# How do you solve x^2-4x-2=0 by using the Quadratic Formula?

Jul 25, 2015

The solutions are:
color(blue)(x=2+sqrt6
color(blue)(x=2-sqrt6

#### Explanation:

The equation: ${x}^{2} - 4 x - 2$ is of the form color(blue)(ax^2+bx+c=0 where:
$a = 1 , b = - 4 , c = - 2$

The Discriminant is given by:

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

$= {\left(- 4\right)}^{2} - \left(4\right) \cdot \left(1\right) \cdot \left(- 2\right)$
$= 16 + 8 = 24$

As $\Delta > 0$ there are two solutions,

The solutions are found using the formula
$x = \frac{- b \pm \sqrt{\Delta}}{2 \cdot a}$

$x = \frac{- \left(- 4\right) \pm \sqrt{24}}{2 \cdot 1} = \frac{4 \pm \sqrt{24}}{2}$

(color(blue)(sqrt24 = sqrt(2*2*2*3) = 2sqrt6)

So,
$x = \frac{4 \pm 2 \sqrt{6}}{2}$

Taking $2$ outside the bracket as it is common to both terms of the numerator

$x = \frac{\cancel{2} \left(2 \pm \sqrt{6}\right)}{\cancel{2}}$
the solutions are:
color(blue)(x=2+sqrt6
color(blue)(x=2-sqrt6