0e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#116;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#56;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"86\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149095532\">\n<div data-type=\"problem\" id=\"fs-id1169149095534\">\n<p id=\"fs-id1169149028835\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2a221685f1e31d2958d9bdc4e276b5c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#123;&#119;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#48;&#61;&#51;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"111\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149000087\">\n<p id=\"fs-id1169148993617\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8eccdcae622ee3e7aca56740e37e27c9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#119;&#61;&plusmn;&#53;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"69\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148870393\">\n<div data-type=\"problem\" id=\"fs-id1169149287688\">\n<p id=\"fs-id1169149287690\">11. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ff3992cb5d56100fad704a32a45ae20b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#123;&#99;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#61;&#49;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"96\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169146617977\"><strong data-effect=\"bold\">Solve Quadratic Equations of the Form <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;\" \/> Using the Square Root Property<\/strong><\/p>\n<p>In the following exercises, solve using the Square Root Property.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169146744166\">\n<div data-type=\"problem\" id=\"fs-id1169146744168\">\n<p id=\"fs-id1169149344983\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0c14710149ae30863bbeac7a13d89d74_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#112;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#61;&#49;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"132\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148957520\">\n<p id=\"fs-id1169148957522\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d94e31eeae4bbda6456225472b952e64_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#112;&#61;&#45;&#49;&#44;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"73\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169146731705\">\n<div data-type=\"problem\" id=\"fs-id1169146647727\">\n<p id=\"fs-id1169146647729\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9663a290a42242d34a071947593c9959_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#117;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#52;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"102\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148970873\">\n<div data-type=\"problem\" id=\"fs-id1169148970875\">\n<p id=\"fs-id1169149158618\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c25227cceba500f29f6e78dbb3c80716_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;&#51;&#125;&#123;&#49;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"104\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147086922\">\n<p id=\"fs-id1169147086924\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-acb282181924c663a5f27fcd7d04d1d1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&plusmn;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#125;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"65\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148890807\">\n<div data-type=\"problem\" id=\"fs-id1169146657246\">\n<p id=\"fs-id1169146657248\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f8beb40d73627b83b825c52038cdb588_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"96\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169146742770\">\n<div data-type=\"problem\" id=\"fs-id1169146644219\">\n<p id=\"fs-id1169146644221\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-866abca9a3b1a6581d52d46be97c0d51_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#110;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#48;&#61;&#49;&#53;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"152\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149123657\">\n<p id=\"fs-id1169149123659\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-366005e072aa2350350085476837c92a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#61;&#52;&plusmn;&#49;&#48;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"85\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148926362\">\n<div data-type=\"problem\" id=\"fs-id1169149319656\">\n<p id=\"fs-id1169149319659\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6922eda640f353be1e5d88ded6dcff98_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#99;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#45;&#49;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"123\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169146627605\">\n<div data-type=\"problem\" id=\"fs-id1169146627607\">\n<p id=\"fs-id1169146627609\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3e8b07ef5fb06d6966c4f930912a7128_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#48;&#110;&#43;&#50;&#53;&#61;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"148\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169144377735\">\n<p id=\"fs-id1169149141082\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4621c53ee37104e72006f56ed07ff568_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#61;&#45;&#53;&plusmn;&#50;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"90\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148963466\">\n<div data-type=\"problem\" id=\"fs-id1169148963469\">\n<p id=\"fs-id1169148963471\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4cf7113e4838e4fa6e24734898206f07_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#52;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#56;&#97;&#43;&#57;&#61;&#56;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"155\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1169149329012\">\n<h4 data-type=\"title\"><a href=\"\/contents\/3e7f365d-4885-4b7a-bb2a-4358cbc00d2e\" class=\"target-chapter\">Solve Quadratic Equations by Completing the Square<\/a><\/h4>\n<p id=\"fs-id1169148968740\"><strong data-effect=\"bold\">Solve Quadratic Equations Using Completing the Square<\/strong><\/p>\n<p id=\"fs-id1169148968745\">In the following exercises, complete the square to make a perfect square trinomial. Then write the result as a binomial squared.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169148984228\">\n<div data-type=\"problem\" id=\"fs-id1169148984230\">\n<p id=\"fs-id1169149112748\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ad866c2a810a519154c9d1f670ed9043_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#50;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"67\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149100741\">\n<p id=\"fs-id1169148843264\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-31594bb4437f87c8b416be7f93936074_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#49;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"69\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169144564417\">\n<div data-type=\"problem\" id=\"fs-id1169148966854\">\n<p id=\"fs-id1169148966856\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-311c036e18d0343c814379bc34be5da3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#45;&#56;&#109;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"69\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149011925\">\n<div data-type=\"problem\" id=\"fs-id1169149011928\">\n<p id=\"fs-id1169149011930\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4611041ba38c92618be28f13f6067419_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"57\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169146662472\">\n<p id=\"fs-id1169146662475\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ec4374fc8c4c7b62f59a017b44ddc151_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"63\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169146660855\">\n<div data-type=\"problem\" id=\"fs-id1169146660857\">\n<p id=\"fs-id1169146660859\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-22a734d7009b6210007e0bada52ea23e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#51;&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"63\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169149087768\">In the following exercises, solve by completing the square.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169149087771\">\n<div data-type=\"problem\" id=\"fs-id1169149087773\">\n<p id=\"fs-id1169149219038\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fba0d96fc7e79be023a52f6c2d09c26d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#100;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#52;&#100;&#61;&#45;&#49;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"121\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148926998\">\n<p id=\"fs-id1169148927000\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5266131dfe262baaa00bb47d8a114d97_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#100;&#61;&#45;&#49;&#51;&#44;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"94\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148948193\">\n<div data-type=\"problem\" id=\"fs-id1169148948196\">\n<p id=\"fs-id1169148970671\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b5c1f6fb05fc8eebedbd249de880eb7a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#121;&#61;&#51;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"99\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148951536\">\n<div data-type=\"problem\" id=\"fs-id1169148951538\">\n<p id=\"fs-id1169148951540\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f9cecb3918cf620b5371b4f70cca6a9e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#109;&#61;&#45;&#49;&#48;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"134\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149143890\">\n<p id=\"fs-id1169149143892\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c002f5b2e0fb85d955999db1ade0c2e9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#45;&#51;&plusmn;&#49;&#48;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"86\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148889866\">\n<div data-type=\"problem\" id=\"fs-id1169148889868\">\n<p id=\"fs-id1169148995968\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6b4c2caff3235133165db49db1f21d25_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#116;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#50;&#116;&#61;&#45;&#52;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"116\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147107888\">\n<div data-type=\"problem\" id=\"fs-id1169146653503\">\n<p id=\"fs-id1169146653505\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-39480910d862254a8851ef573cddb4aa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#52;&#118;&#61;&#45;&#51;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"120\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149361822\">\n<p id=\"fs-id1169146607654\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d39b66d517d801f7dc3fbcd0b9ec2db0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;&#61;&#55;&plusmn;&#51;&#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-id1169149018178\">\n<div data-type=\"problem\" id=\"fs-id1169149018181\">\n<p id=\"fs-id1169149018183\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-44be3c623c885d1566f381417199f486_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#119;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#48;&#119;&#61;&#49;&#48;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"124\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169146662389\">\n<div data-type=\"problem\" id=\"fs-id1169148989123\">\n<p id=\"fs-id1169148989125\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0cd2166494ff5e0768ebbfe5d6a6e058_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#48;&#109;&#45;&#52;&#61;&#45;&#49;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"164\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169146662548\">\n<p