Find All Complex Solutions -2x^2+7x-8=0

$- 2 x^{2} + 7 x - 8 = 0$

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 $a = - 2$ , $b = 7$ , and $c = - 8$ into the quadratic formula and solve for $x$ .

$\frac{- 7 \pm \sqrt{7^{2} - 4 \cdot (- 2 \cdot - 8)}}{2 \cdot - 2}$

Step 3

Simplify.

Simplify the numerator.

Raise $7$ to the power of $2$ .

$x = \frac{- 7 \pm \sqrt{49 - 4 \cdot - 2 \cdot - 8}}{2 \cdot - 2}$

Multiply $- 4 \cdot - 2 \cdot - 8$ .

Multiply $- 4$ by $- 2$ .

$x = \frac{- 7 \pm \sqrt{49 + 8 \cdot - 8}}{2 \cdot - 2}$

Multiply $8$ by $- 8$ .

$x = \frac{- 7 \pm \sqrt{49 - 64}}{2 \cdot - 2}$

Subtract $64$ from $49$ .

$x = \frac{- 7 \pm \sqrt{- 15}}{2 \cdot - 2}$

Rewrite $- 15$ as $- 1 (15)$ .

$x = \frac{- 7 \pm \sqrt{- 1 \cdot 15}}{2 \cdot - 2}$

Rewrite $\sqrt{- 1 (15)}$ as $\sqrt{- 1} \cdot \sqrt{15}$ .

$x = \frac{- 7 \pm \sqrt{- 1} \cdot \sqrt{15}}{2 \cdot - 2}$

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

$x = \frac{- 7 \pm i \sqrt{15}}{2 \cdot - 2}$

Multiply $2$ by $- 2$ .

$x = \frac{- 7 \pm i \sqrt{15}}{- 4}$

Simplify $\frac{- 7 \pm i \sqrt{15}}{- 4}$ .

$x = \frac{7 \pm i \sqrt{15}}{4}$