Solve Using the Quadratic Formula x^2-18x+67=0

$x^{2} - 18 x + 67 = 0$

Step 1

Use the quadratic formula to find the solutions.

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

Step 2

Substitute the values $a = 1$ , $b = - 18$ , and $c = 67$ into the quadratic formula and solve for $x$ .

$\frac{18 \pm \sqrt{{(- 18)}^{2} - 4 \cdot (1 \cdot 67)}}{2 \cdot 1}$

Step 3

Simplify.

Simplify the numerator.

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

$x = \frac{18 \pm \sqrt{324 - 4 \cdot 1 \cdot 67}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 67$ .

Multiply $- 4$ by $1$ .

$x = \frac{18 \pm \sqrt{324 - 4 \cdot 67}}{2 \cdot 1}$

Multiply $- 4$ by $67$ .

$x = \frac{18 \pm \sqrt{324 - 268}}{2 \cdot 1}$

Subtract $268$ from $324$ .

$x = \frac{18 \pm \sqrt{56}}{2 \cdot 1}$

Rewrite $56$ as $2^{2} \cdot 14$ .

Factor $4$ out of $56$ .

$x = \frac{18 \pm \sqrt{4 (14)}}{2 \cdot 1}$

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

$x = \frac{18 \pm \sqrt{2^{2} \cdot 14}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{18 \pm 2 \sqrt{14}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{18 \pm 2 \sqrt{14}}{2}$

Simplify $\frac{18 \pm 2 \sqrt{14}}{2}$ .

$x = 9 \pm \sqrt{14}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = 9 \pm \sqrt{14}$

Decimal Form:

$x = 12.74165738 \dots, 5.25834261 \dots$