id=\"fs-id1169146662550\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-33d62ac150c8ae3fdfd56eaff681b9c5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#45;&#57;&#44;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"92\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148884917\">\n<div data-type=\"problem\" id=\"fs-id1169148884919\">\n<p id=\"fs-id1169149027712\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-145a8c046c764467bbeb3a64b18ba8f9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#110;&#43;&#49;&#49;&#61;&#51;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"141\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169146648896\">\n<div data-type=\"problem\" id=\"fs-id1169146648898\">\n<p id=\"fs-id1169146653908\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bd94d05dac21ea31b2263fb44b2e6c3a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#61;&#51;&#97;&#43;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"90\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149215120\">\n<p id=\"fs-id1169149215122\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c966c8e6875534271a3e6e963cc0f4a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#125;&plusmn;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#52;&#49;&#125;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"72\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149309533\">\n<div data-type=\"problem\" id=\"fs-id1169149309535\">\n<p id=\"fs-id1169149309537\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5ffd9bc49b595cd5fd1990ff0e43787c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#49;&#98;&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"94\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169146744234\">\n<div data-type=\"problem\" id=\"fs-id1169149007007\">\n<p id=\"fs-id1169149007010\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b1b68315d5d2fc4ab1b783ebe43912af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#117;&#43;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#117;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#49;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"153\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148998766\">\n<p id=\"fs-id1169146594818\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-11b3cce588fca140641a06c0605d02e3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#61;&#45;&#54;&plusmn;&#50;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#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-id1169149112785\">\n<div data-type=\"problem\" id=\"fs-id1169149112787\">\n<p id=\"fs-id1169149007107\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a7b0a97bb01b96606f4f8f46ce1d0194_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#122;&#45;&#49;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#122;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"159\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169149091903\"><strong data-effect=\"bold\">Solve Quadratic Equations of the form <em data-effect=\"italics\">ax<\/em><sup>2<\/sup> + <em data-effect=\"italics\">bx<\/em> + <em data-effect=\"italics\">c<\/em> = 0 by Completing the Square<\/strong><\/p>\n<p id=\"fs-id1169148927233\">In the following exercises, solve by completing the square.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169144566586\">\n<div data-type=\"problem\" id=\"fs-id1169144566588\">\n<p id=\"fs-id1169144566590\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d41375aa9f668294480dd544ae0c5933_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#56;&#112;&#43;&#49;&#53;&#61;&#49;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"154\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169146637275\">\n<p id=\"fs-id1169146637277\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cb87c99f521fd05760aaf2dbd4a44495_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#112;&#61;&#48;&#44;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"59\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149373207\">\n<div data-type=\"problem\" id=\"fs-id1169149373210\">\n<p id=\"fs-id1169149373212\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7af3061dd28433700973353b7cdb2313_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#123;&#113;&#125;&#94;&#123;&#50;&#125;&#43;&#55;&#48;&#113;&#43;&#50;&#48;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"145\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149376741\">\n<div data-type=\"problem\" id=\"fs-id1169148988547\">\n<p id=\"fs-id1169148988549\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-68bbf0c7e5993162bc008f90c3e5f54e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#121;&#61;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"98\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149007395\">\n<p id=\"fs-id1169149007398\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-50730d1bf7834e519206be1e5d1e0d2f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#44;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"73\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148925211\">\n<div data-type=\"problem\" id=\"fs-id1169149335772\">\n<p id=\"fs-id1169149335774\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6aaea746c148e70f33f0a2022d95f4d0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#120;&#61;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"100\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148879311\">\n<div data-type=\"problem\" id=\"fs-id1169148879313\">\n<p id=\"fs-id1169148879315\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1db219d7fd3d86998ad12eb82cd1edc7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#123;&#99;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#99;&#61;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"95\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148908999\">\n<p id=\"fs-id1169148909001\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a565490ae18647523014278a3ef95c9e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&plusmn;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#92;&#115;&#113;&#114;&#116;&#123;&#55;&#125;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"84\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169144564314\">\n<div data-type=\"problem\" id=\"fs-id1169144564317\">\n<p id=\"fs-id1169149121477\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e7dfe9442a860fdc859c4277f076cfb5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#100;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#100;&#61;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"98\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149003467\">\n<div data-type=\"problem\" id=\"fs-id1169149003469\">\n<p id=\"fs-id1169144379514\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-06f01da8aa138b9c852eb86c5c61b237_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#120;&#61;&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"113\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148995457\">\n<p id=\"fs-id1169149124152\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-65e0095bb5a4e92e8f7935b19bfe2c6e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#125;&plusmn;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"61\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148871025\">\n<div data-type=\"problem\" id=\"fs-id1169148871027\">\n<p id=\"fs-id1169148871029\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e1b5382ac878cc2c718ea153466fac55_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#120;&#61;&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"113\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1169149005906\">\n<h4 data-type=\"title\"><a href=\"\/contents\/f045a37e-bf7c-4818-95a1-e29172da48b4\" class=\"target-chapter\">Solve Quadratic Equations Using the Quadratic Formula<\/a><\/h4>\n<p id=\"fs-id1169148983922\">In the following exercises, solve by using the Quadratic Formula.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169148983925\">\n<div data-type=\"problem\" id=\"fs-id1169148983927\">\n<p id=\"fs-id1169149037735\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a29f775c698e278932d9863866748228_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#120;&#43;&#49;&#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-id1169148891295\">\n<p id=\"fs-id1169148891297\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9af923dc732590592e1487782ec1e511_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#44;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"60\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148984100\">\n<div data-type=\"problem\" id=\"fs-id1169148984102\">\n<p id=\"fs-id1169148984104\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-da18b1030458590e77cf5ec1809b48af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#121;&#45;&#51;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"129\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169146740292\">\n<div data-type=\"problem\" id=\"fs-id1169146740294\">\n<p id=\"fs-id1169144545760\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-75be3e0579d2b8da77eb742df34d8f7f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#114;&#125;&#94;&#123;&#50;&#125;&#45;&#114;&#45;&#52;&#50;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"118\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149121257\">\n<p id=\"fs-id1169149014927\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-57bc7a9d4fd2979f7394da80b1eaeb4a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#114;&#61;&#45;&#54;&#44;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"72\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149013093\">\n<div data-type=\"problem\" id=\"fs-id1169149121079\">\n<p id=\"fs-id1169149121081\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-97a9defe75e1c8cedcdee499eead79ba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#116;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#51;&#116;&#43;&#50;&#50;&#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-id1169148968116\">\n<div data-type=\"problem\" id=\"fs-id1169148968118\">\n<p id=\"fs-id1169148867792\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f994fa6c3582e90d7e456877f51726b3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#118;&#125;&#94;&#123;&#50;&#125;&#43;&#118;&#45;&#53;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"120\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148947416\">\n<p id=\"fs-id1169148947418\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fea34abc84b28ff94445757ab9a36499_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#49;&plusmn;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#49;&#125;&#125;&#123;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"78\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169146651115\">\n<div data-type=\"problem\" id=\"fs-id1169146651117\">\n<p id=\"fs-id1169146651119\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6571ba9bdafd6062d14490050c06e825_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#123;&#119;&#125;&#94;&#123;&#50;&#125;&#43;&#57;&#119;&#43;&#50;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"137\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148938507\">\n<div data-type=\"problem\" id=\"fs-id1169148938509\">\n<p id=\"fs-id1169148938511\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4fb077644b5df857dc144a859016898a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#43;&#56;&#109;&#43;&#50;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"141\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169146654033\">\n<p id=\"fs-id1169146654035\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5693d626f8c8bef74759df65c9f14cc4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#52;&plusmn;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#48;&#125;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"84\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148912005\">\n<div data-type=\"problem\" id=\"fs-id1169148912007\">\n<p id=\"fs-id1169148912009\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-92b365f1aeebfb62b84792773cafca75_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#110;&#45;&#49;&#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-id1169149114464\">\n<div data-type=\"problem\" id=\"fs-id1169149114466\">\n<p id=\"fs-id1169149114468\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3a00a4be117b47d6f3375903950479f0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#97;&#43;&#50;&#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-id1169149156972\">\n<p