# How do you solve the inequality 2x^2-4x-1>0 and write your answer in interval notation?

Mar 6, 2017

The solution is x in ]-oo,(1-sqrt6/2)[ uu ] (1+sqrt6/2), +oo[

#### Explanation:

First, we must solve the equation

$2 {x}^{2} - 4 x - 1 = 0$

The discriminant is

$\Delta = {b}^{2} - 4 a c = {\left(- 4\right)}^{2} - 4 \left(2\right) \left(- 1\right) = 16 + 8 = 24$

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

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

${x}_{1} = \frac{4 - \sqrt{24}}{4} = \frac{4 - 2 \sqrt{6}}{4} = 1 - \frac{\sqrt{6}}{2} = - 0.225$

${x}_{2} = \frac{4 + \sqrt{24}}{4} = \frac{4 + 2 \sqrt{6}}{4} = 1 + \frac{\sqrt{6}}{2} = 2.225$

Let $f \left(x\right) = 2 {x}^{2} - 4 x - 1$

Now, we build the 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}$${x}_{1}$$\textcolor{w h i t e}{a a a a}$${x}_{2}$$\textcolor{w h i t e}{a a a a}$$+ \infty$

$\textcolor{w h i t e}{a a a a}$$x - {x}_{1}$$\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}$$x - {x}_{2}$$\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}$$f \left(x\right)$$\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,

$f \left(x\right) > 0$, when x in ]-oo,(1-sqrt6/2)[ uu ] (1+sqrt6/2), +oo[