Find the Roots (Zeros) x^2-18x=-31

$x^{2} - 18 x = - 31$

Step 1

Add $31$ to both sides of the equation.

$x^{2} - 18 x + 31 = 0$

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 $a = 1$ , $b = - 18$ , and $c = 31$ into the quadratic formula and solve for $x$ .

$\frac{18 \pm \sqrt{{(- 18)}^{2} - 4 \cdot (1 \cdot 31)}}{2 \cdot 1}$

Step 4

Simplify.

Simplify the numerator.

Raise $- 18$ to the power of $2$ .

$x = \frac{18 \pm \sqrt{324 - 4 \cdot 1 \cdot 31}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 31$ .

Multiply $- 4$ by $1$ .

$x = \frac{18 \pm \sqrt{324 - 4 \cdot 31}}{2 \cdot 1}$

Multiply $- 4$ by $31$ .

$x = \frac{18 \pm \sqrt{324 - 124}}{2 \cdot 1}$

Subtract $124$ from $324$ .

$x = \frac{18 \pm \sqrt{200}}{2 \cdot 1}$

Rewrite $200$ as $10^{2} \cdot 2$ .

Factor $100$ out of $200$ .

$x = \frac{18 \pm \sqrt{100 (2)}}{2 \cdot 1}$

Rewrite $100$ as $10^{2}$ .

$x = \frac{18 \pm \sqrt{10^{2} \cdot 2}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{18 \pm 10 \sqrt{2}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{18 \pm 10 \sqrt{2}}{2}$

Simplify $\frac{18 \pm 10 \sqrt{2}}{2}$ .

$x = 9 \pm 5 \sqrt{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = 9 \pm 5 \sqrt{2}$

Decimal Form:

$x = 16.07106781 \dots, 1.92893218 \dots$

Step 6