id=\"fs-id1169149156974\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bc3207968ae647dfb2e96d45db6b3715_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#49;&#50;&#125;&plusmn;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#51;&#125;&#125;&#123;&#49;&#50;&#125;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"86\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149116251\">\n<div data-type=\"problem\" id=\"fs-id1169149108716\">\n<p id=\"fs-id1169149108718\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5054e66e167131dc9647a3c99195967f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#45;&#98;&#43;&#56;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"117\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149313134\">\n<div data-type=\"problem\" id=\"fs-id1169149220735\">\n<p id=\"fs-id1169149220737\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ba2029d4630325491c2a45b2127cdd57_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#92;&#108;&#101;&#102;&#116;&#40;&#117;&#45;&#49;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#51;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"140\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148926228\">\n<p id=\"fs-id1169148926230\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9408aa418e8ce3fe4de8400da9d9f3bb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#61;&#53;&plusmn;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"75\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149293737\">\n<div data-type=\"problem\" id=\"fs-id1169149293739\">\n<p id=\"fs-id1169149293741\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bf413ce036ec1258033d6cbc12e8bb9e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#122;&#92;&#108;&#101;&#102;&#116;&#40;&#122;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"107\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148971298\">\n<div data-type=\"problem\" id=\"fs-id1169148971300\">\n<p id=\"fs-id1169148971302\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c0cddcda9443f87ec6bb3043e108ba6f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#56;&#125;&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#53;&#125;&#112;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"120\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149025180\">\n<p id=\"fs-id1169149025182\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9b548ac9a1661427e9ebde85b69ed695_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#112;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&plusmn;&#92;&#115;&#113;&#114;&#116;&#123;&#53;&#125;&#125;&#123;&#53;&#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-id1169149308978\">\n<div data-type=\"problem\" id=\"fs-id1169149065758\">\n<p id=\"fs-id1169149065761\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cc5c2e81882ca91ddd66ee890bfd76e9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#53;&#125;&#123;&#113;&#125;&#94;&#123;&#50;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#49;&#48;&#125;&#113;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"112\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148870452\">\n<div data-type=\"problem\" id=\"fs-id1169148870454\">\n<p id=\"fs-id1169148870456\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a83bdefe8ee205a92a6f1cb8aec7b358_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#99;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#99;&#43;&#49;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"126\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149154819\">\n<p id=\"fs-id1169149154821\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b725f8ce0609963cd45880ad77bc101d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#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-id1169144603954\">\n<div data-type=\"problem\" id=\"fs-id1169144603956\">\n<p id=\"fs-id1169144603958\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e802372b72c6f72767c9615ec7916a8b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#123;&#100;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#50;&#100;&#61;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"121\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169149285662\"><strong data-effect=\"bold\">Use the Discriminant to Predict the Number of Solutions of a Quadratic Equation<\/strong><\/p>\n<p id=\"fs-id1169149285668\">In the following exercises, determine the number of solutions for each quadratic equation.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169146651299\">\n<div data-type=\"problem\" id=\"fs-id1169146651301\">\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-6e4d68607c115cd36ac66b4e47e2041c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#120;&#43;&#49;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"131\" 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-389ec3059081f38f8671099de5f03285_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#56;&#121;&#43;&#49;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"129\" 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-87bfabb03eb141dafad5bdb4b66ec155_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#50;&#109;&#43;&#52;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"150\" style=\"vertical-align: -2px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d3<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-74d71cd26519f261c9d6720c054ba77d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#45;&#110;&#43;&#49;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"123\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149024795\">\n<p id=\"fs-id1169149024797\"><span class=\"token\">\u24d0<\/span> 1 <span class=\"token\">\u24d1<\/span> 2 <span class=\"token\">\u24d2<\/span> 2 <span class=\"token\">\u24d3<\/span> 2<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169146606787\">\n<div data-type=\"problem\" id=\"fs-id1169146606789\">\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-86c2359dbd3ef29abc97d4fc5cb52d02_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#55;&#120;&#45;&#56;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"131\" style=\"vertical-align: 0px;\" \/><\/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-f2ef3600b198e84f37c70810f42b4754_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#48;&#120;&#43;&#53;&#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\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-44d71b26d6d8fdcfb02e6ebfc59d1126_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#53;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#57;&#48;&#120;&#43;&#56;&#49;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"157\" style=\"vertical-align: -2px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d3<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-41068f1f12c58d5b3480aa23421441f2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#53;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#56;&#120;&#43;&#52;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"138\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169149349068\"><strong data-effect=\"bold\">Identify the Most Appropriate Method to Use to Solve a Quadratic Equation<\/strong><\/p>\n<p id=\"fs-id1169149109241\">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-id1169149109246\">\n<div data-type=\"problem\" id=\"fs-id1169144365602\">\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-0170958ff5871fea42194a16aa31a086_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#54;&#123;&#114;&#125;&#94;&#123;&#50;&#125;&#45;&#56;&#114;&#43;&#49;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"135\" 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-89fc0215811240dd4f4f904ff92e14f2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#123;&#116;&#125;&#94;&#123;&#50;&#125;&#45;&#56;&#116;&#43;&#51;&#61;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"123\" 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-f329c77ee66e5e240fc5892db3575f88_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#99;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"109\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149027651\">\n<p id=\"fs-id1169149027654\"><span class=\"token\">\u24d0<\/span> factor <span class=\"token\">\u24d1<\/span> Quadratic Formula <span class=\"token\">\u24d2<\/span> square root<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149302439\">\n<div data-type=\"problem\" id=\"fs-id1169149302441\">\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-8220108b2cbcc3585cab1845259e3a4b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#100;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#48;&#100;&#45;&#53;&#61;&#50;&#49;\" 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-fdc41b551c7a804be0cd9173c3388a8c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#53;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#48;&#120;&#43;&#51;&#54;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"157\" 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-6278d57bdd27cf5514b953f02f1292c1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#118;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#53;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"130\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1169149109187\">\n<h4 data-type=\"title\"><a href=\"\/contents\/48820a47-b89e-428f-9534-a71207245a16\" class=\"target-chapter\">Solve Equations in Quadratic Form<\/a><\/h4>\n<p id=\"fs-id1169149003863\"><strong data-effect=\"bold\">Solve Equations in Quadratic Form<\/strong><\/p>\n<p id=\"fs-id1169149370495\">In the following exercises, solve.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169149370498\">\n<div data-type=\"problem\" id=\"fs-id1169149370500\">\n<p id=\"fs-id1169149370290\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-64ef9a29e7505daf6e4f214934da16e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#45;&#49;&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#52;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"147\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149373720\">\n<p id=\"fs-id1169149373722\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1514bce3dca266cb5626ef0143f0485c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&plusmn;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#125;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#61;&plusmn;&#50;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"139\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169146661162\">\n<div data-type=\"problem\" id=\"fs-id1169146661722\">\n<p id=\"fs-id1169146661724\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fc6ec321eb150cb41c97e1f16467ae49_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#43;&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#50;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"138\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169146656250\">\n<div data-type=\"problem\" id=\"fs-id1169146656252\">\n<p id=\"fs-id1169146656254\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-32dbb7975812cd20031684a90ed82f13_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#45;&#53;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"138\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169146657125\">\n<p id=\"fs-id1169146657127\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5bfaf41cf17114565b37294b9adcf286_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&plusmn;&#49;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#61;&plusmn;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"100\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149376033\">\n<div data-type=\"problem\" id=\"fs-id1169149376035\">\n<p id=\"fs-id1169149375166\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-78f09f51bcd9e8ea939c5458065a4e21_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#121;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#121;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#50;&#56;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"237\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169146637673\">\n<div data-type=\"problem\" id=\"fs-id1169146637675\">\n<p