Solve by Completing the Square x^2+18x+67=0

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

Step 1

Subtract $67$ from both sides of the equation.

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

Step 2

To create a trinomial square on the left side of the equation, find a value that is equal to the square of half of $b$ .

${(\frac{b}{2})}^{2} = {(9)}^{2}$

Step 3

Add the term to each side of the equation.

$x^{2} + 18 x + {(9)}^{2} = - 67 + {(9)}^{2}$

Step 4

Simplify the equation.

Simplify the left side.

Raise $9$ to the power of $2$ .

$x^{2} + 18 x + 81 = - 67 + {(9)}^{2}$

Simplify the right side.

Simplify $- 67 + {(9)}^{2}$ .

Raise $9$ to the power of $2$ .

$x^{2} + 18 x + 81 = - 67 + 81$

Add $- 67$ and $81$ .

$x^{2} + 18 x + 81 = 14$

Step 5

Factor the perfect trinomial square into ${(x + 9)}^{2}$ .

${(x + 9)}^{2} = 14$

Step 6

Solve the equation for $x$ .

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

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

Subtract $9$ from both sides of the equation.

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

Step 7

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = - 5.25834261 \dots, - 12.74165738 \dots$