# How do you solve y^2-3y-9<=0 using a sign chart?

Dec 16, 2016

The answer is $y \in \left[\frac{3 - \sqrt{45}}{2} , \frac{3 + \sqrt{45}}{2}\right]$

#### Explanation:

Let $f \left(y\right) = {y}^{2} - 3 y - 9$

First we need the roots of the equation

${y}^{2} - 3 y - 9 = 0$

We calculate the discriminant

$\Delta = {b}^{2} - 4 a c = {\left(- 3\right)}^{2} - 4 \cdot 1 + \left(- 9\right) = 45$

As Delta>0#. we have 2 real roots

$y = \frac{3 \pm \sqrt{\Delta}}{2}$

${y}_{2} = \frac{3 + \sqrt{45}}{2}$

${y}_{1} = \frac{3 - \sqrt{45}}{2}$

Now we can make the sign chart

$\textcolor{w h i t e}{a a a a}$$y$$\textcolor{w h i t e}{a a a a}$$- \infty$$\textcolor{w h i t e}{a a a a}$${y}_{1}$$\textcolor{w h i t e}{a a a a}$${y}_{2}$$\textcolor{w h i t e}{a a a a}$$+ \infty$

$\textcolor{w h i t e}{a a a a}$$y - {y}_{1}$$\textcolor{w h i t e}{a a a a}$$-$$\textcolor{w h i t e}{a a a a}$$+$$\textcolor{w h i t e}{a a a a}$$+$

$\textcolor{w h i t e}{a a a a}$$y - {y}_{2}$$\textcolor{w h i t e}{a a a a}$$-$$\textcolor{w h i t e}{a a a a}$$-$$\textcolor{w h i t e}{a a a a}$$+$

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

Therefore,

$f \left(y \le 0\right)$ when $y \in \left[\frac{3 - \sqrt{45}}{2} , \frac{3 + \sqrt{45}}{2}\right]$