id=\"fs-id1169146637677\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9a561c92f20310eaeb572edbae399804_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#43;&#51;&#92;&#115;&#113;&#114;&#116;&#123;&#120;&#125;&#45;&#50;&#56;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"138\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169146637854\">\n<p id=\"fs-id1169146637856\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d64a5cf83814d9888bfab8bf27403b02_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#49;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"52\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169146643865\">\n<div data-type=\"problem\" id=\"fs-id1169146643867\">\n<p id=\"fs-id1169146643869\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a8bc5bcc0d47a10110eca4f8f828f53f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#120;&#43;&#53;&#92;&#115;&#113;&#114;&#116;&#123;&#120;&#125;&#45;&#54;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"138\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169146644998\">\n<div data-type=\"problem\" id=\"fs-id1169146645048\">\n<p id=\"fs-id1169146645050\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cf9c13d101bb0e412adb19e4f314e566_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#125;&#45;&#49;&#48;&#123;&#120;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#125;&#43;&#50;&#52;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"153\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148934926\">\n<p id=\"fs-id1169146652899\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-91c35c2bab78d384b35da78cde0f7507_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#54;&#52;&#44;&#120;&#61;&#50;&#49;&#54;\" 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-id1169149367711\">\n<div data-type=\"problem\" id=\"fs-id1169149367713\">\n<p id=\"fs-id1169149367688\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b327be6c928a7f5fe50843366cf02d80_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#43;&#55;&#123;&#120;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#125;&#43;&#54;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"125\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149376147\">\n<div data-type=\"problem\" id=\"fs-id1169149376149\">\n<p id=\"fs-id1169149376151\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e2018b793c6f89d2554bee3ceddbe4d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#123;&#120;&#125;&#94;&#123;&#45;&#50;&#125;&#45;&#50;&#123;&#120;&#125;&#94;&#123;&#45;&#49;&#125;&#45;&#51;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"160\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149003331\">\n<p id=\"fs-id1169149330215\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c3a5bf3bd0efa2b1e3cabf2dc9dba920_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#50;&#44;&#120;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"107\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1169149009036\">\n<h4 data-type=\"title\"><a href=\"\/contents\/b6bba63f-4f97-49ab-ac8a-01c3927118a7\" class=\"target-chapter\">Solve Applications of Quadratic Equations<\/a><\/h4>\n<p id=\"fs-id1169146731708\"><strong data-effect=\"bold\">Solve Applications Modeled by Quadratic Equations<\/strong><\/p>\n<p id=\"fs-id1169146731714\">In the following exercises, solve by using the method of factoring, the square root principle, or the Quadratic Formula. Round your answers to the nearest tenth, if needed.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169149349822\">\n<div data-type=\"problem\" id=\"fs-id1169149349824\">\n<p id=\"fs-id1169149349826\">Find two consecutive odd numbers whose product is 323.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149335626\">\n<div data-type=\"problem\" id=\"fs-id1169149335628\">\n<p id=\"fs-id1169149335630\">Find two consecutive even numbers whose product is 624.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149156873\">\n<p id=\"fs-id1169149156875\">Two consecutive even numbers whose product is 624 are 24 and 26, and \u221224 and \u221226.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149094500\">\n<div data-type=\"problem\" id=\"fs-id1169149094502\">\n<p id=\"fs-id1169149094504\">A triangular banner has an area of 351 square centimeters. The length of the base is two centimeters longer than four times the height. Find the height and length of the base.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149123561\">\n<div data-type=\"problem\" id=\"fs-id1169149123563\">\n<p id=\"fs-id1169149123565\">Julius built a triangular display case for his coin collection. The height of the display case is six inches less than twice the width of the base. The area of the of the back of the case is 70 square inches. Find the height and width of the case.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148895521\">\n<p id=\"fs-id1169148895523\">The height is 14 inches and the width is 10 inches.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148989338\">\n<div data-type=\"problem\" id=\"fs-id1169148989340\">\n<p id=\"fs-id1169148989342\">A tile mosaic in the shape of a right triangle is used as the corner of a rectangular pathway. The hypotenuse of the mosaic is 5 feet. One side of the mosaic is twice as long as the other side. What are the lengths of the sides? Round to the nearest tenth.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1169147029064\" data-alt=\"A rectangle is shown is a right triangle in the corner. The hypotenuse of the triangle is 5 feet, the longer leg is 2 times s and the shorter leg is s.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_202_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"A rectangle is shown is a right triangle in the corner. The hypotenuse of the triangle is 5 feet, the longer leg is 2 times s and the shorter leg is s.\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149370246\">\n<div data-type=\"problem\" id=\"fs-id1169149370248\">\n<p id=\"fs-id1169148962548\">A rectangular piece of plywood has a diagonal which measures two feet more than the width. The length of the plywood is twice the width. What is the length of the plywood\u2019s diagonal? Round to the nearest tenth.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149152558\">\n<p id=\"fs-id1169149152561\">The length of the diagonal is 3.6 feet.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149152566\">\n<div data-type=\"problem\" id=\"fs-id1169149116008\">\n<p id=\"fs-id1169149116010\">The front walk from the street to Pam\u2019s house has an area of 250 square feet. Its length is two less than four times its width. Find the length and width of the sidewalk. Round to the nearest tenth.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149338846\">\n<div data-type=\"problem\" id=\"fs-id1169149338848\">\n<p id=\"fs-id1169149338850\">For Sophia\u2019s graduation party, several tables of the same width will be arranged end to end to give serving table with a total area of 75 square feet. The total length of the tables will be two more than three times the width. Find the length and width of the serving table so Sophia can purchase the correct size tablecloth . Round answer to the nearest tenth.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149087629\">\n<p id=\"fs-id1169149087631\">The width of the serving table is 4.7 feet and the length is 16.1 feet.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1169149087636\" data-alt=\"Four tables arranged end-to-end are shown. Together, they have an area of 75 feet. The short side measures w and the long side measures 3 times w plus 2.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_203_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Four tables arranged end-to-end are shown. Together, they have an area of 75 feet. The short side measures w and the long side measures 3 times w plus 2.\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149017955\">\n<div data-type=\"problem\" id=\"fs-id1169149017957\">\n<p id=\"fs-id1169149017959\">A ball is thrown vertically in the air with a velocity of 160 ft\/sec. Use the formula <em data-effect=\"italics\">h<\/em> = \u221216<em data-effect=\"italics\">t<\/em><sup>2<\/sup> + <em data-effect=\"italics\">v<\/em><sub>0<\/sub><em data-effect=\"italics\">t<\/em> to determine when the ball will be 384 feet from the ground. Round to the nearest tenth.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148927292\">\n<div data-type=\"problem\" id=\"fs-id1169148927295\">\n<p id=\"fs-id1169148927297\">The couple took a small airplane for a quick flight up to the wine country for a romantic dinner and then returned home. The plane flew a total of 5 hours and each way the trip was 360 miles. If the plane was flying at 150 mph, what was the speed of the wind that affected the plane?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149011762\">\n<p id=\"fs-id1169149011764\">The speed of the wind was 30 mph.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149004012\">\n<div data-type=\"problem\" id=\"fs-id1169149004014\">\n<p id=\"fs-id1169149004016\">Ezra kayaked up the river and then back in a total time of 6 hours. The trip was 4 miles each way and the current was difficult. If Roy kayaked at a speed of 5 mph, what was the speed of the current?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149033290\">\n<div data-type=\"problem\" id=\"fs-id1169149033292\">\n<p id=\"fs-id1169149033294\">Two handymen can do a home repair in 2 hours if they work together. One of the men takes 3 hours more than the other man to finish the job by himself. How long does it take for each handyman to do the home repair individually?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169144382246\">\n<p>One man takes 3 hours and the other man 6 hours to finish the repair alone.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1169148969732\">\n<h4 data-type=\"title\"><a href=\"\/contents\/3de1ae34-6225-4751-be75-a17b3e0e665b\" class=\"target-chapter\">Graph Quadratic Functions Using Properties<\/a><\/h4>\n<p id=\"fs-id1169148929691\"><strong data-effect=\"bold\">Recognize the Graph of a Quadratic Function<\/strong><\/p>\n<p id=\"fs-id1169148929696\">In the following exercises, graph by plotting point.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169144551404\">\n<div data-type=\"problem\" id=\"fs-id1169144551406\">\n<p id=\"fs-id1169144551408\">Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0f2ee10627b4218eb9d908345891e998_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"80\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148991591\">\n<div data-type=\"problem\" id=\"fs-id1169148991593\">\n<p id=\"fs-id1169148991595\">Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f702e64f794ed2487c3a39f60323567e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"81\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149089320\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1169149089323\" data-alt=\"This figure shows a downward-opening parabola on the x y-coordinate plane. It has a vertex of (3, 0) and other points of (negative 2, negative 1) and (2, negative 1).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_314_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a downward-opening parabola on the x y-coordinate plane. It has a vertex of (3, 0) and other points of (negative 2, negative 1) and (2, negative 1).\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169149065516\">In the following exercises, determine if the following parabolas open up or down.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169149065519\">\n<div data-type=\"problem\" id=\"fs-id1169146655769\">\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-e98599c0384d6177d31b6f1f11e4afb5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#120;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"144\" 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-1957ce16abf99359080cc36395cd1f78_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#53;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#120;&#43;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"131\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149230086\">\n<div data-type=\"problem\" id=\"fs-id1169149230089\">\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-56cd314db7454711c085901fa63d6dbf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#56;&#120;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"121\" 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-ee9f2af0a2d3cf034c90adb2bd4e73d7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#55;&#120;&#43;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"144\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149095454\">\n<p id=\"fs-id1169149095457\"><span class=\"token\">\u24d0<\/span> up <span class=\"token\">\u24d1<\/span> down<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169149095361\"><strong data-effect=\"bold\">Find the Axis of Symmetry and Vertex of a Parabola<\/strong><\/p>\n<p id=\"fs-id1169148994045\">In the following exercises, find <span class=\"token\">\u24d0<\/span> the equation of the axis of symmetry and <span class=\"token\">\u24d1<\/span> the vertex.