Solve for x x^2+14x+17=-96

$x^{2} + 14 x + 17 = - 96$

Step 1

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

Add $96$ to both sides of the equation.

$x^{2} + 14 x + 17 + 96 = 0$

Add $17$ and $96$ .

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

Step 2

Use the quadratic formula to find the solutions.

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

Step 3

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 4

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

$x = - 7 \pm 8 i$

Step 5

The final answer is the combination of both solutions.

$x = - 7 + 8 i, - 7 - 8 i$