u^2-2u-1=0

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

Simplifying
u2 + -2u + -1 = 0

Reorder the terms:
-1 + -2u + u2 = 0

Solving
-1 + -2u + u2 = 0

Solving for variable 'u'.

Begin completing the square.

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + -2u + 1 + u2 = 0 + 1

Reorder the terms:
-1 + 1 + -2u + u2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + -2u + u2 = 0 + 1
-2u + u2 = 0 + 1

Combine like terms: 0 + 1 = 1
-2u + u2 = 1

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

Add '1' to each side of the equation.
-2u + 1 + u2 = 1 + 1

Reorder the terms:
1 + -2u + u2 = 1 + 1

Combine like terms: 1 + 1 = 2
1 + -2u + u2 = 2

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

Calculate the square root of the right side: 1.414213562

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

Subproblem 1
u + -1 = 1.414213562

Simplifying
u + -1 = 1.414213562

Reorder the terms:
-1 + u = 1.414213562

Solving
-1 + u = 1.414213562

Solving for variable 'u'.

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

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

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

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

Simplifying
u = 2.414213562

Subproblem 2
u + -1 = -1.414213562

Simplifying
u + -1 = -1.414213562

Reorder the terms:
-1 + u = -1.414213562

Solving
-1 + u = -1.414213562

Solving for variable 'u'.

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

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

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

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

Simplifying
u = -0.414213562

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