y^2-2y-1=0

Solution for y^2-2y-1=0 equation:

Simplifying
y2 + -2y + -1 = 0

Reorder the terms:
-1 + -2y + y2 = 0

Solving
-1 + -2y + y2 = 0

Solving for variable 'y'.

Begin completing the square.

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + -2y + 1 + y2 = 0 + 1

Reorder the terms:
-1 + 1 + -2y + y2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + -2y + y2 = 0 + 1
-2y + y2 = 0 + 1

Combine like terms: 0 + 1 = 1
-2y + y2 = 1

The y term is -2y.  Take half its coefficient (-1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
-2y + 1 + y2 = 1 + 1

Reorder the terms:
1 + -2y + y2 = 1 + 1

Combine like terms: 1 + 1 = 2
1 + -2y + y2 = 2

Factor a perfect square on the left side:
(y + -1)(y + -1) = 2

Calculate the square root of the right side: 1.414213562

Break this problem into two subproblems by setting 
(y + -1) equal to 1.414213562 and -1.414213562.

Subproblem 1
y + -1 = 1.414213562

Simplifying
y + -1 = 1.414213562

Reorder the terms:
-1 + y = 1.414213562

Solving
-1 + y = 1.414213562

Solving for variable 'y'.

Move all terms containing y to the left, all other terms to the right.

Add '1' to each side of the equation.
-1 + 1 + y = 1.414213562 + 1

Combine like terms: -1 + 1 = 0
0 + y = 1.414213562 + 1
y = 1.414213562 + 1

Combine like terms: 1.414213562 + 1 = 2.414213562
y = 2.414213562

Simplifying
y = 2.414213562

Subproblem 2
y + -1 = -1.414213562

Simplifying
y + -1 = -1.414213562

Reorder the terms:
-1 + y = -1.414213562

Solving
-1 + y = -1.414213562

Solving for variable 'y'.

Move all terms containing y to the left, all other terms to the right.

Add '1' to each side of the equation.
-1 + 1 + y = -1.414213562 + 1

Combine like terms: -1 + 1 = 0
0 + y = -1.414213562 + 1
y = -1.414213562 + 1

Combine like terms: -1.414213562 + 1 = -0.414213562
y = -0.414213562

Simplifying
y = -0.414213562

Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {2.414213562, -0.414213562}