Learning Objectives
By the end of this section, you will be able to:
 Solve quadratic inequalities graphically
 Solve quadratic inequalities algebraically
Be Prepared 9.22
Before you get started, take this readiness quiz.
Solve: $2x3=0.$
If you missed this problem, review Example 2.2.
Be Prepared 9.23
Solve: $2{y}^{2}+y=15$.
If you missed this problem, review Example 6.45.
Be Prepared 9.24
Solve $\frac{1}{{x}^{2}+2x8}>0$
If you missed this problem, review Example 7.56.
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.
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.
We will solve quadratic inequalities two ways—both graphically and algebraically.
Solve Quadratic Inequalities Graphically
A quadratic equation is in standard form when written as ax^{2} + bx + c = 0. If we replace the equal sign with an inequality sign, we have a quadratic inequality in standard form.
Quadratic Inequality
A quadratic inequality is an inequality that contains a quadratic expression.
The standard form of a quadratic inequality is written:
The graph of a quadratic function f(x) = ax^{2} + bx + c = 0 is a parabola. When we ask when is ax^{2} + bx + c < 0, we are asking when is f(x) < 0. We want to know when the parabola is below the xaxis.
When we ask when is ax^{2} + bx + c > 0, we are asking when is f(x) > 0. We want to know when the parabola is above the xaxis.
Example 9.64
How to Solve a Quadratic Inequality Graphically
Solve ${x}^{2}6x+8<0$ graphically. Write the solution in interval notation.
Solution
Try It 9.127
ⓐ Solve ${x}^{2}+2x8<0$ graphically and ⓑ write the solution in interval notation.
Try It 9.128
ⓐ Solve ${x}^{2}8x+12\ge 0$ graphically and ⓑ write the solution in interval notation.
We list the steps to take to solve a quadratic inequality graphically.
How To
Solve a quadratic inequality graphically.
 Step 1. Write the quadratic inequality in standard form.
 Step 2. Graph the function $f\left(x\right)=a{x}^{2}+bx+c.$
 Step 3. Determine the solution from the graph.
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 xaxis but note how the position of the parabola affects the solution.
Example 9.65
Solve $\text{\u2212}{x}^{2}8x12\le 0$ graphically. Write the solution in interval notation.
Solution
The quadratic inequality in standard form.  ${x}^{2}8x12\le 0$  
Graph the function $f\left(x\right)=\text{\u2212}{x}^{2}8x12$.  The parabola opens downward. 

Find the line of symmetry.  $\phantom{\rule{1.8em}{0ex}}x=\frac{b}{2a}$ $\phantom{\rule{1.8em}{0ex}}x=\frac{8}{2(\mathrm{1})}$ $\phantom{\rule{1.8em}{0ex}}x=\mathrm{4}$ 

Find the vertex.  $\phantom{\rule{0.65em}{0ex}}f(x)=\text{\u2212}{x}^{2}8x12$ $f(\mathrm{4})=\text{\u2212}{(\mathrm{4})}^{2}8(\mathrm{4})12$ $f(\mathrm{4})=\mathrm{16}+3212$ $f(\mathrm{4})=4$ Vertex $\left(\mathrm{4},4\right)$ 

Find the xintercepts. Let $f\left(x\right)=0$.  $\phantom{\rule{0.75em}{0ex}}f(x)=\text{\u2212}{x}^{2}8x12$ $\phantom{\rule{1.95em}{0ex}}0=\text{\u2212}{x}^{2}8x12$ 

Factor. Use the Zero Product Property. 
$\phantom{\rule{1.95em}{0ex}}0=\mathrm{1}(x+6)(x+2)$ $\phantom{\rule{2em}{0ex}}x=\mathrm{6}\phantom{\rule{1.5em}{0ex}}x=\mathrm{2}$ 

Graph the parabola.  xintercepts $\left(\mathrm{6},0\right),\left(\mathrm{2},0\right)$ 

