Solve by Completing the Square x^2-24x+81=0

$x^{2} - 24 x + 81 = 0$

Step 1

Subtract $81$ from both sides of the equation.

$x^{2} - 24 x = - 81$

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 $b$ .

${(\frac{b}{2})}^{2} = {(- 12)}^{2}$

Step 3

Add the term to each side of the equation.

$x^{2} - 24 x + {(- 12)}^{2} = - 81 + {(- 12)}^{2}$

Step 4

Simplify the equation.

Simplify the left side.

Raise $- 12$ to the power of $2$ .

$x^{2} - 24 x + 144 = - 81 + {(- 12)}^{2}$

Simplify the right side.

Simplify $- 81 + {(- 12)}^{2}$ .

Raise $- 12$ to the power of $2$ .

$x^{2} - 24 x + 144 = - 81 + 144$

Add $- 81$ and $144$ .

$x^{2} - 24 x + 144 = 63$

Step 5

Factor the perfect trinomial square into ${(x - 12)}^{2}$ .

${(x - 12)}^{2} = 63$

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.

$x - 12 = \pm \sqrt{63}$

Simplify $\pm \sqrt{63}$ .

Rewrite $63$ as $3^{2} \cdot 7$ .

Factor $9$ out of $63$ .

$x - 12 = \pm \sqrt{9 (7)}$

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

$x - 12 = \pm \sqrt{3^{2} \cdot 7}$

Pull terms out from under the radical.

$x - 12 = \pm 3 \sqrt{7}$

Add $12$ to both sides of the equation.

$x = \pm 3 \sqrt{7} + 12$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \pm 3 \sqrt{7} + 12$

Decimal Form:

$x = 19.93725393 \dots, 4.06274606 \dots$