Find All Complex Solutions 2x^2-4x+5=-3

$2 x^{2} - 4 x + 5 = - 3$

Move all terms to the left side of the equation and simplify.

Add $3$ to both sides of the equation.

$2 x^{2} - 4 x + 5 + 3 = 0$

Add $5$ and $3$ .

$2 x^{2} - 4 x + 8 = 0$

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 2$ , $b = - 4$ , and $c = 8$ into the quadratic formula and solve for $x$ .

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

Simplify.

Simplify the numerator.

Raise $- 4$ to the power of $2$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 2 \cdot 8}}{2 \cdot 2}$

Multiply $- 4$ by $2$ .

$x = \frac{4 \pm \sqrt{16 - 8 \cdot 8}}{2 \cdot 2}$

Multiply $- 8$ by $8$ .

$x = \frac{4 \pm \sqrt{16 - 64}}{2 \cdot 2}$

Subtract $64$ from $16$ .

$x = \frac{4 \pm \sqrt{- 48}}{2 \cdot 2}$

Rewrite $- 48$ as $- 1 (48)$ .

$x = \frac{4 \pm \sqrt{- 1 \cdot 48}}{2 \cdot 2}$

Rewrite $\sqrt{- 1 (48)}$ as $\sqrt{- 1} \cdot \sqrt{48}$ .

$x = \frac{4 \pm \sqrt{- 1} \cdot \sqrt{48}}{2 \cdot 2}$

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

$x = \frac{4 \pm i \cdot \sqrt{48}}{2 \cdot 2}$

Rewrite $48$ as $4^{2} \cdot 3$ .

Factor $16$ out of $48$ .

$x = \frac{4 \pm i \cdot \sqrt{16 (3)}}{2 \cdot 2}$

Rewrite $16$ as $4^{2}$ .

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

Pull terms out from under the radical.

$x = \frac{4 \pm i \cdot (4 \sqrt{3})}{2 \cdot 2}$

Move $4$ to the left of $i$ .

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

Multiply $2$ by $2$ .

$x = \frac{4 \pm 4 i \sqrt{3}}{4}$

Simplify $\frac{4 \pm 4 i \sqrt{3}}{4}$ .

$x = 1 \pm i \sqrt{3}$