# How do you solve x^2 - 6x + 6 = 0?

Aug 2, 2015

The solutions are:
color(blue)(x=3+sqrt(3)
color(blue)(x=3-sqrt(3)

#### Explanation:

${x}^{2} - 6 x + 6$

The equation is of the form color(blue)(ax^2+bx+c=0 where:
$a = 1 , b = - 6 , c = 6$

The Discriminant is given by:
$\Delta = {b}^{2} - 4 \cdot a \cdot c$
$= {\left(- 6\right)}^{2} - \left(4 \cdot \left(1\right) \cdot 6\right)$
$= 36 - 24 = 12$

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

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

$x = \frac{- \left(- 6\right) \pm \sqrt{12}}{2 \cdot 1} = \frac{6 \pm \sqrt{12}}{2}$

$\sqrt{12} = \sqrt{2 \cdot 2 \cdot 3} = 2 \sqrt{3}$

$x = \frac{6 \pm 2 \sqrt{3}}{2}$
$x = \frac{\cancel{2} \left(3 \pm \sqrt{3}\right)}{\cancel{2}}$
$x = 3 \pm \sqrt{3}$

The solutions are:
color(blue)(x=3+sqrt(3)
color(blue)(x=3-sqrt(3)