Solve by Completing the Square x^2+12x+13=0

$x^{2} + 12 x + 13 = 0$

Step 1

Subtract $13$ from both sides of the equation.

$x^{2} + 12 x = - 13$

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.

$x^{2} + 12 x + {(6)}^{2} = - 13 + {(6)}^{2}$

Step 4

Simplify the equation.

Simplify the left side.

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

$x^{2} + 12 x + 36 = - 13 + {(6)}^{2}$

Simplify the right side.

Simplify $- 13 + {(6)}^{2}$ .

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

$x^{2} + 12 x + 36 = - 13 + 36$

Add $- 13$ and $36$ .

$x^{2} + 12 x + 36 = 23$

Step 5

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

${(x + 6)}^{2} = 23$

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.

$x + 6 = \pm \sqrt{23}$

Subtract $6$ from both sides of the equation.

$x = \pm \sqrt{23} - 6$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \pm \sqrt{23} - 6$

Decimal Form:

$x = - 1.20416847 \dots, - 10.79583152 \dots$