# How do you solve x²-6x+3=0 using the quadratic formula?

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Jun 28, 2016

The solutions are:
color(blue)( x = 3 + sqrt6

color(blue)( x = 3 - sqrt6

#### Explanation:

${x}^{2} - 6 x + 3 = 0$

The equation is of the form color(blue)(ax^2+bx+c=0 where:
$a = 1 , b = - 6 , c = 3$

The Discriminant is given by:

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

$= {\left(- 6\right)}^{2} - \left(4 \cdot 1 \cdot 3\right)$

$= 36 - 12 = 24$

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

$x = \frac{- \left(- 6\right) \pm \sqrt{24}}{2 \cdot 1} = \frac{6 \pm \sqrt{24}}{2}$

$\sqrt{24}$ can be further simplified as follows:

$\sqrt{24} = \sqrt{2 \cdot 2 \cdot 2 \cdot 3} = \sqrt{{2}^{2} \cdot 2 \cdot 3} = 2 \sqrt{6}$

$x = \frac{6 \pm 2 \sqrt{6}}{2}$

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

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

The solutions are:
color(blue)( x = 3 + sqrt6
color(blue)( x = 3 - sqrt6

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