Solve by Completing the Square x^2-2x-1=8

$x^{2} - 2 x - 1 = 8$

Step 1

Move all terms not containing $x$ to the right side of the equation.

Add $1$ to both sides of the equation.

$x^{2} - 2 x = 8 + 1$

Add $8$ and $1$ .

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

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 $b$ .

${(\frac{b}{2})}^{2} = {(- 1)}^{2}$

Step 3

Add the term to each side of the equation.

$x^{2} - 2 x + {(- 1)}^{2} = 9 + {(- 1)}^{2}$

Step 4

Simplify the equation.

Simplify the left side.

Raise $- 1$ to the power of $2$ .

$x^{2} - 2 x + 1 = 9 + {(- 1)}^{2}$

Simplify the right side.

Simplify $9 + {(- 1)}^{2}$ .

Raise $- 1$ to the power of $2$ .

$x^{2} - 2 x + 1 = 9 + 1$

Add $9$ and $1$ .

$x^{2} - 2 x + 1 = 10$

Step 5

Factor the perfect trinomial square into ${(x - 1)}^{2}$ .

${(x - 1)}^{2} = 10$

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.

$x - 1 = \pm \sqrt{10}$

Add $1$ to both sides of the equation.

$x = \pm \sqrt{10} + 1$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \pm \sqrt{10} + 1$

Decimal Form:

$x = 4.16227766 \dots, - 2.16227766 \dots$