Solve Using the Quadratic Formula x^2+13=8x+37

$x^{2} + 13 = 8 x + 37$

Step 1

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

Move all the expressions to the left side of the equation.

Subtract $8 x$ from both sides of the equation.

$x^{2} + 13 - 8 x = 37$

Subtract $37$ from both sides of the equation.

$x^{2} + 13 - 8 x - 37 = 0$

Subtract $37$ from $13$ .

$x^{2} - 8 x - 24 = 0$

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 $a = 1$ , $b = - 8$ , and $c = - 24$ into the quadratic formula and solve for $x$ .

$\frac{8 \pm \sqrt{{(- 8)}^{2} - 4 \cdot (1 \cdot - 24)}}{2 \cdot 1}$

Step 4

Simplify.

Simplify the numerator.

Raise $- 8$ to the power of $2$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 1 \cdot - 24}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 24$ .

Multiply $- 4$ by $1$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot - 24}}{2 \cdot 1}$

Multiply $- 4$ by $- 24$ .

$x = \frac{8 \pm \sqrt{64 + 96}}{2 \cdot 1}$

Add $64$ and $96$ .

$x = \frac{8 \pm \sqrt{160}}{2 \cdot 1}$

Rewrite $160$ as $4^{2} \cdot 10$ .

Factor $16$ out of $160$ .

$x = \frac{8 \pm \sqrt{16 (10)}}{2 \cdot 1}$

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

$x = \frac{8 \pm \sqrt{4^{2} \cdot 10}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{8 \pm 4 \sqrt{10}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{8 \pm 4 \sqrt{10}}{2}$

Simplify $\frac{8 \pm 4 \sqrt{10}}{2}$ .

$x = 4 \pm 2 \sqrt{10}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = 4 \pm 2 \sqrt{10}$

Decimal Form:

$x = 10.32455532 \dots, - 2.32455532 \dots$