q^2-2q-1=0

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

Simplifying
q2 + -2q + -1 = 0

Reorder the terms:
-1 + -2q + q2 = 0

Solving
-1 + -2q + q2 = 0

Solving for variable 'q'.

Begin completing the square.

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + -2q + 1 + q2 = 0 + 1

Reorder the terms:
-1 + 1 + -2q + q2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + -2q + q2 = 0 + 1
-2q + q2 = 0 + 1

Combine like terms: 0 + 1 = 1
-2q + q2 = 1

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

Add '1' to each side of the equation.
-2q + 1 + q2 = 1 + 1

Reorder the terms:
1 + -2q + q2 = 1 + 1

Combine like terms: 1 + 1 = 2
1 + -2q + q2 = 2

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

Calculate the square root of the right side: 1.414213562

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

Subproblem 1
q + -1 = 1.414213562

Simplifying
q + -1 = 1.414213562

Reorder the terms:
-1 + q = 1.414213562

Solving
-1 + q = 1.414213562

Solving for variable 'q'.

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

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

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

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

Simplifying
q = 2.414213562

Subproblem 2
q + -1 = -1.414213562

Simplifying
q + -1 = -1.414213562

Reorder the terms:
-1 + q = -1.414213562

Solving
-1 + q = -1.414213562

Solving for variable 'q'.

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

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

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

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

Simplifying
q = -0.414213562

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