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

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

Step 1

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

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

Step 2

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

Subtract $1$ from both sides of the equation.

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

Subtract $1$ from $10$ .

$x - x^{2} + 9 = 0$

Step 3

Use the quadratic formula to find the solutions.

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

Step 4

Substitute the values $a = - 1$ , $b = 1$ , and $c = 9$ into the quadratic formula and solve for $x$ .

$\frac{- 1 \pm \sqrt{1^{2} - 4 \cdot (- 1 \cdot 9)}}{2 \cdot - 1}$

Step 5

Simplify.

Simplify the numerator.

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot - 1 \cdot 9}}{2 \cdot - 1}$

Multiply $- 4 \cdot - 1 \cdot 9$ .

Multiply $- 4$ by $- 1$ .

$x = \frac{- 1 \pm \sqrt{1 + 4 \cdot 9}}{2 \cdot - 1}$

Multiply $4$ by $9$ .

$x = \frac{- 1 \pm \sqrt{1 + 36}}{2 \cdot - 1}$

Add $1$ and $36$ .

$x = \frac{- 1 \pm \sqrt{37}}{2 \cdot - 1}$

Multiply $2$ by $- 1$ .

$x = \frac{- 1 \pm \sqrt{37}}{- 2}$

Simplify $\frac{- 1 \pm \sqrt{37}}{- 2}$ .

$x = \frac{1 \pm \sqrt{37}}{2}$

Step 6

The final answer is the combination of both solutions.

$x = \frac{1 + \sqrt{37}}{2}, \frac{1 - \sqrt{37}}{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1 + \sqrt{37}}{2}, \frac{1 - \sqrt{37}}{2}$

Decimal Form:

$x = 3.54138126 \dots, - 2.54138126 \dots$