Solve for n 8n^2+4n-16=-n^2

$8 n^{2} + 4 n - 16 = - n^{2}$

Move all terms containing $n$ to the left side of the equation.

Add $n^{2}$ to both sides of the equation.

$8 n^{2} + 4 n - 16 + n^{2} = 0$

Add $8 n^{2}$ and $n^{2}$ .

$9 n^{2} + 4 n - 16 = 0$

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 9$ , $b = 4$ , and $c = - 16$ into the quadratic formula and solve for $n$ .

$\frac{- 4 \pm \sqrt{4^{2} - 4 \cdot (9 \cdot - 16)}}{2 \cdot 9}$

Simplify.

Simplify the numerator.

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

$n = \frac{- 4 \pm \sqrt{16 - 4 \cdot 9 \cdot - 16}}{2 \cdot 9}$

Multiply $- 4$ by $9$ .

$n = \frac{- 4 \pm \sqrt{16 - 36 \cdot - 16}}{2 \cdot 9}$

Multiply $- 36$ by $- 16$ .

$n = \frac{- 4 \pm \sqrt{16 + 576}}{2 \cdot 9}$

Add $16$ and $576$ .

$n = \frac{- 4 \pm \sqrt{592}}{2 \cdot 9}$

Rewrite $592$ as $4^{2} \cdot 37$ .

Factor $16$ out of $592$ .

$n = \frac{- 4 \pm \sqrt{16 (37)}}{2 \cdot 9}$

Rewrite $16$ as $4^{2}$ .

$n = \frac{- 4 \pm \sqrt{4^{2} \cdot 37}}{2 \cdot 9}$

Pull terms out from under the radical.

$n = \frac{- 4 \pm 4 \sqrt{37}}{2 \cdot 9}$

Multiply $2$ by $9$ .

$n = \frac{- 4 \pm 4 \sqrt{37}}{18}$

Simplify $\frac{- 4 \pm 4 \sqrt{37}}{18}$ .

$n = \frac{- 2 \pm 2 \sqrt{37}}{9}$

The final answer is the combination of both solutions.

$n = - \frac{2 - 2 \sqrt{37}}{9}, - \frac{2 + 2 \sqrt{37}}{9}$

The result can be shown in multiple forms.

Exact Form:

$n = - \frac{2 - 2 \sqrt{37}}{9}, - \frac{2 + 2 \sqrt{37}}{9}$

Decimal Form:

$n = 1.12950278 \dots, - 1.57394722 \dots$