Find All Complex Solutions x^2+7=0

$x^{2} + 7 = 0$

Step 1

Use the quadratic formula to find the solutions.

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

Step 2

Substitute the values $a = 1$ , $b = 0$ , and $c = 7$ into the quadratic formula and solve for $x$ .

$\frac{0 \pm \sqrt{0^{2} - 4 \cdot (1 \cdot 7)}}{2 \cdot 1}$

Step 3

Simplify.

Simplify the numerator.

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

$x = \frac{0 \pm \sqrt{0 - 4 \cdot 1 \cdot 7}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 7$ .

Multiply $- 4$ by $1$ .

$x = \frac{0 \pm \sqrt{0 - 4 \cdot 7}}{2 \cdot 1}$

Multiply $- 4$ by $7$ .

$x = \frac{0 \pm \sqrt{0 - 28}}{2 \cdot 1}$

Subtract $28$ from $0$ .

$x = \frac{0 \pm \sqrt{- 28}}{2 \cdot 1}$

Rewrite $- 28$ as $- 1 (28)$ .

$x = \frac{0 \pm \sqrt{- 1 \cdot 28}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (28)}$ as $\sqrt{- 1} \cdot \sqrt{28}$ .

$x = \frac{0 \pm \sqrt{- 1} \cdot \sqrt{28}}{2 \cdot 1}$

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

$x = \frac{0 \pm i \cdot \sqrt{28}}{2 \cdot 1}$

Rewrite $28$ as $2^{2} \cdot 7$ .

Factor $4$ out of $28$ .

$x = \frac{0 \pm i \cdot \sqrt{4 (7)}}{2 \cdot 1}$

Rewrite $4$ as $2^{2}$ .

$x = \frac{0 \pm i \cdot \sqrt{2^{2} \cdot 7}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{0 \pm i \cdot (2 \sqrt{7})}{2 \cdot 1}$

Move $2$ to the left of $i$ .

$x = \frac{0 \pm 2 i \sqrt{7}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{0 \pm 2 i \sqrt{7}}{2}$

Simplify $\frac{0 \pm 2 i \sqrt{7}}{2}$ .

$x = \pm i \sqrt{7}$