Solve by Completing the Square x^2+16x=41

$x^{2} + 16 x = 41$

Step 1

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 2

Add the term to each side of the equation.

$x^{2} + 16 x + {(8)}^{2} = 41 + {(8)}^{2}$

Step 3

Simplify the equation.

Simplify the left side.

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

$x^{2} + 16 x + 64 = 41 + {(8)}^{2}$

Simplify the right side.

Simplify $41 + {(8)}^{2}$ .

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

$x^{2} + 16 x + 64 = 41 + 64$

Add $41$ and $64$ .

$x^{2} + 16 x + 64 = 105$

Step 4

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

${(x + 8)}^{2} = 105$

Step 5

Solve the equation for $x$ .

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x + 8 = \pm \sqrt{105}$

Subtract $8$ from both sides of the equation.

$x = \pm \sqrt{105} - 8$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \pm \sqrt{105} - 8$

Decimal Form:

$x = 2.24695076 \dots, - 18.24695076 \dots$