Find All Complex Solutions x^2-14x+113=0

$x^{2} - 14 x + 113 = 0$

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 $a = 1$ , $b = - 14$ , and $c = 113$ into the quadratic formula and solve for $x$ .

$\frac{14 \pm \sqrt{{(- 14)}^{2} - 4 \cdot (1 \cdot 113)}}{2 \cdot 1}$

Step 3

Simplify.

Simplify the numerator.

Raise $- 14$ to the power of $2$ .

$x = \frac{14 \pm \sqrt{196 - 4 \cdot 1 \cdot 113}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 113$ .

Multiply $- 4$ by $1$ .

$x = \frac{14 \pm \sqrt{196 - 4 \cdot 113}}{2 \cdot 1}$

Multiply $- 4$ by $113$ .

$x = \frac{14 \pm \sqrt{196 - 452}}{2 \cdot 1}$

Subtract $452$ from $196$ .

$x = \frac{14 \pm \sqrt{- 256}}{2 \cdot 1}$

Rewrite $- 256$ as $- 1 (256)$ .

$x = \frac{14 \pm \sqrt{- 1 \cdot 256}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (256)}$ as $\sqrt{- 1} \cdot \sqrt{256}$ .

$x = \frac{14 \pm \sqrt{- 1} \cdot \sqrt{256}}{2 \cdot 1}$

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

$x = \frac{14 \pm i \cdot \sqrt{256}}{2 \cdot 1}$

Rewrite $256$ as $16^{2}$ .

$x = \frac{14 \pm i \cdot \sqrt{16^{2}}}{2 \cdot 1}$

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

$x = \frac{14 \pm i \cdot 16}{2 \cdot 1}$

Move $16$ to the left of $i$ .

$x = \frac{14 \pm 16 i}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{14 \pm 16 i}{2}$

Simplify $\frac{14 \pm 16 i}{2}$ .

$x = 7 \pm 8 i$