Solve Using the Quadratic Formula x^2+27=0

$x^{2} + 27 = 0$

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 $c = 27$ into the quadratic formula and solve for $x$ .

$\frac{0 \pm \sqrt{0^{2} - 4 \cdot (1 \cdot 27)}}{2 \cdot 1}$

Step 3

Simplify.

Simplify the numerator.

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

$x = \frac{0 \pm \sqrt{0 - 4 \cdot 1 \cdot 27}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 27$ .

Multiply $- 4$ by $1$ .

$x = \frac{0 \pm \sqrt{0 - 4 \cdot 27}}{2 \cdot 1}$

Multiply $- 4$ by $27$ .

$x = \frac{0 \pm \sqrt{0 - 108}}{2 \cdot 1}$

Subtract $108$ from $0$ .

$x = \frac{0 \pm \sqrt{- 108}}{2 \cdot 1}$

Rewrite $- 108$ as $- 1 (108)$ .

$x = \frac{0 \pm \sqrt{- 1 \cdot 108}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (108)}$ as $\sqrt{- 1} \cdot \sqrt{108}$ .

$x = \frac{0 \pm \sqrt{- 1} \cdot \sqrt{108}}{2 \cdot 1}$

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

$x = \frac{0 \pm i \cdot \sqrt{108}}{2 \cdot 1}$

Rewrite $108$ as $6^{2} \cdot 3$ .

Factor $36$ out of $108$ .

$x = \frac{0 \pm i \cdot \sqrt{36 (3)}}{2 \cdot 1}$

Rewrite $36$ as $6^{2}$ .

$x = \frac{0 \pm i \cdot \sqrt{6^{2} \cdot 3}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{0 \pm i \cdot (6 \sqrt{3})}{2 \cdot 1}$

Move $6$ to the left of $i$ .

$x = \frac{0 \pm 6 i \sqrt{3}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{0 \pm 6 i \sqrt{3}}{2}$

Simplify $\frac{0 \pm 6 i \sqrt{3}}{2}$ .

$x = \pm 3 i \sqrt{3}$