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169148955071\">\n<div data-type=\"problem\" id=\"fs-id1169148955073\">\n<p id=\"fs-id1169148955075\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-788758f574a5c4fa0b8126d8956f0e25_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#120;&#43;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"122\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149285597\">\n<div data-type=\"problem\" id=\"fs-id1169149369843\">\n<p id=\"fs-id1169149369845\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-301d95c3ed149cdb2feabec73ee903a5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#56;&#120;&#43;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"130\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149211429\">\n<p id=\"fs-id1169148939472\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d409841138637fb7212bba05d7694ccc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#50;&#59;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"103\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169146633836\"><strong data-effect=\"bold\">Find the Intercepts of a Parabola<\/strong><\/p>\n<p id=\"fs-id1169144603974\">In the following exercises, find the <em data-effect=\"italics\">x<\/em>&#8211; and <em data-effect=\"italics\">y<\/em>-intercepts.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169149007215\">\n<div data-type=\"problem\" id=\"fs-id1169149007217\">\n<p id=\"fs-id1169149007219\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-21f8e4c325a30eee5e0abc953f04c840_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#120;&#43;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"121\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148869070\">\n<div data-type=\"problem\" id=\"fs-id1169148869072\">\n<p id=\"fs-id1169148869075\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7d9cc581ea0379f47e5213612adc1f2b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#56;&#120;&#43;&#49;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"130\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149341584\">\n<p id=\"fs-id1169146617675\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-64174122eff202cf439bd46e4e871107_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;&#121;&#58;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#49;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#58;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#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=\"114\" style=\"vertical-align: -15px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148959755\">\n<div data-type=\"problem\" id=\"fs-id1169148959757\">\n<p id=\"fs-id1169148984708\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9392ab6cefe8afa3722982c22ce0976a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#120;&#43;&#49;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"131\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149038744\">\n<div data-type=\"problem\" id=\"fs-id1169149038746\">\n<p id=\"fs-id1169149038748\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a9465e4281afd857b462615cd6f20fad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#53;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#48;&#120;&#45;&#52;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"163\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149025866\">\n<p id=\"fs-id1169149025868\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-03dde4ad338691bb4ed9a707ea3b72b1_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;&#121;&#58;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#45;&#52;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#58;&#92;&#116;&#101;&#120;&#116;&#123;&#110;&#111;&#110;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"86\" style=\"vertical-align: -11px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148963942\">\n<div data-type=\"problem\" id=\"fs-id1169148963944\">\n<p id=\"fs-id1169148963946\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-29d428a504a8c23f4a37ffcbc00d84e8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#49;&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#56;&#120;&#43;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"139\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149285285\">\n<div data-type=\"problem\" id=\"fs-id1169149285288\">\n<p id=\"fs-id1169149285290\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ea34edcd9e31bbc47f2da7ed6c7ecc4e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#54;&#120;&#43;&#54;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"140\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149109297\">\n<p id=\"fs-id1169149109299\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fed65829ef901d7eddd49ec07ed758ba_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;&#121;&#58;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#45;&#54;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#58;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#56;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#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=\"86\" style=\"vertical-align: -15px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169148963400\"><strong data-effect=\"bold\">Graph Quadratic Functions Using Properties<\/strong><\/p>\n<p id=\"fs-id1169148924491\">In the following exercises, graph by using its properties.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169148924494\">\n<div data-type=\"problem\" id=\"fs-id1169148924496\">\n<p id=\"fs-id1169148924498\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8e1f5f5b736d532f45046414d9f59bc4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#56;&#120;&#43;&#49;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"130\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149370164\">\n<div data-type=\"problem\" id=\"fs-id1169149370167\">\n<p id=\"fs-id1169149370169\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-35c43a723513c318b60f862e960f2b35_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#120;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"122\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169146657085\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1169146657090\" data-alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (1, negative 4) and a y-intercept of (0, negative 3).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_316_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (1, negative 4) and a y-intercept of (0, negative 3).\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149024631\">\n<div data-type=\"problem\" id=\"fs-id1169149024633\">\n<p id=\"fs-id1169144564487\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0a9ffd37dcf6d28b93c5dc684836f904_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#56;&#120;&#45;&#49;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"131\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149309999\">\n<div data-type=\"problem\" id=\"fs-id1169149310001\">\n<p id=\"fs-id1169149310003\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ac0c7c1c61105f72f2e95dda8940c2a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#120;&#43;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"130\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148995767\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1169148995770\" data-alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (one-half, 0) and a y-intercept of (0, 1).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_318_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (one-half, 0) and a y-intercept of (0, 1).\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149349320\">\n<div data-type=\"problem\" id=\"fs-id1169149349322\">\n<p id=\"fs-id1169149349324\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-87f5347d689b45f9ac7bce29ad348f42_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#120;&#43;&#49;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"131\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148996221\">\n<div data-type=\"problem\" id=\"fs-id1169148996223\">\n<p id=\"fs-id1169148996225\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4e1c141b5239e0cce00fbfe6e730374a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#56;&#120;&#45;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"153\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169146655944\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1169146655947\" data-alt=\"This figure shows a downward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 2, negative 4) and a y-intercept of (0, negative 12).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_320_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a downward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 2, negative 4) and a y-intercept of (0, negative 12).\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169149291277\"><strong data-effect=\"bold\">Solve Maximum and Minimum Applications<\/strong><\/p>\n<p id=\"fs-id1169149039976\">In the following exercises, find the minimum or maximum value.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169149039979\">\n<div data-type=\"problem\" id=\"fs-id1169149039981\">\n<p id=\"fs-id1169149039983\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5aab7328c5d286fd2ca8ce09b6fe7c35_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#55;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#52;&#120;&#43;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"140\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149029750\">\n<div data-type=\"problem\" id=\"fs-id1169149029752\">\n<p id=\"fs-id1169146660811\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9ab0e3b6ac9f45aa2e4a1fbc45ef3344_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#50;&#120;&#45;&#49;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"163\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147028313\">\n<p id=\"fs-id1169147028316\">The maximum value is 2 when <em data-effect=\"italics\">x<\/em> = 2.<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169148957096\">In the following exercises, solve. Rounding answers to the nearest tenth.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169149336292\">\n<div data-type=\"problem\" id=\"fs-id1169149336294\">\n<p id=\"fs-id1169149336296\">A ball is thrown upward from the ground with an initial velocity of 112 ft\/sec. Use the quadratic equation <em data-effect=\"italics\">h<\/em> = \u221216<em data-effect=\"italics\">t<\/em><sup>2<\/sup> + 112<em data-effect=\"italics\">t<\/em> to find how long it will take the ball to reach maximum height, and then find the maximum height.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148972750\">\n<div data-type=\"problem\" id=\"fs-id1169148972752\">\n<p id=\"fs-id1169148969278\">A daycare facility is enclosing a rectangular area along the side of their building for the children to play outdoors. They need to maximize the area using 180 feet of fencing on three sides of the yard. The quadratic equation <em data-effect=\"italics\">A<\/em> = \u22122<em data-effect=\"italics\">x<\/em><sup>2<\/sup> + 180<em data-effect=\"italics\">x<\/em> gives the area, <em data-effect=\"italics\">A<\/em>, of the yard for the length, <em data-effect=\"italics\">x<\/em>, of the building that will border the yard. Find the length of the building that should border the yard to maximize the area, and then find the maximum area.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1169149292285\" data-alt=\"An odd-shaped figure is given. 3 sides of a rectangle are attached to the right side of the figure.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_204_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"An odd-shaped figure is given. 