Find All Complex Solutions 169x^2+36=0

$169 x^{2} + 36 = 0$

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 $a = 169$ , $b = 0$ , and $c = 36$ into the quadratic formula and solve for $x$ .

$\frac{0 \pm \sqrt{0^{2} - 4 \cdot (169 \cdot 36)}}{2 \cdot 169}$

Step 3

Simplify.

Simplify the numerator.

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

$x = \frac{0 \pm \sqrt{0 - 4 \cdot 169 \cdot 36}}{2 \cdot 169}$

Multiply $- 4 \cdot 169 \cdot 36$ .

Multiply $- 4$ by $169$ .

$x = \frac{0 \pm \sqrt{0 - 676 \cdot 36}}{2 \cdot 169}$

Multiply $- 676$ by $36$ .

$x = \frac{0 \pm \sqrt{0 - 24336}}{2 \cdot 169}$

Subtract $24336$ from $0$ .

$x = \frac{0 \pm \sqrt{- 24336}}{2 \cdot 169}$

Rewrite $- 24336$ as $- 1 (24336)$ .

$x = \frac{0 \pm \sqrt{- 1 \cdot 24336}}{2 \cdot 169}$

Rewrite $\sqrt{- 1 (24336)}$ as $\sqrt{- 1} \cdot \sqrt{24336}$ .

$x = \frac{0 \pm \sqrt{- 1} \cdot \sqrt{24336}}{2 \cdot 169}$

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

$x = \frac{0 \pm i \cdot \sqrt{24336}}{2 \cdot 169}$

Rewrite $24336$ as $156^{2}$ .

$x = \frac{0 \pm i \cdot \sqrt{156^{2}}}{2 \cdot 169}$

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

$x = \frac{0 \pm i \cdot 156}{2 \cdot 169}$

Move $156$ to the left of $i$ .

$x = \frac{0 \pm 156 i}{2 \cdot 169}$

Multiply $2$ by $169$ .

$x = \frac{0 \pm 156 i}{338}$

Simplify $\frac{0 \pm 156 i}{338}$ .

$x = \frac{\pm 6 i}{13}$