Find All Complex Solutions 3x^2=-12

$3 x^{2} = - 12$

Add $12$ to both sides of the equation.

$3 x^{2} + 12 = 0$

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Substitute the values $a = 3$ , $b = 0$ , and $c = 12$ into the quadratic formula and solve for $x$ .

$\frac{0 \pm \sqrt{0^{2} - 4 \cdot (3 \cdot 12)}}{2 \cdot 3}$

Simplify.

Simplify the numerator.

Raising $0$ to any positive power yields $0$ .

$x = \frac{0 \pm \sqrt{0 - 4 \cdot 3 \cdot 12}}{2 \cdot 3}$

Multiply $- 4$ by $3$ .

$x = \frac{0 \pm \sqrt{0 - 12 \cdot 12}}{2 \cdot 3}$

Multiply $- 12$ by $12$ .

$x = \frac{0 \pm \sqrt{0 - 144}}{2 \cdot 3}$

Subtract $144$ from $0$ .

$x = \frac{0 \pm \sqrt{- 144}}{2 \cdot 3}$

Rewrite $- 144$ as $- 1 (144)$ .

$x = \frac{0 \pm \sqrt{- 1 \cdot 144}}{2 \cdot 3}$

Rewrite $\sqrt{- 1 (144)}$ as $\sqrt{- 1} \cdot \sqrt{144}$ .

$x = \frac{0 \pm \sqrt{- 1} \cdot \sqrt{144}}{2 \cdot 3}$

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{0 \pm i \cdot \sqrt{144}}{2 \cdot 3}$

Rewrite $144$ as $12^{2}$ .

$x = \frac{0 \pm i \cdot \sqrt{12^{2}}}{2 \cdot 3}$

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{0 \pm i \cdot 12}{2 \cdot 3}$

Move $12$ to the left of $i$ .

$x = \frac{0 \pm 12 i}{2 \cdot 3}$

Multiply $2$ by $3$ .

$x = \frac{0 \pm 12 i}{6}$

Simplify $\frac{0 \pm 12 i}{6}$ .

$x = \pm 2 i$