3 sides of a rectangle are attached to the right side of the figure.\" \/><\/span><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149114447\">\n<p id=\"fs-id1169149114449\">The length adjacent to the building is 90 feet giving a maximum area of 4,050 square feet.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1169144376789\">\n<h4 data-type=\"title\"><a href=\"\/contents\/9054900c-f191-40ed-8ceb-e4f051bbbf2b\" class=\"target-chapter\">Graph Quadratic Functions Using Transformations<\/a><\/h4>\n<p id=\"fs-id1169148909034\"><strong data-effect=\"bold\">Graph Quadratic Functions of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b95f9d4f6bb7a0e3c081b6c10cf2355c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#107;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"110\" style=\"vertical-align: -4px;\" \/><\/strong><\/p>\n<p id=\"fs-id1169149030734\">In the following exercises, graph each function using a vertical shift.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169149030738\">\n<div data-type=\"problem\" id=\"fs-id1169149030740\">\n<p id=\"fs-id1169149214960\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-13dcead0d565a36990c4814adee92a10_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"108\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169146740423\">\n<div data-type=\"problem\" id=\"fs-id1169146744005\">\n<p id=\"fs-id1169146744007\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-91cf66e3e69af2e0399e37bebe15729d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"109\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148939925\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1169148939929\" data-alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 3, 0) and other points of (negative 1, negative 2) and (1, negative 2).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_322_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 3, 0) and other points of (negative 1, negative 2) and (1, negative 2).\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169146744134\">In the following exercises, graph each function using a horizontal shift.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169146744137\">\n<div data-type=\"problem\" id=\"fs-id1169146744139\">\n<p id=\"fs-id1169146744141\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ff11c96722d86aa2cd983ab3718cd00b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"123\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147085150\">\n<div data-type=\"problem\" id=\"fs-id1169147085152\">\n<p id=\"fs-id1169147085155\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-caad0418523c5ad535f196eb8182b0f7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"121\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149230329\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1169148923644\" data-alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (3, 0) and other points of (2, 1) and (4,1).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_324_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (3, 0) and other points of (2, 1) and (4,1).\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169149345843\">In the following exercises, graph each function using transformations.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169149345846\">\n<div data-type=\"problem\" id=\"fs-id1169149345848\">\n<p id=\"fs-id1169149345850\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c16e0ae8c0d08949bc04bb1ad8d446a0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"154\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149114367\">\n<div data-type=\"problem\" id=\"fs-id1169149114369\">\n<p id=\"fs-id1169149114372\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-84d67c11a1d2ed787150fee30d180706_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"153\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149220778\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1169146740834\" data-alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 3, negative 2) and other points of (negative 5, 2) and (negative 1, 2).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_326_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 3, negative 2) and other points of (negative 5, 2) and (negative 1, 2).\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148843323\">\n<div data-type=\"problem\" id=\"fs-id1169148843325\">\n<p id=\"fs-id1169148843327\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-70101a80c71bce7e50d00610770ee337_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"154\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149043690\">\n<div data-type=\"problem\" id=\"fs-id1169149043692\">\n<p id=\"fs-id1169149043694\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d2e513acee98fc835ff36953d9069d6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"154\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148937884\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1169149222724\" data-alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (4, negative 3) and other points of (3, negative 2) and (5, negative 2).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_328_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (4, negative 3) and other points of (3, negative 2) and (5, negative 2).\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169144376807\"><strong data-effect=\"bold\">Graph Quadratic Functions of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0e5ce7ec07dcf04f843c0416726078af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#97;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"87\" style=\"vertical-align: -4px;\" \/><\/strong><\/p>\n<p id=\"fs-id1169148972924\">In the following exercises, graph each function.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169148972927\">\n<div data-type=\"problem\" id=\"fs-id1169149032750\">\n<p id=\"fs-id1169149032752\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-71b9184acf1857940c069ac7852062d9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"87\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147107658\">\n<div data-type=\"problem\" id=\"fs-id1169148963022\">\n<p id=\"fs-id1169148963024\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1be1d57a2ab45be04aa1b45d2427657e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"78\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147028990\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1169147028995\" data-alt=\"This figure shows a downward-opening parabola on the x y-coordinate plane. It has a vertex of (0, 0) and other points of (negative 1, negative 1) and (1, negative 1).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_330_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a downward-opening parabola on the x y-coordinate plane. It has a vertex of (0, 0) and other points of (negative 1, negative 1) and (1, negative 1).\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169146744334\">\n<div data-type=\"problem\" id=\"fs-id1169147028998\">\n<p id=\"fs-id1169149304791\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-395f327411241134f1dd2a8ec1f36227_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"89\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169149287025\"><strong data-effect=\"bold\">Graph Quadratic Functions Using Transformations<\/strong><\/p>\n<p id=\"fs-id1169149172194\">In the following exercises, rewrite each function in the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-06540d3a6d2073911c6201af3d16da60_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#97;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#104;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#107;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"165\" style=\"vertical-align: -4px;\" \/> form by completing the square.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169148969375\">\n<div data-type=\"problem\" id=\"fs-id1169148969377\">\n<p id=\"fs-id1169148875162\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a9b78ddbba38f755fcaa4c490acb538d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#120;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"159\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149038778\">\n<p id=\"fs-id1169149038780\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-51cce692e104c77b95f94b34516fa030_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#45;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"163\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148990578\">\n<div data-type=\"problem\" id=\"fs-id1169148990580\">\n<p id=\"fs-id1169148990582\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7fe421afd42a94e13c500a6edb57b5bc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#50;&#120;&#43;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"168\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169149110351\">In the following exercises, <span class=\"token\">\u24d0<\/span> rewrite each function in <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-06540d3a6d2073911c6201af3d16da60_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#97;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#104;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#107;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"165\" style=\"vertical-align: -4px;\" \/> form and <span class=\"token\">\u24d1<\/span> graph it by using transformations.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169148994595\">\n<div data-type=\"problem\" id=\"fs-id1169148994597\">\n<p id=\"fs-id1169146815184\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5f15a9be6bdd54556c4507265027a79a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#120;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"158\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149155501\">\n<p id=\"fs-id1169146631384\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9b0b8d52f6401661c3b184da62c87a43_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#51;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"163\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1169148965135\" data-alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (1, negative 4) and other points of (0, negative 1) and (2, negative 1).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_332_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (1, negative 4) and other points of (0, negative 1) and (2, negative 1).\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148987024\">\n<div data-type=\"problem\" id=\"fs-id1169148987026\">\n<p id=\"fs-id1169148987028\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e8515e413ef90f7b26400c6775ce2328_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#50;&#120;&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"181\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149346707\">\n<div data-type=\"problem\" id=\"fs-id1169149346709\">\n<p id=\"fs-id1169149346711\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c8b3c6d94be5e15cc3037ab977b91fcd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#120;&#43;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"159\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149341018\">\n<p id=\"fs-id1169149341020\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a1be3d09803199c5e3c4f5c107797700_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"163\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1169149293605\" data-alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 1, 4) and other points of (negative 2, 6) and (0, 6).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_334_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 1, 4) and other points of (negative 2, 6) and (0, 6).\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148947324\">\n<div data-type=\"problem\" id=\"fs-id1169148947326\">\n<p id=\"fs-id1169148947329\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-eb892015195b7a94a6365d651ca86282_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#50;&#120;&#43;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"168\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169149195662\">In the following exercises, <span class=\"token\">\u24d0<\/span> rewrite each function in <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-06540d3a6d2073911c6201af3d16da60_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#97;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#104;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#107;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"165\" style=\"vertical-align: -4px;\" \/> form and <span class=\"token\">\u24d1<\/span> graph it using properties.