2w^2-4w-1=0

Solution for 2w^2-4w-1=0 equation:

Simplifying
2w2 + -4w + -1 = 0

Reorder the terms:
-1 + -4w + 2w2 = 0

Solving
-1 + -4w + 2w2 = 0

Solving for variable 'w'.

Begin completing the square.  Divide all terms by
2 the coefficient of the squared term: 

Divide each side by '2'.
-0.5 + -2w + w2 = 0

Move the constant term to the right:

Add '0.5' to each side of the equation.
-0.5 + -2w + 0.5 + w2 = 0 + 0.5

Reorder the terms:
-0.5 + 0.5 + -2w + w2 = 0 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + -2w + w2 = 0 + 0.5
-2w + w2 = 0 + 0.5

Combine like terms: 0 + 0.5 = 0.5
-2w + w2 = 0.5

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

Add '1' to each side of the equation.
-2w + 1 + w2 = 0.5 + 1

Reorder the terms:
1 + -2w + w2 = 0.5 + 1

Combine like terms: 0.5 + 1 = 1.5
1 + -2w + w2 = 1.5

Factor a perfect square on the left side:
(w + -1)(w + -1) = 1.5

Calculate the square root of the right side: 1.224744871

Break this problem into two subproblems by setting 
(w + -1) equal to 1.224744871 and -1.224744871.

Subproblem 1
w + -1 = 1.224744871

Simplifying
w + -1 = 1.224744871

Reorder the terms:
-1 + w = 1.224744871

Solving
-1 + w = 1.224744871

Solving for variable 'w'.

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

Add '1' to each side of the equation.
-1 + 1 + w = 1.224744871 + 1

Combine like terms: -1 + 1 = 0
0 + w = 1.224744871 + 1
w = 1.224744871 + 1

Combine like terms: 1.224744871 + 1 = 2.224744871
w = 2.224744871

Simplifying
w = 2.224744871

Subproblem 2
w + -1 = -1.224744871

Simplifying
w + -1 = -1.224744871

Reorder the terms:
-1 + w = -1.224744871

Solving
-1 + w = -1.224744871

Solving for variable 'w'.

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

Add '1' to each side of the equation.
-1 + 1 + w = -1.224744871 + 1

Combine like terms: -1 + 1 = 0
0 + w = -1.224744871 + 1
w = -1.224744871 + 1

Combine like terms: -1.224744871 + 1 = -0.224744871
w = -0.224744871

Simplifying
w = -0.224744871

Solution
The solution to the problem is based on the solutions
from the subproblems.
w = {2.224744871, -0.224744871}