8y^2=8y-1

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Solution for 8y^2=8y-1 equation:


Simplifying
8y2 = 8y + -1

Reorder the terms:
8y2 = -1 + 8y

Solving
8y2 = -1 + 8y

Solving for variable 'y'.

Reorder the terms:
1 + -8y + 8y2 = -1 + 8y + 1 + -8y

Reorder the terms:
1 + -8y + 8y2 = -1 + 1 + 8y + -8y

Combine like terms: -1 + 1 = 0
1 + -8y + 8y2 = 0 + 8y + -8y
1 + -8y + 8y2 = 8y + -8y

Combine like terms: 8y + -8y = 0
1 + -8y + 8y2 = 0

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

Divide each side by '8'.
0.125 + -1y + y2 = 0

Move the constant term to the right:

Add '-0.125' to each side of the equation.
0.125 + -1y + -0.125 + y2 = 0 + -0.125

Reorder the terms:
0.125 + -0.125 + -1y + y2 = 0 + -0.125

Combine like terms: 0.125 + -0.125 = 0.000
0.000 + -1y + y2 = 0 + -0.125
-1y + y2 = 0 + -0.125

Combine like terms: 0 + -0.125 = -0.125
-1y + y2 = -0.125

The y term is -1y.  Take half its coefficient (-0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
-1y + 0.25 + y2 = -0.125 + 0.25

Reorder the terms:
0.25 + -1y + y2 = -0.125 + 0.25

Combine like terms: -0.125 + 0.25 = 0.125
0.25 + -1y + y2 = 0.125

Factor a perfect square on the left side:
(y + -0.5)(y + -0.5) = 0.125

Calculate the square root of the right side: 0.353553391

Break this problem into two subproblems by setting 
(y + -0.5) equal to 0.353553391 and -0.353553391.

Subproblem 1

y + -0.5 = 0.353553391 Simplifying y + -0.5 = 0.353553391 Reorder the terms: -0.5 + y = 0.353553391 Solving -0.5 + y = 0.353553391 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + y = 0.353553391 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + y = 0.353553391 + 0.5 y = 0.353553391 + 0.5 Combine like terms: 0.353553391 + 0.5 = 0.853553391 y = 0.853553391 Simplifying y = 0.853553391

Subproblem 2

y + -0.5 = -0.353553391 Simplifying y + -0.5 = -0.353553391 Reorder the terms: -0.5 + y = -0.353553391 Solving -0.5 + y = -0.353553391 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + y = -0.353553391 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + y = -0.353553391 + 0.5 y = -0.353553391 + 0.5 Combine like terms: -0.353553391 + 0.5 = 0.146446609 y = 0.146446609 Simplifying y = 0.146446609

Solution

The solution to the problem is based on the solutions from the subproblems. y = {0.853553391, 0.146446609}

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