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169149220010\">\n<div data-type=\"problem\" id=\"fs-id1169149220012\">\n<p id=\"fs-id1169149220014\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9a36bb1e62f1b6c7787181f4bb1a5536_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#50;&#120;&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"181\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149012893\">\n<p id=\"fs-id1169149012896\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-42f763c91a397e23fe0efd8687375c77_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#51;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"177\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1169149001060\" data-alt=\"This figure shows a downward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 2, 7) and other points of (negative 4, negative 5) and (0, negative 5).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_336_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a downward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 2, 7) and other points of (negative 4, negative 5) and (0, negative 5).\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149015898\">\n<div data-type=\"problem\" id=\"fs-id1169149015900\">\n<p id=\"fs-id1169149000933\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ec581d44e31ac7e9d86bea17b858ad67_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#50;&#120;&#43;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"168\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169149104832\"><strong data-effect=\"bold\">Find a Quadratic Function from its Graph<\/strong><\/p>\n<p id=\"fs-id1171791302843\">In the following exercises, write the quadratic function in <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-06540d3a6d2073911c6201af3d16da60_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#97;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#104;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#107;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"165\" style=\"vertical-align: -4px;\" \/> form.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169148911498\">\n<div data-type=\"problem\" id=\"fs-id1169149007350\"><span data-type=\"media\" id=\"fs-id1169149007352\" data-alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 1, negative 1) and other points of (negative 2, negative 4) and (0, negative 4).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_205_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 1, negative 1) and other points of (negative 2, negative 4) and (0, negative 4).\" \/><\/span><\/div>\n<div data-type=\"solution\" id=\"fs-id1169144566540\">\n<p id=\"fs-id1169144566542\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-03edc2357a2e442a1d968de8ffc04044_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"153\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148938157\">\n<div data-type=\"problem\" id=\"fs-id1169148938159\"><span data-type=\"media\" id=\"fs-id1169148938161\" data-alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (2, 4) and other points of (0, 8) and (4, 8).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_206_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (2, 4) and other points of (0, 8) and (4, 8).\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1169148924238\">\n<h4 data-type=\"title\"><a href=\"\/contents\/4f54d544-fb3e-4336-9f8c-57e6b8489257\" class=\"target-chapter\">Solve Quadratic Inequalities<\/a><\/h4>\n<p id=\"fs-id1169144565221\"><strong data-effect=\"bold\">Solve Quadratic Inequalities Graphically<\/strong><\/p>\n<p id=\"fs-id1169144565227\">In the following exercises, solve graphically and write the solution in interval notation.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169144565232\">\n<div data-type=\"problem\" id=\"fs-id1169149008556\">\n<p id=\"fs-id1169149008559\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4e5429d8538c31ceec0d0f36228999b0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#120;&#45;&#54;&#62;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"113\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149107840\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1169149107850\" data-alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (one-half, negative 6 and one-fourth) and other points of (0, negative 6) and (1, negative 6).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_338_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (one-half, negative 6 and one-fourth) and other points of (0, negative 6) and (1, negative 6).\" \/><\/span><\/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-eca6b75ba39af6abaefa0e2d64b7c35c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#44;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#99;&#117;&#112;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" 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-id1169149115800\">\n<div data-type=\"problem\" id=\"fs-id1169149115802\">\n<p id=\"fs-id1169149115804\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7211043c9125c152733b46138983814e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#120;&#43;&#51;&#92;&#108;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"122\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149100215\">\n<div data-type=\"problem\" id=\"fs-id1169149100218\">\n<p id=\"fs-id1169149100220\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a58aa233457478bcb48357ddf429066c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#120;&#43;&#50;&#92;&#103;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"113\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148952525\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1169149008950\" data-alt=\"This figure shows a downward-opening parabola on the x y-coordinate plane. It has a vertex of (negative one-half, 2 and one-fourth) and other points of (negative 2, 0) and (1, 0).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_340_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a downward-opening parabola on the x y-coordinate plane. It has a vertex of (negative one-half, 2 and one-fourth) and other points of (negative 2, 0) and (1, 0).\" \/><\/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-8901d72060072320fe2225af30739ea6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#45;&#50;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#93;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"45\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149003165\">\n<div data-type=\"problem\" id=\"fs-id1169149003167\">\n<p id=\"fs-id1169149006401\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-97e382dcaedf8ab7791fe8bceffc099e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#120;&#43;&#51;&#60;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"122\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169149042015\">In the following exercises, solve each inequality algebraically and write any solution in interval notation.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169149042019\">\n<div data-type=\"problem\" id=\"fs-id1169149042022\">\n<p id=\"fs-id1169149042024\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bf33ffa63a06dafad984244d3005dc16_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#120;&#43;&#56;&#60;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"122\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169146659084\">\n<p id=\"fs-id1169146669955\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-65c84a7e1d884e51e9ff8e8338318a74_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149015035\">\n<div data-type=\"problem\" id=\"fs-id1169149015037\">\n<p id=\"fs-id1169149015040\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-308404c6340f5035301319ea1d9e7157_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#120;&#62;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"90\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148962536\">\n<div data-type=\"problem\" id=\"fs-id1169148962538\">\n<p id=\"fs-id1169148962540\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-95b5fee331ae765a04e33b7a02e4e004_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#120;&#43;&#52;&#92;&#108;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"122\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149096065\">\n<p id=\"fs-id1169149096067\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8ca40d2ff2b8f251519a9104a1ecaa13_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#51;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#53;&#125;&#44;&#51;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#53;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#93;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"125\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149112768\">\n<div data-type=\"problem\" id=\"fs-id1169149112770\">\n<p id=\"fs-id1169149112772\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-11982b93417b1604407eeab3a53778bd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#55;&#120;&#45;&#52;&#62;&#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-id1169149028228\">\n<div data-type=\"problem\" id=\"fs-id1169149033349\">\n<p id=\"fs-id1169149033351\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-18f64f2627f337ffd2462e6e739830f2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#120;&#45;&#54;&#62;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"113\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149144392\">\n<p id=\"fs-id1169149144394\">no solution<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149144400\">\n<div data-type=\"problem\" id=\"fs-id1169148962968\">\n<p id=\"fs-id1169148962970\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-56afa5b7390419b7a09162609b6f11ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#120;&#43;&#52;&#92;&#103;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"122\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"practice-test\" data-depth=\"1\" id=\"fs-id1169149012734\">\n<h3 data-type=\"title\">Practice Test<\/h3>\n<div data-type=\"exercise\" id=\"fs-id1169149012741\">\n<div data-type=\"problem\" id=\"fs-id1169146742654\">\n<p id=\"fs-id1169146742656\">Use the Square Root Property to solve the quadratic equation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0670f154cd91de5e6e65d74403fdf531_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#119;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#50;&#55;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"120\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149143992\">\n<p id=\"fs-id1169149143994\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3e0507077c28d05fcd0e1ada130902f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#119;&#61;&#45;&#50;&#44;&#119;&#61;&#45;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"127\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169144565579\">\n<div data-type=\"problem\" id=\"fs-id1169144565582\">\n<p id=\"fs-id1169144565584\">Use Completing the Square to solve the quadratic equation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-132aea6f1e978b21ea38638aedc2afee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#56;&#97;&#43;&#55;&#61;&#50;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"133\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148879638\">\n<div data-type=\"problem\" id=\"fs-id1169149094530\">\n<p id=\"fs-id1169149094532\">Use the Quadratic Formula to solve the quadratic equation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-edede2b2cd5e289c415d5ed0c948bf97_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#109;&#43;&#51;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"145\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169146741118\">\n<p id=\"fs-id1169146741120\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ad7605136a9887fd07721ef11d621b69_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#49;&#44;&#109;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"104\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169149279941\">Solve the following quadratic equations. Use any method.