Solve by Completing the Square x^2-6x-9=0

$x^{2} - 6 x - 9 = 0$

Step 1

Add $9$ to both sides of the equation.

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

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.

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

Step 4

Simplify the equation.

Simplify the left side.

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

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

Simplify the right side.

Simplify $9 + {(- 3)}^{2}$ .

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

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

Add $9$ and $9$ .

$x^{2} - 6 x + 9 = 18$

Step 5

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

${(x - 3)}^{2} = 18$

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.

$x - 3 = \pm \sqrt{18}$

Simplify $\pm \sqrt{18}$ .

Rewrite $18$ as $3^{2} \cdot 2$ .

Factor $9$ out of $18$ .

$x - 3 = \pm \sqrt{9 (2)}$

Rewrite $9$ as $3^{2}$ .

$x - 3 = \pm \sqrt{3^{2} \cdot 2}$

Pull terms out from under the radical.

$x - 3 = \pm 3 \sqrt{2}$

Add $3$ to both sides of the equation.

$x = \pm 3 \sqrt{2} + 3$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \pm 3 \sqrt{2} + 3$

Decimal Form:

$x = 7.24264068 \dots, - 1.24264068 \dots$