Solve by Completing the Square x^2-12x+31=0

$x^{2} - 12 x + 31 = 0$

Step 1

Subtract $31$ from both sides of the equation.

$x^{2} - 12 x = - 31$

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 $b$ .

${(\frac{b}{2})}^{2} = {(- 6)}^{2}$

Step 3

Add the term to each side of the equation.

$x^{2} - 12 x + {(- 6)}^{2} = - 31 + {(- 6)}^{2}$

Step 4

Simplify the equation.

Simplify the left side.

Raise $- 6$ to the power of $2$ .

$x^{2} - 12 x + 36 = - 31 + {(- 6)}^{2}$

Simplify the right side.

Simplify $- 31 + {(- 6)}^{2}$ .

Raise $- 6$ to the power of $2$ .

$x^{2} - 12 x + 36 = - 31 + 36$

Add $- 31$ and $36$ .

$x^{2} - 12 x + 36 = 5$

Step 5

Factor the perfect trinomial square into ${(x - 6)}^{2}$ .

${(x - 6)}^{2} = 5$

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.

$x - 6 = \pm \sqrt{5}$

Add $6$ to both sides of the equation.

$x = \pm \sqrt{5} + 6$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \pm \sqrt{5} + 6$

Decimal Form:

$x = 8.23606797 \dots, 3.76393202 \dots$