Find All Complex Solutions 0=4x^2+9

$0 = 4 x^{2} + 9$

Step 1

Move all the expressions to the left side of the equation.

Subtract $4 x^{2}$ from both sides of the equation.

$- 4 x^{2} = 9$

Subtract $9$ from both sides of the equation.

$- 4 x^{2} - 9 = 0$

Step 2

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 3

Substitute the values $a = - 4$ , $b = 0$ , and $c = - 9$ into the quadratic formula and solve for $x$ .

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

Step 4

Simplify.

Simplify the numerator.

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

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

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

Multiply $- 4$ by $- 4$ .

$x = \frac{0 \pm \sqrt{0 + 16 \cdot - 9}}{2 \cdot - 4}$

Multiply $16$ by $- 9$ .

$x = \frac{0 \pm \sqrt{0 - 144}}{2 \cdot - 4}$

Subtract $144$ from $0$ .

$x = \frac{0 \pm \sqrt{- 144}}{2 \cdot - 4}$

Rewrite $- 144$ as $- 1 (144)$ .

$x = \frac{0 \pm \sqrt{- 1 \cdot 144}}{2 \cdot - 4}$

Rewrite $\sqrt{- 1 (144)}$ as $\sqrt{- 1} \cdot \sqrt{144}$ .

$x = \frac{0 \pm \sqrt{- 1} \cdot \sqrt{144}}{2 \cdot - 4}$

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

$x = \frac{0 \pm i \cdot \sqrt{144}}{2 \cdot - 4}$

Rewrite $144$ as $12^{2}$ .

$x = \frac{0 \pm i \cdot \sqrt{12^{2}}}{2 \cdot - 4}$

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

$x = \frac{0 \pm i \cdot 12}{2 \cdot - 4}$

Move $12$ to the left of $i$ .

$x = \frac{0 \pm 12 i}{2 \cdot - 4}$

Multiply $2$ by $- 4$ .

$x = \frac{0 \pm 12 i}{- 8}$

Simplify $\frac{0 \pm 12 i}{- 8}$ .

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