Solve Using the Square Root Property x^2-18x+81=5

$x^{2} - 18 x + 81 = 5$

Step 1

Move all terms to the left side of the equation and simplify.

Subtract $5$ from both sides of the equation.

$x^{2} - 18 x + 81 - 5 = 0$

Subtract $5$ from $81$ .

$x^{2} - 18 x + 76 = 0$

Step 2

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 3

Substitute the values $a = 1$ , $b = - 18$ , and $c = 76$ into the quadratic formula and solve for $x$ .

$\frac{18 \pm \sqrt{{(- 18)}^{2} - 4 \cdot (1 \cdot 76)}}{2 \cdot 1}$

Step 4

Simplify.

Simplify the numerator.

Raise $- 18$ to the power of $2$ .

$x = \frac{18 \pm \sqrt{324 - 4 \cdot 1 \cdot 76}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 76$ .

Multiply $- 4$ by $1$ .

$x = \frac{18 \pm \sqrt{324 - 4 \cdot 76}}{2 \cdot 1}$

Multiply $- 4$ by $76$ .

$x = \frac{18 \pm \sqrt{324 - 304}}{2 \cdot 1}$

Subtract $304$ from $324$ .

$x = \frac{18 \pm \sqrt{20}}{2 \cdot 1}$

Rewrite $20$ as $2^{2} \cdot 5$ .

Factor $4$ out of $20$ .

$x = \frac{18 \pm \sqrt{4 (5)}}{2 \cdot 1}$

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

$x = \frac{18 \pm \sqrt{2^{2} \cdot 5}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{18 \pm 2 \sqrt{5}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{18 \pm 2 \sqrt{5}}{2}$

Simplify $\frac{18 \pm 2 \sqrt{5}}{2}$ .

$x = 9 \pm \sqrt{5}$

Step 5

Simplify the expression to solve for the $+$ portion of the $\pm$ .

Simplify the numerator.

Raise $- 18$ to the power of $2$ .

$x = \frac{18 \pm \sqrt{324 - 4 \cdot 1 \cdot 76}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 76$ .

Multiply $- 4$ by $1$ .

$x = \frac{18 \pm \sqrt{324 - 4 \cdot 76}}{2 \cdot 1}$

Multiply $- 4$ by $76$ .

$x = \frac{18 \pm \sqrt{324 - 304}}{2 \cdot 1}$

Subtract $304$ from $324$ .

$x = \frac{18 \pm \sqrt{20}}{2 \cdot 1}$

Rewrite $20$ as $2^{2} \cdot 5$ .

Factor $4$ out of $20$ .

$x = \frac{18 \pm \sqrt{4 (5)}}{2 \cdot 1}$

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

$x = \frac{18 \pm \sqrt{2^{2} \cdot 5}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{18 \pm 2 \sqrt{5}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{18 \pm 2 \sqrt{5}}{2}$

Simplify $\frac{18 \pm 2 \sqrt{5}}{2}$ .

$x = 9 \pm \sqrt{5}$

Change the $\pm$ to $+$ .

$x = 9 + \sqrt{5}$

Step 6

Simplify the expression to solve for the $-$ portion of the $\pm$ .

Simplify the numerator.

Raise $- 18$ to the power of $2$ .

$x = \frac{18 \pm \sqrt{324 - 4 \cdot 1 \cdot 76}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 76$ .

Multiply $- 4$ by $1$ .

$x = \frac{18 \pm \sqrt{324 - 4 \cdot 76}}{2 \cdot 1}$

Multiply $- 4$ by $76$ .

$x = \frac{18 \pm \sqrt{324 - 304}}{2 \cdot 1}$

Subtract $304$ from $324$ .

$x = \frac{18 \pm \sqrt{20}}{2 \cdot 1}$

Rewrite $20$ as $2^{2} \cdot 5$ .

Factor $4$ out of $20$ .

$x = \frac{18 \pm \sqrt{4 (5)}}{2 \cdot 1}$

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

$x = \frac{18 \pm \sqrt{2^{2} \cdot 5}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{18 \pm 2 \sqrt{5}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{18 \pm 2 \sqrt{5}}{2}$

Simplify $\frac{18 \pm 2 \sqrt{5}}{2}$ .

$x = 9 \pm \sqrt{5}$

Change the $\pm$ to $-$ .

$x = 9 - \sqrt{5}$

Step 7

The final answer is the combination of both solutions.

$x = 9 + \sqrt{5}, 9 - \sqrt{5}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = 9 + \sqrt{5}, 9 - \sqrt{5}$

Decimal Form:

$x = 11.23606797 \dots, 6.76393202 \dots$

Solve Using the Square Root Property x^2-18x+81=5

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