1.1.4 Solving Quadratic Inequalities

In this topic we will learn how to:

solve quadratic inequalities in one unknown

To be able to solve a quadratic inequality, you should be able to solve a quadratic equation. The first step in solving a quadratic inequality is finding the roots of the quadratic, by solving it. Once you have the roots of the quadratic, you can then solve the inequality. There are four possible quadratic inequalities:

1. When the coefficient of $x^{2}$ is positive and we are looking for the less than region.
$ax^{2} + bx + c < 0$ Let’s look at an example.

Example 1
Solve the quadratic inequality $x^{2} - 6x + 8 < 0$
$x^{2} - 6x + 8 = 0$ Start by finding the roots of the quadratic,
$(x - 2)(x - 4) = 0$ $x = 2, x = 4$ Sketch the graph of the quadratic inequality to identify the wanted region,

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Therefore, the solution is,
$2 < x < 4$

2. When the coefficient of $x^{2}$ is positive and we are looking for the greater than region.
$ax^{2} + bx + c > 0$ Let’s look at an example.

Example 2
Solve the quadratic inequality $x^{2} - 6x + 8 > 0$
$x^{2} - 6x + 8 = 0$ Start by finding the roots of the quadratic,
$(x - 2)(x - 4) = 0$ $x = 2, x = 4$ Sketch the graph of the quadratic inequality to identify the wanted region,

Rendered by QuickLaTeX.com

Therefore, the solution is,
$x < 2,\ \ x > 4$

3. When the coefficient of $x^{2}$ is negative and we are looking for the greater than region.
$-ax^{2} + bx + c > 0$ Let’s look at an example.

Example 3
Solve the quadratic inequality $-x^{2} + 6x - 8 > 0$
$-x^{2} + 6x - 8 = 0$ Start by finding the roots of the quadratic,
$(x - 2)(4 - x) = 0$ $x = 2, x = 4$ Sketch the graph of the quadratic inequality to identify the wanted region,

Rendered by QuickLaTeX.com

Therefore, the solution is,
$2 < x < 4$

4. When the coefficient of $x^{2}$ is negative and we are looking for the less than region.
$-ax^{2} + bx + c < 0$ Let’s look at an example.

Example 4
Solve the quadratic inequality $-x^{2} + 6x - 8 < 0$
$-x^{2} + 6x - 8 = 0$ Start by finding the roots of the quadratic,
$(x - 2)(4 - x) = 0$ $x = 2, x = 4$ Sketch the graph of the quadratic inequality to identify the wanted region,

Rendered by QuickLaTeX.com

Therefore, the solution is,
$x < 2,\ \ x > 4$ Note: The third and fourth cases can be converted into the first and second cases respectively, by multiplying the whole quadratic inequality by $-1$ . So in reality, there are only two scenarios at play.

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