Solve by Completing the Square x^2+6x+6=0

$x^{2} + 6 x + 6 = 0$

Step 1

Subtract $6$ from both sides of the equation.

$x^{2} + 6 x = - 6$

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 $b$ .

${(\frac{b}{2})}^{2} = {(3)}^{2}$

Step 3

Add the term to each side of the equation.

$x^{2} + 6 x + {(3)}^{2} = - 6 + {(3)}^{2}$

Step 4

Simplify the equation.

Simplify the left side.

Raise $3$ to the power of $2$ .

$x^{2} + 6 x + 9 = - 6 + {(3)}^{2}$

Simplify the right side.

Simplify $- 6 + {(3)}^{2}$ .

Raise $3$ to the power of $2$ .

$x^{2} + 6 x + 9 = - 6 + 9$

Add $- 6$ and $9$ .

$x^{2} + 6 x + 9 = 3$

Step 5

Factor the perfect trinomial square into ${(x + 3)}^{2}$ .

${(x + 3)}^{2} = 3$

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.

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

Subtract $3$ from both sides of the equation.

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

Step 7

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = - 1.26794919 \dots, - 4.73205080 \dots$