Find All Complex Solutions -2x^2+6x-10=0

$- 2 x^{2} + 6 x - 10 = 0$

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 $a = - 2$ , $b = 6$ , and $c = - 10$ into the quadratic formula and solve for $x$ .

$\frac{- 6 \pm \sqrt{6^{2} - 4 \cdot (- 2 \cdot - 10)}}{2 \cdot - 2}$

Step 3

Simplify.

Simplify the numerator.

Raise $6$ to the power of $2$ .

$x = \frac{- 6 \pm \sqrt{36 - 4 \cdot - 2 \cdot - 10}}{2 \cdot - 2}$

Multiply $- 4 \cdot - 2 \cdot - 10$ .

Multiply $- 4$ by $- 2$ .

$x = \frac{- 6 \pm \sqrt{36 + 8 \cdot - 10}}{2 \cdot - 2}$

Multiply $8$ by $- 10$ .

$x = \frac{- 6 \pm \sqrt{36 - 80}}{2 \cdot - 2}$

Subtract $80$ from $36$ .

$x = \frac{- 6 \pm \sqrt{- 44}}{2 \cdot - 2}$

Rewrite $- 44$ as $- 1 (44)$ .

$x = \frac{- 6 \pm \sqrt{- 1 \cdot 44}}{2 \cdot - 2}$

Rewrite $\sqrt{- 1 (44)}$ as $\sqrt{- 1} \cdot \sqrt{44}$ .

$x = \frac{- 6 \pm \sqrt{- 1} \cdot \sqrt{44}}{2 \cdot - 2}$

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

$x = \frac{- 6 \pm i \cdot \sqrt{44}}{2 \cdot - 2}$

Rewrite $44$ as $2^{2} \cdot 11$ .

Factor $4$ out of $44$ .

$x = \frac{- 6 \pm i \cdot \sqrt{4 (11)}}{2 \cdot - 2}$

Rewrite $4$ as $2^{2}$ .

$x = \frac{- 6 \pm i \cdot \sqrt{2^{2} \cdot 11}}{2 \cdot - 2}$

Pull terms out from under the radical.

$x = \frac{- 6 \pm i \cdot (2 \sqrt{11})}{2 \cdot - 2}$

Move $2$ to the left of $i$ .

$x = \frac{- 6 \pm 2 i \sqrt{11}}{2 \cdot - 2}$

Multiply $2$ by $- 2$ .

$x = \frac{- 6 \pm 2 i \sqrt{11}}{- 4}$

Simplify $\frac{- 6 \pm 2 i \sqrt{11}}{- 4}$ .

$x = \frac{3 \pm i \sqrt{11}}{2}$