Find All Complex Solutions -4x^2+3x-4=0

$- 4 x^{2} + 3 x - 4 = 0$

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 $a = - 4$ , $b = 3$ , and $c = - 4$ into the quadratic formula and solve for $x$ .

$\frac{- 3 \pm \sqrt{3^{2} - 4 \cdot (- 4 \cdot - 4)}}{2 \cdot - 4}$

Step 3

Simplify.

Simplify the numerator.

Raise $3$ to the power of $2$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot - 4 \cdot - 4}}{2 \cdot - 4}$

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

Multiply $- 4$ by $- 4$ .

$x = \frac{- 3 \pm \sqrt{9 + 16 \cdot - 4}}{2 \cdot - 4}$

Multiply $16$ by $- 4$ .

$x = \frac{- 3 \pm \sqrt{9 - 64}}{2 \cdot - 4}$

Subtract $64$ from $9$ .

$x = \frac{- 3 \pm \sqrt{- 55}}{2 \cdot - 4}$

Rewrite $- 55$ as $- 1 (55)$ .

$x = \frac{- 3 \pm \sqrt{- 1 \cdot 55}}{2 \cdot - 4}$

Rewrite $\sqrt{- 1 (55)}$ as $\sqrt{- 1} \cdot \sqrt{55}$ .

$x = \frac{- 3 \pm \sqrt{- 1} \cdot \sqrt{55}}{2 \cdot - 4}$

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

$x = \frac{- 3 \pm i \sqrt{55}}{2 \cdot - 4}$

Multiply $2$ by $- 4$ .

$x = \frac{- 3 \pm i \sqrt{55}}{- 8}$

Simplify $\frac{- 3 \pm i \sqrt{55}}{- 8}$ .

$x = \frac{3 \pm i \sqrt{55}}{8}$