Solve by Completing the Square x^2+16x+53=0

$x^{2} + 16 x + 53 = 0$

Step 1

Subtract $53$ from both sides of the equation.

$x^{2} + 16 x = - 53$

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 $b$ .

${(\frac{b}{2})}^{2} = {(8)}^{2}$

Step 3

Add the term to each side of the equation.

$x^{2} + 16 x + {(8)}^{2} = - 53 + {(8)}^{2}$

Step 4

Simplify the equation.

Simplify the left side.

Raise $8$ to the power of $2$ .

$x^{2} + 16 x + 64 = - 53 + {(8)}^{2}$

Simplify the right side.

Simplify $- 53 + {(8)}^{2}$ .

Raise $8$ to the power of $2$ .

$x^{2} + 16 x + 64 = - 53 + 64$

Add $- 53$ and $64$ .

$x^{2} + 16 x + 64 = 11$

Step 5

Factor the perfect trinomial square into ${(x + 8)}^{2}$ .

${(x + 8)}^{2} = 11$

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.

$x + 8 = \pm \sqrt{11}$

Subtract $8$ from both sides of the equation.

$x = \pm \sqrt{11} - 8$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \pm \sqrt{11} - 8$

Decimal Form:

$x = - 4.68337520 \dots, - 11.31662479 \dots$