Find All Complex Solutions x^2-2x+50=0

$x^{2} - 2 x + 50 = 0$

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 $a = 1$ , $b = - 2$ , and $c = 50$ into the quadratic formula and solve for $x$ .

$\frac{2 \pm \sqrt{{(- 2)}^{2} - 4 \cdot (1 \cdot 50)}}{2 \cdot 1}$

Step 3

Simplify.

Simplify the numerator.

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

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 1 \cdot 50}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 50$ .

Multiply $- 4$ by $1$ .

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 50}}{2 \cdot 1}$

Multiply $- 4$ by $50$ .

$x = \frac{2 \pm \sqrt{4 - 200}}{2 \cdot 1}$

Subtract $200$ from $4$ .

$x = \frac{2 \pm \sqrt{- 196}}{2 \cdot 1}$

Rewrite $- 196$ as $- 1 (196)$ .

$x = \frac{2 \pm \sqrt{- 1 \cdot 196}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (196)}$ as $\sqrt{- 1} \cdot \sqrt{196}$ .

$x = \frac{2 \pm \sqrt{- 1} \cdot \sqrt{196}}{2 \cdot 1}$

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

$x = \frac{2 \pm i \cdot \sqrt{196}}{2 \cdot 1}$

Rewrite $196$ as $14^{2}$ .

$x = \frac{2 \pm i \cdot \sqrt{14^{2}}}{2 \cdot 1}$

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

$x = \frac{2 \pm i \cdot 14}{2 \cdot 1}$

Move $14$ to the left of $i$ .

$x = \frac{2 \pm 14 i}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{2 \pm 14 i}{2}$

Simplify $\frac{2 \pm 14 i}{2}$ .

$x = 1 \pm 7 i$