Determine the solution from the graph. We include the xintercepts as the inequality is “less than or equal to.” 
$(\text{\u2212}\infty ,\phantom{\rule{0.2em}{0ex}}\text{\u2212}6]\cup [\text{\u2212}2,\phantom{\rule{0.2em}{0ex}}\infty )$ 
Try It 9.129
ⓐ Solve $\text{\u2212}{x}^{2}6x5>0$ graphically and ⓑ write the solution in interval notation.
Try It 9.130
ⓐ Solve $\text{\u2212}{x}^{2}+10x16\le 0$ graphically and ⓑ write the solution in interval notation.
Solve Quadratic Inequalities Algebraically
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.
We will use the critical points 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.
Example 9.66
How To Solve Quadratic Inequalities Algebraically
Solve ${x}^{2}x12\ge 0$ algebraically. Write the solution in interval notation.
Solution
Try It 9.131
Solve ${x}^{2}+2x8\ge 0$ algebraically. Write the solution in interval notation.
Try It 9.132
Solve ${x}^{2}2x15\le 0$ algebraically. Write the solution in interval notation.
In this example, since the expression ${x}^{2}x12$ 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:
The result is the same as we found using the other method.
We summarize the steps here.
How To
Solve a quadratic inequality algebraically.
 Step 1. Write the quadratic inequality in standard form.
 Step 2. Determine the critical points—the solutions to the related quadratic equation.
 Step 3. Use the critical points to divide the number line into intervals.
 Step 4. Above the number line show the sign of each quadratic expression using test points from each interval substituted into the original inequality.
 Step 5. Determine the intervals where the inequality is correct. Write the solution in interval notation.
Example 9.67
Solve ${x}^{2}+6x7\ge 0$ algebraically. Write the solution in interval notation.
Solution
Write the quadratic inequality in standard form.  ${x}^{2}+6x7\ge 0$ 
Multiply both sides of the inequality by $\mathrm{1}$. Remember to reverse the inequality sign. 
$\phantom{\rule{0.7em}{0ex}}{x}^{2}6x+7\le 0$ 
Determine the critical points by solving the related quadratic equation. 
$\phantom{\rule{0.7em}{0ex}}{x}^{2}6x+7=0$ 
Write the Quadratic Formula.  $\phantom{\rule{4.9em}{0ex}}x=\frac{b\pm \sqrt{{b}^{2}4ac}}{2a}$ 
Then substitute in the values of $a,b,c$.  $\phantom{\rule{4.9em}{0ex}}x=\frac{(\mathrm{6})\pm \sqrt{{(\mathrm{6})}^{2}4\cdot 1\cdot (7)}}{2\cdot 1}$ 
Simplify.  $\phantom{\rule{4.9em}{0ex}}x=\frac{6\pm \sqrt{8}}{2}$ 
Simplify the radical.  $\phantom{\rule{4.9em}{0ex}}x=\frac{6\pm 2\sqrt{2}}{2}$ 
Remove the common factor, 2.  $\phantom{\rule{4.9em}{0ex}}x=\frac{2\left(3\pm \sqrt{2}\right)}{2}$ $\phantom{\rule{4.9em}{0ex}}x=3\pm \sqrt{2}$ $\phantom{\rule{4.9em}{0ex}}x=3+\sqrt{2}\phantom{\rule{2em}{0ex}}x=3\sqrt{2}$ $\phantom{\rule{4.9em}{0ex}}x\approx 1.6\phantom{\rule{3.35em}{0ex}}x\approx 4.4$ 
Use the critical points to divide the number line into intervals. Test numbers from each interval in the original inequality. 

