Solve Using the Quadratic Formula 3x^2+81=0

$3 x^{2} + 81 = 0$

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 $a = 3$ , $b = 0$ , and $c = 81$ into the quadratic formula and solve for $x$ .

$\frac{0 \pm \sqrt{0^{2} - 4 \cdot (3 \cdot 81)}}{2 \cdot 3}$

Step 3

Simplify.

Simplify the numerator.

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

$x = \frac{0 \pm \sqrt{0 - 4 \cdot 3 \cdot 81}}{2 \cdot 3}$

Multiply $- 4 \cdot 3 \cdot 81$ .

Multiply $- 4$ by $3$ .

$x = \frac{0 \pm \sqrt{0 - 12 \cdot 81}}{2 \cdot 3}$

Multiply $- 12$ by $81$ .

$x = \frac{0 \pm \sqrt{0 - 972}}{2 \cdot 3}$

Subtract $972$ from $0$ .

$x = \frac{0 \pm \sqrt{- 972}}{2 \cdot 3}$

Rewrite $- 972$ as $- 1 (972)$ .

$x = \frac{0 \pm \sqrt{- 1 \cdot 972}}{2 \cdot 3}$

Rewrite $\sqrt{- 1 (972)}$ as $\sqrt{- 1} \cdot \sqrt{972}$ .

$x = \frac{0 \pm \sqrt{- 1} \cdot \sqrt{972}}{2 \cdot 3}$

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

$x = \frac{0 \pm i \cdot \sqrt{972}}{2 \cdot 3}$

Rewrite $972$ as $18^{2} \cdot 3$ .

Factor $324$ out of $972$ .

$x = \frac{0 \pm i \cdot \sqrt{324 (3)}}{2 \cdot 3}$

Rewrite $324$ as $18^{2}$ .

$x = \frac{0 \pm i \cdot \sqrt{18^{2} \cdot 3}}{2 \cdot 3}$

Pull terms out from under the radical.

$x = \frac{0 \pm i \cdot (18 \sqrt{3})}{2 \cdot 3}$

Move $18$ to the left of $i$ .

$x = \frac{0 \pm 18 i \sqrt{3}}{2 \cdot 3}$

Multiply $2$ by $3$ .

$x = \frac{0 \pm 18 i \sqrt{3}}{6}$

Simplify $\frac{0 \pm 18 i \sqrt{3}}{6}$ .

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