Solve by Completing the Square x^2-2x-1=0

$x^{2} - 2 x - 1 = 0$

Step 1

Add $1$ to both sides of the equation.

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

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.

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

Step 4

Simplify the equation.

Simplify the left side.

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

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

Simplify the right side.

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

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

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

Add $1$ and $1$ .

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

Step 5

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

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

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.

$x - 1 = \pm \sqrt{2}$

Add $1$ to both sides of the equation.

$x = \pm \sqrt{2} + 1$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \pm \sqrt{2} + 1$

Decimal Form:

$x = 2.41421356 \dots, - 0.41421356 \dots$