Find All Complex Solutions x^2-4x+85=0

$x^{2} - 4 x + 85 = 0$

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 $a = 1$ , $b = - 4$ , and $c = 85$ into the quadratic formula and solve for $x$ .

$\frac{4 \pm \sqrt{{(- 4)}^{2} - 4 \cdot (1 \cdot 85)}}{2 \cdot 1}$

Step 3

Simplify.

Simplify the numerator.

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

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 1 \cdot 85}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 85$ .

Multiply $- 4$ by $1$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 85}}{2 \cdot 1}$

Multiply $- 4$ by $85$ .

$x = \frac{4 \pm \sqrt{16 - 340}}{2 \cdot 1}$

Subtract $340$ from $16$ .

$x = \frac{4 \pm \sqrt{- 324}}{2 \cdot 1}$

Rewrite $- 324$ as $- 1 (324)$ .

$x = \frac{4 \pm \sqrt{- 1 \cdot 324}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (324)}$ as $\sqrt{- 1} \cdot \sqrt{324}$ .

$x = \frac{4 \pm \sqrt{- 1} \cdot \sqrt{324}}{2 \cdot 1}$

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

$x = \frac{4 \pm i \cdot \sqrt{324}}{2 \cdot 1}$

Rewrite $324$ as $18^{2}$ .

$x = \frac{4 \pm i \cdot \sqrt{18^{2}}}{2 \cdot 1}$

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

$x = \frac{4 \pm i \cdot 18}{2 \cdot 1}$

Move $18$ to the left of $i$ .

$x = \frac{4 \pm 18 i}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{4 \pm 18 i}{2}$

Simplify $\frac{4 \pm 18 i}{2}$ .

$x = 2 \pm 9 i$