2y^2=4y-1

Solution for 2y^2=4y-1 equation:

Simplifying
2y2 = 4y + -1

Reorder the terms:
2y2 = -1 + 4y

Solving
2y2 = -1 + 4y

Solving for variable 'y'.

Reorder the terms:
1 + -4y + 2y2 = -1 + 4y + 1 + -4y

Reorder the terms:
1 + -4y + 2y2 = -1 + 1 + 4y + -4y

Combine like terms: -1 + 1 = 0
1 + -4y + 2y2 = 0 + 4y + -4y
1 + -4y + 2y2 = 4y + -4y

Combine like terms: 4y + -4y = 0
1 + -4y + 2y2 = 0

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

Divide each side by '2'.
0.5 + -2y + y2 = 0

Move the constant term to the right:

Add '-0.5' to each side of the equation.
0.5 + -2y + -0.5 + y2 = 0 + -0.5

Reorder the terms:
0.5 + -0.5 + -2y + y2 = 0 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + -2y + y2 = 0 + -0.5
-2y + y2 = 0 + -0.5

Combine like terms: 0 + -0.5 = -0.5
-2y + y2 = -0.5

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 = -0.5 + 1

Reorder the terms:
1 + -2y + y2 = -0.5 + 1

Combine like terms: -0.5 + 1 = 0.5
1 + -2y + y2 = 0.5

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

Calculate the square root of the right side: 0.707106781

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

Subproblem 1
y + -1 = 0.707106781

Simplifying
y + -1 = 0.707106781

Reorder the terms:
-1 + y = 0.707106781

Solving
-1 + y = 0.707106781

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 = 0.707106781 + 1

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

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

Simplifying
y = 1.707106781

Subproblem 2
y + -1 = -0.707106781

Simplifying
y + -1 = -0.707106781

Reorder the terms:
-1 + y = -0.707106781

Solving
-1 + y = -0.707106781

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 = -0.707106781 + 1

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

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

Simplifying
y = 0.292893219

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