# How do you solve x² + 2x - 15 ≤ 0?

Apr 15, 2018

The solution is $x \in \left[- 5 , 3\right]$

#### Explanation:

The inequality is

${x}^{2} + 2 x - 15 \le 0$

$\left(x + 5\right) \left(x - 3\right) \le 0$

Let $f \left(x\right) = \left(x + 5\right) \left(x - 3\right)$

Build a sign chart

$\textcolor{w h i t e}{a a a a}$$x$$\textcolor{w h i t e}{a a a a}$$- \infty$$\textcolor{w h i t e}{a a a a a}$$- 5$$\textcolor{w h i t e}{a a a a a a a a}$$3$$\textcolor{w h i t e}{a a a a a a}$$+ \infty$

$\textcolor{w h i t e}{a a a a}$$x + 5$$\textcolor{w h i t e}{a a a a a}$$-$$\textcolor{w h i t e}{a a a}$$0$$\textcolor{w h i t e}{a a a}$$+$$\textcolor{w h i t e}{a a a a a a a}$$+$

$\textcolor{w h i t e}{a a a a}$$x - 3$$\textcolor{w h i t e}{a a a a a}$$-$$\textcolor{w h i t e}{a a a a}$color(white)(aaa)-$\textcolor{w h i t e}{a a a}$$0$$\textcolor{w h i t e}{a a a}$$+$

$\textcolor{w h i t e}{a a a a}$$f \left(x\right)$$\textcolor{w h i t e}{a a a a a a}$$+$$\textcolor{w h i t e}{a a a a}$$0$$\textcolor{w h i t e}{a a}$$-$$\textcolor{w h i t e}{a a a}$$0$$\textcolor{w h i t e}{a a a}$$+$

Therefore,

$f \left(x\right) \le 0$ when $x \in \left[- 5 , 3\right]$

graph{x^2+2x-15 [-36.52, 36.52, -18.26, 18.28]}