Solve Using the Quadratic Formula 7x^2+84=0

$7 x^{2} + 84 = 0$

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 $a = 7$ , $b = 0$ , and $c = 84$ into the quadratic formula and solve for $x$ .

$\frac{0 \pm \sqrt{0^{2} - 4 \cdot (7 \cdot 84)}}{2 \cdot 7}$

Step 3

Simplify.

Simplify the numerator.

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

$x = \frac{0 \pm \sqrt{0 - 4 \cdot 7 \cdot 84}}{2 \cdot 7}$

Multiply $- 4 \cdot 7 \cdot 84$ .

Multiply $- 4$ by $7$ .

$x = \frac{0 \pm \sqrt{0 - 28 \cdot 84}}{2 \cdot 7}$

Multiply $- 28$ by $84$ .

$x = \frac{0 \pm \sqrt{0 - 2352}}{2 \cdot 7}$

Subtract $2352$ from $0$ .

$x = \frac{0 \pm \sqrt{- 2352}}{2 \cdot 7}$

Rewrite $- 2352$ as $- 1 (2352)$ .

$x = \frac{0 \pm \sqrt{- 1 \cdot 2352}}{2 \cdot 7}$

Rewrite $\sqrt{- 1 (2352)}$ as $\sqrt{- 1} \cdot \sqrt{2352}$ .

$x = \frac{0 \pm \sqrt{- 1} \cdot \sqrt{2352}}{2 \cdot 7}$

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

$x = \frac{0 \pm i \cdot \sqrt{2352}}{2 \cdot 7}$

Rewrite $2352$ as $28^{2} \cdot 3$ .

Factor $784$ out of $2352$ .

$x = \frac{0 \pm i \cdot \sqrt{784 (3)}}{2 \cdot 7}$

Rewrite $784$ as $28^{2}$ .

$x = \frac{0 \pm i \cdot \sqrt{28^{2} \cdot 3}}{2 \cdot 7}$

Pull terms out from under the radical.

$x = \frac{0 \pm i \cdot (28 \sqrt{3})}{2 \cdot 7}$

Move $28$ to the left of $i$ .

$x = \frac{0 \pm 28 i \sqrt{3}}{2 \cdot 7}$

Multiply $2$ by $7$ .

$x = \frac{0 \pm 28 i \sqrt{3}}{14}$

Simplify $\frac{0 \pm 28 i \sqrt{3}}{14}$ .

$x = \pm 2 i \sqrt{3}$