How do you solve this for x by completing the square?

x² - 2x -1 = 0 => x² - 2x + 1 = 1 => (x - 1)² = 2 => x - 1 = ± √2
=> x = 1 ± √2

AJM

x² - 2x = 1
x² - 2x + 1 = 1 + 1
(x - 1)² = 2
x - 1 = ±√2
x = 1 ± √2

Answer: x = 1 ± √2

x² - 2x - 1 = 0
x² - x = 1 + (- 1)²
x² - x = 1 + 1
(x - 1)² = 2
x - 1 = 1.4142135

x = 1.4142135 + 1, x = 2.4142135
x = - 1.4142135 + 1, x = - 0.4142135

Answer: x = 2.4142135, x = - 0.4142135

Proof (x = 2.4142135):
2.4142135² - 2(2.4142135) = 1
5.828427 - 4.828427 = 1
1 = 1

Proof (x = - 0.4142135):
(- 0.4142135)² - 2(- 0.4142135) = 1
0.171573 + 0.828427 = 1
1 = 1

x² - 2x = 1
x² - 2x + 4 = 1 + 4
(x - 2)² = 5
x + 2 = ±√5
x = - 2 ± √5

perhaps you meant: x² -2x + 1 = 0
(x - 1)(x -1)=0
x=1

To solve by completing the square, you need to know the fact that

a² + 2ab + b² = (a + b).

By substituting x for a, the formula can be rewritten as

x² + 2b*x + b² = (x + b)

If you can manipulate your original equation into this form:

x² + 2b*x + b² = d

then you can also say

(x + b) = d which is much easier to solve.

Your original equation is

x² - 2x - 1 = 0 => x² - 2x = 1

In this case b = -2/2 = -1 so b² = 1 . You need to add b² = 1 to both sides to convert the equation into the 'square' form.

x² - 2x = 1 => x² - 2x +1 = 2

Now we can apply the 'completing the square formula' and say:

(x - 1)² = 2 which is easy to solve.

To make the method clearer here is another example.

Solve x² - 6x + 5 = 0

Rearrange to x² - 6x = -5

In this case b = (-6)/2 = -3 so b² = (-3)² = 9

Add 9 to both sides => x² - 6x +9 = 4

Now use the 'completing the square formula' and remembering that b = (-3) the equation can be written as

(x - 3)² = 4 which is also easy to solve.

x^2 - 2x - 1 = 0
x^2 - 2x = 1
x^2 - x - x = 1
x^2 - x - x + 1 = 1 + 1
(x^2 - x) - (x - 1) = 2
x(x - 1) - 1(x - 1) = 2
(x - 1)(x - 1) = 2
(x - 1)^2 = 2
x - 1 = ±√2
x = ±√2 + 1

x²-2x-1=0
x²-2x+1-2=0
x²-2x+1=2
(x-1)²=2
x-1= + or - sqrt of 2
x=1+ or - sqrt of 2

x² - 2x - 1 = 0

Add 1 to both sides.
x² - 2x - 1 + 1 = 0 + 1
x² - 2x = 1

Group.
(x² - 2x) = 1

Add placeholders.
(x² - 2x + ___) = 1 + ___

Take the coefficient of the x term: -2
Divide it by 2: -2 / 2 = -1
Square it: (-1)² = 1

Add 1 to both blanks.
(x² - 2x + 1) = 1 + 1

x² - 2x + 1 is the expanded form of a perfect square binomial.

Remember that (a - b)² = a² - 2ab + b². Apply this to what you have.
(x² - 2x + 1) = 1 + 1
(x - 1)² = 1 + 1

Simplify the rest.
(x - 1)² = 2 <== completed square

Take the square root of both sides
√(x - 1)² = √2
x - 1 = ± √2

Add 1 from both sides.
x - 1 + 1 = 1 ± √2
x = 1 ± √2

ANSWER: x = 1 ± √2

CHECK USING QUADRATIC FORMULA:

Given: ax² + bx + c = 0
Quadratic Formula: x = [-b ± √(b² - 4ac)] / 2a

Given: x² - 2x - 1 = 0
Means: a = 1, b = -2, c = -1

x = [-b ± √(b² - 4ac)] / 2a
x = [-(-2) ± √((-2)² - 4(1)(-1))] / 2(1)
x = [2 ± √(4 + 4] / 2
x = [2 ± √8] / 2
x = [2 ± √(4 * 2)] / 2
x = [2 ± 2√2] / 2
x = 1 ± √2
true

Generally, ensure that the coefficient of x² is 1
x² - 2x = 1
Add the square of half the coefficient of x to both sides
x² - 2x + [-2/2]² = 1 + [-2/2]²
Simplify
x² - 2x + (-1)² = 1 + (-1)²
Recall a² + 2ab + b² = (a + b)²
(x)² - 2(x)(1) + (1)² = 1 + 1
(x - 1)² = 2
x - 1 = ±√2
x = 1 ± √2