Solve by Completing the Square x^2-8x+13=0

$x^{2} - 8 x + 13 = 0$

Step 1

Subtract $13$ from both sides of the equation.

$x^{2} - 8 x = - 13$

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 $b$ .

${(\frac{b}{2})}^{2} = {(- 4)}^{2}$

Step 3

Add the term to each side of the equation.

$x^{2} - 8 x + {(- 4)}^{2} = - 13 + {(- 4)}^{2}$

Step 4

Simplify the equation.

Simplify the left side.

Raise $- 4$ to the power of $2$ .

$x^{2} - 8 x + 16 = - 13 + {(- 4)}^{2}$

Simplify the right side.

Simplify $- 13 + {(- 4)}^{2}$ .

Raise $- 4$ to the power of $2$ .

$x^{2} - 8 x + 16 = - 13 + 16$

Add $- 13$ and $16$ .

$x^{2} - 8 x + 16 = 3$

Step 5

Factor the perfect trinomial square into ${(x - 4)}^{2}$ .

${(x - 4)}^{2} = 3$

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.

$x - 4 = \pm \sqrt{3}$

Add $4$ to both sides of the equation.

$x = \pm \sqrt{3} + 4$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \pm \sqrt{3} + 4$

Decimal Form:

$x = 5.73205080 \dots, 2.26794919 \dots$