Find All Complex Solutions 225x^2+4=0

$225 x^{2} + 4 = 0$

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 $a = 225$ , $b = 0$ , and $c = 4$ into the quadratic formula and solve for $x$ .

$\frac{0 \pm \sqrt{0^{2} - 4 \cdot (225 \cdot 4)}}{2 \cdot 225}$

Step 3

Simplify.

Simplify the numerator.

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

$x = \frac{0 \pm \sqrt{0 - 4 \cdot 225 \cdot 4}}{2 \cdot 225}$

Multiply $- 4 \cdot 225 \cdot 4$ .

Multiply $- 4$ by $225$ .

$x = \frac{0 \pm \sqrt{0 - 900 \cdot 4}}{2 \cdot 225}$

Multiply $- 900$ by $4$ .

$x = \frac{0 \pm \sqrt{0 - 3600}}{2 \cdot 225}$

Subtract $3600$ from $0$ .

$x = \frac{0 \pm \sqrt{- 3600}}{2 \cdot 225}$

Rewrite $- 3600$ as $- 1 (3600)$ .

$x = \frac{0 \pm \sqrt{- 1 \cdot 3600}}{2 \cdot 225}$

Rewrite $\sqrt{- 1 (3600)}$ as $\sqrt{- 1} \cdot \sqrt{3600}$ .

$x = \frac{0 \pm \sqrt{- 1} \cdot \sqrt{3600}}{2 \cdot 225}$

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

$x = \frac{0 \pm i \cdot \sqrt{3600}}{2 \cdot 225}$

Rewrite $3600$ as $60^{2}$ .

$x = \frac{0 \pm i \cdot \sqrt{60^{2}}}{2 \cdot 225}$

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

$x = \frac{0 \pm i \cdot 60}{2 \cdot 225}$

Move $60$ to the left of $i$ .

$x = \frac{0 \pm 60 i}{2 \cdot 225}$

Multiply $2$ by $225$ .

$x = \frac{0 \pm 60 i}{450}$

Simplify $\frac{0 \pm 60 i}{450}$ .

$x = \frac{\pm 2 i}{15}$