# How do you solve the quadratic with complex numbers given -6x^2+12x-7=0?

May 6, 2017

$x = \frac{1 + 2 \sqrt{6 i}}{6} , \frac{1 - 2 \sqrt{6 i}}{6}$

#### Explanation:

Solve:

$- 6 {x}^{2} + 12 x - 7 = 0$ is a quadratic equation in standard form: $a {x}^{2} + b x + c$, where $a = - 6$, $b = 12$, and $c = - 7$.

The quadratic formula can be used to solve this equation.

$x = \frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{2 a}$

Substitute the given values into the formula.

$x = \frac{- 12 \pm \sqrt{{12}^{2} - 4 \times - 6 \times - 7}}{2 \times - 6}$

$x = \frac{- 12 \pm \sqrt{144 - 168}}{- 12}$

$x = \frac{- 12 \pm \sqrt{- 24}}{- 12}$

Factor $- 24$.

$x = \frac{- 12 \pm \sqrt{2 \times 2 \times 2 \times 3 i}}{- 12}$

$x = \frac{- 12 \pm 2 \sqrt{6 i}}{- 12}$

Simplify by dividing by $- 12$.

$x = \frac{1 \pm 2 \sqrt{6 i}}{6}$

Solutions for $x$.

$x = \frac{1 + 2 \sqrt{6 i}}{6} , \frac{1 - 2 \sqrt{6 i}}{6}$