# How do you solve 2x^2+5x-12>=0?

Nov 30, 2016

The answer is x in ] -oo,-4] uu [3/2, oo[

#### Explanation:

Let's find the roots of the equation

$2 {x}^{2} + 5 x - 12 = 0$

This is a simultaneous equation, $a {x}^{2} + b x + c = 0$

We calculate the discriminant,

$\Delta = {b}^{2} - 4 a c = 25 - 4 \cdot 2 \cdot - 12 = 121$

$\delta > 0$, so we have 2 real roots

The roots are

$x = \frac{- b \pm \sqrt{\Delta}}{2 a}$

$= \frac{- 5 \pm 11}{4}$

${x}_{1} = - 4$ and ${x}_{2} = \frac{6}{4} = \frac{3}{2}$

Let $f \left(x\right) = 2 {x}^{2} + 5 x - 12$

We do 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}$$- 4$$\textcolor{w h i t e}{a a a a}$$\frac{3}{2}$$\textcolor{w h i t e}{a a a a}$$+ \infty$

$\textcolor{w h i t e}{a a a a}$$x + 4$$\textcolor{w h i t e}{a a a a a a}$$-$$\textcolor{w h i t e}{a a a}$$+$$\textcolor{w h i t e}{a a a a}$$+$

$\textcolor{w h i t e}{a a a a}$$x - \frac{3}{2}$$\textcolor{w h i t e}{a a a a a}$$-$$\textcolor{w h i t e}{a a a}$$-$$\textcolor{w h i t e}{a a a a}$$+$

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

We need $f \left(x\right) \ge 0$

x in ] -oo,-4] uu [3/2, oo[
graph{2x^2+5x-12 [-12.66, 12.66, -6.34, 6.32]}