Solve by Completing the Square x^2-12x-9=0

$x^{2} - 12 x - 9 = 0$

Step 1

Add $9$ to both sides of the equation.

$x^{2} - 12 x = 9$

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 $b$ .

${(\frac{b}{2})}^{2} = {(- 6)}^{2}$

Step 3

Add the term to each side of the equation.

$x^{2} - 12 x + {(- 6)}^{2} = 9 + {(- 6)}^{2}$

Step 4

Simplify the equation.

Simplify the left side.

Raise $- 6$ to the power of $2$ .

$x^{2} - 12 x + 36 = 9 + {(- 6)}^{2}$

Simplify the right side.

Simplify $9 + {(- 6)}^{2}$ .

Raise $- 6$ to the power of $2$ .

$x^{2} - 12 x + 36 = 9 + 36$

Add $9$ and $36$ .

$x^{2} - 12 x + 36 = 45$

Step 5

Factor the perfect trinomial square into ${(x - 6)}^{2}$ .

${(x - 6)}^{2} = 45$

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.

$x - 6 = \pm \sqrt{45}$

Simplify $\pm \sqrt{45}$ .

Rewrite $45$ as $3^{2} \cdot 5$ .

Factor $9$ out of $45$ .

$x - 6 = \pm \sqrt{9 (5)}$

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

$x - 6 = \pm \sqrt{3^{2} \cdot 5}$

Pull terms out from under the radical.

$x - 6 = \pm 3 \sqrt{5}$

Add $6$ to both sides of the equation.

$x = \pm 3 \sqrt{5} + 6$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \pm 3 \sqrt{5} + 6$

Decimal Form:

$x = 12.70820393 \dots, - 0.70820393 \dots$