Solve Using the Quadratic Formula x^2=-28

$x^{2} = - 28$

Step 1

Add $28$ to both sides of the equation.

$x^{2} + 28 = 0$

Step 2

Use the quadratic formula to find the solutions.

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

Step 3

Substitute the values $a = 1$ , $b = 0$ , and $c = 28$ into the quadratic formula and solve for $x$ .

$\frac{0 \pm \sqrt{0^{2} - 4 \cdot (1 \cdot 28)}}{2 \cdot 1}$

Step 4

Simplify.

Simplify the numerator.

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

$x = \frac{0 \pm \sqrt{0 - 4 \cdot 1 \cdot 28}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 28$ .

Multiply $- 4$ by $1$ .

$x = \frac{0 \pm \sqrt{0 - 4 \cdot 28}}{2 \cdot 1}$

Multiply $- 4$ by $28$ .

$x = \frac{0 \pm \sqrt{0 - 112}}{2 \cdot 1}$

Subtract $112$ from $0$ .

$x = \frac{0 \pm \sqrt{- 112}}{2 \cdot 1}$

Rewrite $- 112$ as $- 1 (112)$ .

$x = \frac{0 \pm \sqrt{- 1 \cdot 112}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (112)}$ as $\sqrt{- 1} \cdot \sqrt{112}$ .

$x = \frac{0 \pm \sqrt{- 1} \cdot \sqrt{112}}{2 \cdot 1}$

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

$x = \frac{0 \pm i \cdot \sqrt{112}}{2 \cdot 1}$

Rewrite $112$ as $4^{2} \cdot 7$ .

Factor $16$ out of $112$ .

$x = \frac{0 \pm i \cdot \sqrt{16 (7)}}{2 \cdot 1}$

Rewrite $16$ as $4^{2}$ .

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

Pull terms out from under the radical.

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

Move $4$ to the left of $i$ .

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

Multiply $2$ by $1$ .

$x = \frac{0 \pm 4 i \sqrt{7}}{2}$

Simplify $\frac{0 \pm 4 i \sqrt{7}}{2}$ .

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