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169149295358\">\n<div data-type=\"problem\" id=\"fs-id1169149295360\">\n<p id=\"fs-id1169149295362\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-28d382f2fc41bfb8442466cbdf4c0ce3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#120;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#120;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#49;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"148\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149121563\">\n<div data-type=\"problem\" id=\"fs-id1169149121566\">\n<p id=\"fs-id1169149121568\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-89e98421a7d98ee866c42640dffa7bb5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#125;&#123;&#52;&#125;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#121;&#43;&#49;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"129\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149311826\">\n<p id=\"fs-id1169149156327\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d85ed4bb0138f3c213ad63cf5527e6d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"42\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169149029934\">Use the discriminant to determine the number and type of solutions of each quadratic equation.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169149029938\">\n<div data-type=\"problem\" id=\"fs-id1169149029940\">\n<p id=\"fs-id1169149122254\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e312122438b5c7987af4a519782f4849_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#51;&#112;&#43;&#55;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"137\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149095062\">\n<div data-type=\"problem\" id=\"fs-id1169149095064\">\n<p id=\"fs-id1169149345203\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-019f2d186ff3691019a48523b0a40c5b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#123;&#113;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#48;&#113;&#43;&#49;&#50;&#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-id1169148958290\">\n<p id=\"fs-id1169148958292\">2 complex<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169148958297\">Solve each equation.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169149295913\">\n<div data-type=\"problem\" id=\"fs-id1169149295915\">\n<p id=\"fs-id1169149295917\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c6fe5017255902437ef65243db92e83a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#45;&#49;&#55;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"147\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148879788\">\n<div data-type=\"problem\" id=\"fs-id1169148879790\">\n<p id=\"fs-id1169148879792\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-586b24fb41b033a8e5fd4c290a1f2b07_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#125;&#43;&#50;&#123;&#121;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#125;&#45;&#51;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"134\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148870657\">\n<p id=\"fs-id1169148870659\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b483f385bae5923b6e4ad978e4cbdf13_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#49;&#44;&#121;&#61;&#45;&#50;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"115\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169148995102\">For each parabola, find <span class=\"token\">\u24d0<\/span> which direction it opens, <span class=\"token\">\u24d1<\/span> the equation of the axis of symmetry, <span class=\"token\">\u24d2<\/span> the vertex, <span class=\"token\">\u24d3<\/span> the <em data-effect=\"italics\">x-<\/em> and <em data-effect=\"italics\">y<\/em>-intercepts, and e) the maximum or minimum value.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169146669108\">\n<div data-type=\"problem\" id=\"fs-id1169146669110\">\n<p id=\"fs-id1169146669112\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e7f29fe8ae0bb094eabd948a4d2798f3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#120;&#43;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"131\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148967539\">\n<div data-type=\"problem\" id=\"fs-id1169148967542\">\n<p id=\"fs-id1169148967544\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7a1e78512fe3f06bd90e69d5d4ad04a3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#56;&#120;&#43;&#49;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"131\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148908757\">\n<p id=\"fs-id1169149042133\"><span class=\"token\">\u24d0<\/span> down <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b3dc975a98ccada6f136856736d7df06_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"57\" 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-91357ad9f837f690ca370a0ee126647e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d3<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-638dbfec40453787752f95dbfbb29f9e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#92;&#116;&#101;&#120;&#116;&#123;&#58;&#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;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#49;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#116;&#101;&#120;&#116;&#123;&#59;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#92;&#116;&#101;&#120;&#116;&#123;&#58;&#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;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"154\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d4<\/span> minimum value of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-00b9cce9021441b203ec0271d72e6ba2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"22\" style=\"vertical-align: -1px;\" \/> when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8203ced39e0cdafefa708857c7ec2264_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169149140297\">Graph each quadratic function using intercepts, the vertex, and the equation of the axis of symmetry.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169149140300\">\n<div data-type=\"problem\" id=\"fs-id1169149140302\">\n<p id=\"fs-id1169149140304\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-47db8493bf2e7a6a9b78561328da8867_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#120;&#43;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"150\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149115844\">\n<div data-type=\"problem\" id=\"fs-id1169149103441\">\n<p id=\"fs-id1169149103443\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6b27a25a53d7b42bdcba626a524ccc3e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#56;&#120;&#43;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"173\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147029418\"><span data-type=\"media\" id=\"fs-id1169149033999\" data-alt=\"This figure shows a downward-opening parabola on the x y-coordinate plane. It has a vertex of (2, 12) and other points of (0, 4) and (4, 4).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_343_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a downward-opening parabola on the x y-coordinate plane. It has a vertex of (2, 12) and other points of (0, 4) and (4, 4).\" \/><\/span><\/div>\n<\/div>\n<p id=\"fs-id1169149013976\">In the following exercises, graph each function using transformations.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169149013979\">\n<div data-type=\"problem\" id=\"fs-id1169149013982\">\n<p id=\"fs-id1169149013984\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1ab6b4326151c57ab225866e646937ee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"153\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149306709\">\n<div data-type=\"problem\" id=\"fs-id1169149306711\">\n<p id=\"fs-id1169149306714\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7015a216e57812b39feb70f545a02ce5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#120;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"149\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148951771\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1169149009270\" data-alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (2, negative 5) and other points of (0, negative 1) and (4, negative 1).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_09_08_345_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (2, negative 5) and other points of (0, negative 1) and (4, negative 1).\" \/><\/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-51cce692e104c77b95f94b34516fa030_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#45;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"163\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169148925023\">In the following exercises, solve each inequality algebraically and write any solution in interval notation.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169148925027\">\n<div data-type=\"problem\" id=\"fs-id1169148925029\">\n<p id=\"fs-id1169149279506\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e4f30341ca77fea4c49ccfae480a440c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#120;&#45;&#56;&#92;&#108;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"122\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149110365\">\n<div data-type=\"problem\" id=\"fs-id1169149110367\">\n<p id=\"fs-id1169146742867\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-99aa92e41ed9a993a4d8900db96f0687_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#120;&#45;&#49;&#48;&#62;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"131\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148999276\">\n<p id=\"fs-id1169148999278\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0b053332d4ea35f5a4f21fdb55398f04_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#44;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#99;&#117;&#112;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"132\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169146629166\">Model the situation with a quadratic equation and solve by any method.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169146629169\">\n<div data-type=\"problem\" id=\"fs-id1169146629171\">\n<p id=\"fs-id1169146629173\">Find two consecutive even numbers whose product is 360.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149312273\">\n<div data-type=\"problem\" id=\"fs-id1169149312276\">\n<p id=\"fs-id1169149294719\">The length of a diagonal of a rectangle is three more than the width. The length of the rectangle is three times the width. Find the length of the diagonal. (Round to the nearest tenth.)<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149294725\">\n<p id=\"fs-id1169149294727\">The diagonal is 3.8 units long.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149009528\">\n<div data-type=\"problem\" id=\"fs-id1169149009530\">\n<p id=\"fs-id1169149009532\">A water balloon is launched upward at the rate of 86 ft\/sec. Using the formula <em data-effect=\"italics\">h<\/em> = \u221216<em data-effect=\"italics\">t<\/em><sup>2<\/sup> + 86<em data-effect=\"italics\">t<\/em> find how long it will take the balloon to reach the maximum height, and then find the maximum height. Round to the nearest tenth.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"glossary\" class=\"textbox shaded\">\n<h3 data-type=\"glossary-title\">Glossary<\/h3>\n<dl id=\"fs-id1169148990888\">\n<dt>quadratic inequality<\/dt>\n<dd id=\"fs-id1169148990891\">A quadratic inequality is an inequality that contains a quadratic expression.<\/dd>\n<\/dl>\n<\/div>\n","protected":false},"author":103,"menu_order":9,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-4978","chapter","type-chapter","status-publish","hentry"],"part":3677,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/4978","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":1,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/4978\/revisions"}],"predecessor-version":[{"id":5084,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/4978\/revisions\/5084"}],"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\/4978\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/media?parent=4978"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapter-type?post=4978"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/contributor?post=4978"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/license?post=4978"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}