Determine the intervals where the inequality is correct. Write the solution in interval notation. 
$\text{\u2212}{x}^{2}+6x7\ge 0$ in the middle interval $\left[3\sqrt{2},\phantom{\rule{0.5em}{0ex}}3+\sqrt{2}\right]$ 
Try It 9.133
Solve $\text{\u2212}{x}^{2}+2x+1\ge 0$ algebraically. Write the solution in interval notation.
Try It 9.134
Solve $\text{\u2212}{x}^{2}+8x14<0$ algebraically. Write the solution in interval notation.
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 ax^{2} + bx + c = 0. These two solutions then gave us either the two xintercepts for the graph or the two critical points to divide the number line into intervals.
This correlates to our previous discussion of the number and type of solutions to a quadratic equation using the discriminant.
For a quadratic equation of the form ax^{2} + bx + c = 0, $a\ne 0.$
The last row of the table shows us when the parabolas never intersect the xaxis. Using the Quadratic Formula to solve the quadratic equation, the radicand is a negative. We get two complex solutions.
In the next example, the quadratic inequality solutions will result from the solution of the quadratic equation being complex.
Example 9.68
Solve, writing any solution in interval notation:
ⓐ ${x}^{2}3x+4>0$ ⓑ ${x}^{2}3x+4\le 0$
Solution
ⓐ
Write the quadratic inequality in standard form.  $\phantom{\rule{0.8em}{0ex}}{x}^{2}3x+4>0$ 
Determine the critical points by solving the related quadratic equation. 
$\phantom{\rule{1.5em}{0ex}}{x}^{2}3x+4=0$ 
Write the Quadratic Formula.  $x=\frac{b\pm \sqrt{{b}^{2}4ac}}{2a}$ 
Then substitute in the values of $a,b,c$.  $x=\frac{(\text{\u2212}3)\pm \sqrt{{(\text{\u2212}3)}^{2}4\cdot 1\cdot (4)}}{2\cdot 1}$ 
Simplify.  $x=\frac{3\pm \sqrt{7}}{2}$ 
Simplify the radicand.  $x=\frac{3\pm \sqrt{7}i}{2}$ 
The complex solutions tell us the parabola does not intercept the xaxis. Also, the parabola opens upward. This tells us that the parabola is completely above the xaxis. 
Complex solutions 
We are to find the solution to ${x}^{2}3x+4>0.$ Since for all values of $x$ the graph is above the xaxis, all values of x make the inequality true. In interval notation we write $\left(\text{\u2212}\infty ,\infty \right).$
ⓑ
Write the quadratic inequality in standard form.  ${x}^{2}3x+4\le 0$ 
Determine the critical points by solving the related quadratic equation  ${x}^{2}3x+4=0$ 
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 xaxis—no part of it is below the xaxis.
We are to find the solution to ${x}^{2}3x+4\le 0.$ Since for all values of x the graph is never below the xaxis, no values of x make the inequality true. There is no solution to the inequality.
Try It 9.135
Solve and write any solution in interval notation:
ⓐ $\text{\u2212}{x}^{2}+2x4\le 0$ ⓑ $\text{\u2212}{x}^{2}+2x4\ge 0$
Try It 9.136
Solve and write any solution in interval notation:
ⓐ ${x}^{2}+3x+3<0$ ⓑ ${x}^{2}+3x+3>0$
Section 9.8 Exercises
Practice Makes Perfect
Solve Quadratic Inequalities Graphically
In the following exercises, ⓐ solve graphically and ⓑ write the solution in interval notation.
${x}^{2}+4x12<0$
${x}^{2}6x+8\ge 0$
$\text{\u2212}{x}^{2}+2x+24<0$
$\text{\u2212}{x}^{2}+2x+15>0$
In the following exercises, solve each inequality algebraically and write any solution in interval notation.
${x}^{2}+x6\le 0$
${x}^{2}4x+3>0$
${x}^{2}+8x<12$
$\text{\u2212}{x}^{2}+8x11<0$
${x}^{2}+6x<3$
$\mathrm{3}{x}^{2}4x+4\le 0$
$2{x}^{2}+5x12>0$
${x}^{2}3x+6\le 0$
$\text{\u2212}{x}^{2}4x5<0$
$\text{\u2212}{x}^{2}+2x7\ge 0$
Writing Exercises
Solve ${x}^{2}+2x\ge 8$ both graphically and algebraically. Which method do you prefer, and why?
Describe the steps needed to solve a quadratic inequality algebraically.
Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ On a scale of 110, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?