# How do you solve 2x^2 - 2x - 1 = 0?

Oct 10, 2015

The solutions are:
color(blue)(x= (1+sqrt(3))/2

color(blue)(x= (1-sqrt(3))/2

#### Explanation:

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

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

The Discriminant is given by:
$\Delta = {b}^{2} - 4 \cdot a \cdot c$
$= {\left(- 2\right)}^{2} - \left(4 \cdot 2 \cdot \left(- 1\right)\right)$
$= 4 + 8 = 12$

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

$x = \frac{- \left(- 2\right) \pm \sqrt{12}}{2 \cdot 2} = \frac{2 \pm 2 \sqrt{3}}{4}$

$\frac{2 \pm 2 \sqrt{3}}{4} = \frac{\cancel{2} \left(1 \pm \sqrt{3}\right)}{\cancel{4}}$

$= \frac{1 \pm \sqrt{3}}{2}$

The solutions are:
color(blue)(x= (1+sqrt(3))/2

color(blue)(x= (1-sqrt(3))/2