Find All Complex Solutions 196x^2+9=0

$196 x^{2} + 9 = 0$

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 $a = 196$ , $b = 0$ , and $c = 9$ into the quadratic formula and solve for $x$ .

$\frac{0 \pm \sqrt{0^{2} - 4 \cdot (196 \cdot 9)}}{2 \cdot 196}$

Step 3

Simplify.

Simplify the numerator.

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

$x = \frac{0 \pm \sqrt{0 - 4 \cdot 196 \cdot 9}}{2 \cdot 196}$

Multiply $- 4 \cdot 196 \cdot 9$ .

Multiply $- 4$ by $196$ .

$x = \frac{0 \pm \sqrt{0 - 784 \cdot 9}}{2 \cdot 196}$

Multiply $- 784$ by $9$ .

$x = \frac{0 \pm \sqrt{0 - 7056}}{2 \cdot 196}$

Subtract $7056$ from $0$ .

$x = \frac{0 \pm \sqrt{- 7056}}{2 \cdot 196}$

Rewrite $- 7056$ as $- 1 (7056)$ .

$x = \frac{0 \pm \sqrt{- 1 \cdot 7056}}{2 \cdot 196}$

Rewrite $\sqrt{- 1 (7056)}$ as $\sqrt{- 1} \cdot \sqrt{7056}$ .

$x = \frac{0 \pm \sqrt{- 1} \cdot \sqrt{7056}}{2 \cdot 196}$

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

$x = \frac{0 \pm i \cdot \sqrt{7056}}{2 \cdot 196}$

Rewrite $7056$ as $84^{2}$ .

$x = \frac{0 \pm i \cdot \sqrt{84^{2}}}{2 \cdot 196}$

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{0 \pm i \cdot 84}{2 \cdot 196}$

Move $84$ to the left of $i$ .

$x = \frac{0 \pm 84 i}{2 \cdot 196}$

Multiply $2$ by $196$ .

$x = \frac{0 \pm 84 i}{392}$

Simplify $\frac{0 \pm 84 i}{392}$ .

$x = \frac{\pm 3 i}{14}$