Find All Complex Solutions X^2-16X+89=0

$X^{2} - 16 X + 89 = 0$

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 $a = 1$ , $b = - 16$ , and $c = 89$ into the quadratic formula and solve for $X$ .

$\frac{16 \pm \sqrt{{(- 16)}^{2} - 4 \cdot (1 \cdot 89)}}{2 \cdot 1}$

Step 3

Simplify.

Simplify the numerator.

Raise $- 16$ to the power of $2$ .

$X = \frac{16 \pm \sqrt{256 - 4 \cdot 1 \cdot 89}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 89$ .

Multiply $- 4$ by $1$ .

$X = \frac{16 \pm \sqrt{256 - 4 \cdot 89}}{2 \cdot 1}$

Multiply $- 4$ by $89$ .

$X = \frac{16 \pm \sqrt{256 - 356}}{2 \cdot 1}$

Subtract $356$ from $256$ .

$X = \frac{16 \pm \sqrt{- 100}}{2 \cdot 1}$

Rewrite $- 100$ as $- 1 (100)$ .

$X = \frac{16 \pm \sqrt{- 1 \cdot 100}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (100)}$ as $\sqrt{- 1} \cdot \sqrt{100}$ .

$X = \frac{16 \pm \sqrt{- 1} \cdot \sqrt{100}}{2 \cdot 1}$

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

$X = \frac{16 \pm i \cdot \sqrt{100}}{2 \cdot 1}$

Rewrite $100$ as $10^{2}$ .

$X = \frac{16 \pm i \cdot \sqrt{10^{2}}}{2 \cdot 1}$

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

$X = \frac{16 \pm i \cdot 10}{2 \cdot 1}$

Move $10$ to the left of $i$ .

$X = \frac{16 \pm 10 i}{2 \cdot 1}$

Multiply $2$ by $1$ .

$X = \frac{16 \pm 10 i}{2}$

Simplify $\frac{16 \pm 10 i}{2}$ .

$X = 8 \pm 5 i$