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 = 0.5
1 + -2w + w2 = 0.5

Factor a perfect square on the left side:
(w + -1)(w + -1) = 0.5

Calculate the square root of the right side: 0.707106781

Break this problem into two subproblems by setting 
(w + -1) equal to 0.707106781 and -0.707106781.

Subproblem 1
w + -1 = 0.707106781

Simplifying
w + -1 = 0.707106781

Reorder the terms:
-1 + w = 0.707106781

Solving
-1 + w = 0.707106781

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 = 0.707106781 + 1

Combine like terms: -1 + 1 = 0
0 + w = 0.707106781 + 1
w = 0.707106781 + 1

Combine like terms: 0.707106781 + 1 = 1.707106781
w = 1.707106781

Simplifying
w = 1.707106781

Subproblem 2
w + -1 = -0.707106781

Simplifying
w + -1 = -0.707106781

Reorder the terms:
-1 + w = -0.707106781

Solving
-1 + w = -0.707106781

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 = -0.707106781 + 1

Combine like terms: -1 + 1 = 0
0 + w = -0.707106781 + 1
w = -0.707106781 + 1

Combine like terms: -0.707106781 + 1 = 0.292893219
w = 0.292893219

Simplifying
w = 0.292893219

Solution
The solution to the problem is based on the solutions
from the subproblems.
w = {1.707106781, 0.292893219}