Solve for x x+10=x^2-1

$x + 10 = x^{2} - 1$

Subtract $x^{2}$ from both sides of the equation.

$x + 10 - x^{2} = - 1$

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

Add $1$ to both sides of the equation.

$x + 10 - x^{2} + 1 = 0$

Add $10$ and $1$ .

$x - x^{2} + 11 = 0$

Use the quadratic formula to find the solutions.

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

Substitute the values $a = - 1$ , $b = 1$ , and $c = 11$ into the quadratic formula and solve for $x$ .

$\frac{- 1 \pm \sqrt{1^{2} - 4 \cdot (- 1 \cdot 11)}}{2 \cdot - 1}$

Simplify.

Simplify the numerator.

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot - 1 \cdot 11}}{2 \cdot - 1}$

Multiply $- 4$ by $- 1$ .

$x = \frac{- 1 \pm \sqrt{1 + 4 \cdot 11}}{2 \cdot - 1}$

Multiply $4$ by $11$ .

$x = \frac{- 1 \pm \sqrt{1 + 44}}{2 \cdot - 1}$

Add $1$ and $44$ .

$x = \frac{- 1 \pm \sqrt{45}}{2 \cdot - 1}$

Rewrite $45$ as $3^{2} \cdot 5$ .

Factor $9$ out of $45$ .

$x = \frac{- 1 \pm \sqrt{9 (5)}}{2 \cdot - 1}$

Rewrite $9$ as $3^{2}$ .

$x = \frac{- 1 \pm \sqrt{3^{2} \cdot 5}}{2 \cdot - 1}$

Pull terms out from under the radical.

$x = \frac{- 1 \pm 3 \sqrt{5}}{2 \cdot - 1}$

Multiply $2$ by $- 1$ .

$x = \frac{- 1 \pm 3 \sqrt{5}}{- 2}$

Simplify $\frac{- 1 \pm 3 \sqrt{5}}{- 2}$ .

$x = \frac{1 \pm 3 \sqrt{5}}{2}$

The final answer is the combination of both solutions.

$x = \frac{1 + 3 \sqrt{5}}{2}, \frac{1 - 3 \sqrt{5}}{2}$

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1 + 3 \sqrt{5}}{2}, \frac{1 - 3 \sqrt{5}}{2}$

Decimal Form:

$x = 3.85410196 \dots, - 2.85410196 \dots$