Solve by Completing the Square x^2-18x+86=0

$x^{2} - 18 x + 86 = 0$

Step 1

Subtract $86$ from both sides of the equation.

$x^{2} - 18 x = - 86$

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.

$x^{2} - 18 x + {(- 9)}^{2} = - 86 + {(- 9)}^{2}$

Step 4

Simplify the equation.

Simplify the left side.

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

$x^{2} - 18 x + 81 = - 86 + {(- 9)}^{2}$

Simplify the right side.

Simplify $- 86 + {(- 9)}^{2}$ .

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

$x^{2} - 18 x + 81 = - 86 + 81$

Add $- 86$ and $81$ .

$x^{2} - 18 x + 81 = - 5$

Step 5

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

${(x - 9)}^{2} = - 5$

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.

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

Simplify $\pm \sqrt{- 5}$ .

Rewrite $- 5$ as $- 1 (5)$ .

$x - 9 = \pm \sqrt{- 1 (5)}$

Rewrite $\sqrt{- 1 (5)}$ as $\sqrt{- 1} \cdot \sqrt{5}$ .

$x - 9 = \pm \sqrt{- 1} \cdot \sqrt{5}$

Rewrite $\sqrt{- 1}$ as $i$ .

$x - 9 = \pm i \sqrt{5}$

Move all terms not containing $x$ to the right side of the equation.

Add $9$ to both sides of the equation.

$x = \pm i \sqrt{5} + 9$

Reorder $\pm i \sqrt{5}$ and $9$ .

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