# How do you solve 3x^2 - 2x = 4 using the quadratic formula?

Feb 27, 2016

The solutions are :
color(blue)(x=(1+sqrt13)/3

color(blue)(x=(1-sqrt13)/3

#### Explanation:

$3 {x}^{2} - 2 x - 4 = 0$

The equation is of the form color(blue)(ax^2+bx+c=0 where:
$a = 3 , b = - 2 , c = - 4$

The Discriminant is given by:
$\Delta = {b}^{2} - 4 \cdot a \cdot c$

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

$= 4 + 48 = 52$

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

$x = \frac{- \left(- 2\right) \pm \sqrt{52}}{2 \cdot 3} = \frac{2 \pm \sqrt{52}}{6}$

Upon further simplification $\sqrt{52} = \sqrt{2 \cdot 2 \cdot 13} = 2 \sqrt{13}$

So, $x = \frac{2 \pm 2 \sqrt{13}}{6} = \frac{\cancel{2} \left(1 \pm \sqrt{13}\right)}{\cancel{6}}$
$= \frac{1 \pm \sqrt{13}}{3}$

The solutions are :
color(blue)(x=(1+sqrt13)/3
color(blue)(x=(1-sqrt13)/3