Solve by Completing the Square x^2-18x=3

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

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 $b$ .

${(\frac{b}{2})}^{2} = {(- 9)}^{2}$

Step 2

Add the term to each side of the equation.

$x^{2} - 18 x + {(- 9)}^{2} = 3 + {(- 9)}^{2}$

Step 3

Simplify the equation.

Simplify the left side.

Raise $- 9$ to the power of $2$ .

$x^{2} - 18 x + 81 = 3 + {(- 9)}^{2}$

Simplify the right side.

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

Raise $- 9$ to the power of $2$ .

$x^{2} - 18 x + 81 = 3 + 81$

Add $3$ and $81$ .

$x^{2} - 18 x + 81 = 84$

Step 4

Factor the perfect trinomial square into ${(x - 9)}^{2}$ .

${(x - 9)}^{2} = 84$

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.

$x - 9 = \pm \sqrt{84}$

Simplify $\pm \sqrt{84}$ .

Rewrite $84$ as $2^{2} \cdot 21$ .

Factor $4$ out of $84$ .

$x - 9 = \pm \sqrt{4 (21)}$

Rewrite $4$ as $2^{2}$ .

$x - 9 = \pm \sqrt{2^{2} \cdot 21}$

Pull terms out from under the radical.

$x - 9 = \pm 2 \sqrt{21}$

Add $9$ to both sides of the equation.

$x = \pm 2 \sqrt{21} + 9$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \pm 2 \sqrt{21} + 9$

Decimal Form:

$x = 18.16515138 \dots, - 0.16515138 \dots$