Find All Complex Solutions -8x^2-8=-32

$- 8 x^{2} - 8 = - 32$

Move all terms to the left side of the equation and simplify.

Add $32$ to both sides of the equation.

$- 8 x^{2} - 8 + 32 = 0$

Add $- 8$ and $32$ .

$- 8 x^{2} + 24 = 0$

Use the quadratic formula to find the solutions.

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

Substitute the values $a = - 8$ , $b = 0$ , and $c = 24$ into the quadratic formula and solve for $x$ .

$\frac{0 \pm \sqrt{0^{2} - 4 \cdot (- 8 \cdot 24)}}{2 \cdot - 8}$

Simplify.

Simplify the numerator.

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

$x = \frac{0 \pm \sqrt{0 - 4 \cdot - 8 \cdot 24}}{2 \cdot - 8}$

Multiply $- 4$ by $- 8$ .

$x = \frac{0 \pm \sqrt{0 + 32 \cdot 24}}{2 \cdot - 8}$

Multiply $32$ by $24$ .

$x = \frac{0 \pm \sqrt{0 + 768}}{2 \cdot - 8}$

Add $0$ and $768$ .

$x = \frac{0 \pm \sqrt{768}}{2 \cdot - 8}$

Rewrite $768$ as $16^{2} \cdot 3$ .

Factor $256$ out of $768$ .

$x = \frac{0 \pm \sqrt{256 (3)}}{2 \cdot - 8}$

Rewrite $256$ as $16^{2}$ .

$x = \frac{0 \pm \sqrt{16^{2} \cdot 3}}{2 \cdot - 8}$

Pull terms out from under the radical.

$x = \frac{0 \pm 16 \sqrt{3}}{2 \cdot - 8}$

Multiply $2$ by $- 8$ .

$x = \frac{0 \pm 16 \sqrt{3}}{- 16}$

Simplify $\frac{0 \pm 16 \sqrt{3}}{- 16}$ .

$x = \pm \sqrt{3}$

The final answer is the combination of both solutions.

$x = \sqrt{3}, - \sqrt{3}$

The result can be shown in multiple forms.

Exact Form:

$x = \sqrt{3}, - \sqrt{3}$

Decimal Form:

$x = 1.73205080 \dots, - 1.73205080 \dots$