# How do you solve x^2-2x-5<=0 using a sign chart?

Oct 30, 2016

The answer is $1 - \sqrt{6} \le x \le 1 + \sqrt{6}$

#### Explanation:

First we start by solving $y = {x}^{2} - 2 x - 5 = 0$
We calculate $\Delta = {b}^{2} - 4 a c = 4 + 20 = 24$

So the roots are $= \frac{2 \pm \sqrt{24}}{2} = \frac{2 \pm 2 \sqrt{6}}{2}$
so the roots are $1 + \sqrt{6}$ and $1 - \sqrt{6}$
We can do the sign chart

$x$$\textcolor{w h i t e}{a a a a a a}$$- \infty$$\textcolor{w h i t e}{a a a}$$1 - \sqrt{6}$$\textcolor{w h i t e}{a a a}$$1 + \sqrt{6}$$\textcolor{w h i t e}{a a a}$$+ \infty$
$1 - \sqrt{6}$$\textcolor{w h i t e}{a a a a}$$-$$\textcolor{w h i t e}{a a a}$$0$$\textcolor{w h i t e}{a a a a}$$+$$\textcolor{w h i t e}{a a a a a}$$+$
$1 + \sqrt{6}$$\textcolor{w h i t e}{a a a a}$$-$$\textcolor{w h i t e}{a a a}$color(white)(aaaaa)-$\textcolor{w h i t e}{a a a a a}$$+$
$y$$\textcolor{w h i t e}{a a a a a a a a a}$$+$$\textcolor{w h i t e}{a a a a a a a a}$$-$$\textcolor{w h i t e}{a a a a a}$$+$$\textcolor{w h i t e}{a a a a a}$

So ${x}^{2} - 2 x - 5 \le 0$ when $1 - \sqrt{6} \le x \le 1 + \sqrt{6}$