# How do you solve 2x^2+4x<=3 using a sign chart?

Jan 4, 2017

The answer is $x \in \left[- 2.58 , 0.58\right]$

#### Explanation:

Let`s rewrite the equation

$2 {x}^{2} + 4 x - 3 \le 0$

We need the roots of the equation

$2 {x}^{2} + 4 x - 3 = 0$

We calculate the discriminant

$\Delta = {b}^{2} - 4 a c = 16 + 4 \cdot 2 \cdot 3 = 40$

As $\Delta > 0$, there are 2 real roots

Therefore,

${x}_{1} = \frac{- 4 + \sqrt{40}}{4} = 0.58$

${x}_{2} = \frac{- 4 - \sqrt{40}}{4} = - 2.58$

Let $f \left(x\right) = \left(x + 2.58\right) \left(x - 0.58\right) \le 0$

Now, we can do our 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}$$- 2.58$$\textcolor{w h i t e}{a a a a a}$$0.58$$\textcolor{w h i t e}{a a a a}$$+ \infty$

$\textcolor{w h i t e}{a a a a}$$x + 2.58$$\textcolor{w h i t e}{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 a}$$+$

$\textcolor{w h i t e}{a a a a}$$x - 0.58$$\textcolor{w h i t e}{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 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 a}$$+$$\textcolor{w h i t e}{a a a a a}$$-$$\textcolor{w h i t e}{a a a a a a}$$+$

Therefore,

$f \left(x\right) \le 0$, when $x \in \left[- 2.58 , 0.58\right]$