6y(y-1)=2

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Solution for 6y(y-1)=2 equation: 


Simplifying
6y(y + -1) = 2

Reorder the terms:
6y(-1 + y) = 2
(-1 * 6y + y * 6y) = 2
(-6y + 6y2) = 2

Solving
-6y + 6y2 = 2

Solving for variable 'y'.

Reorder the terms:
-2 + -6y + 6y2 = 2 + -2

Combine like terms: 2 + -2 = 0
-2 + -6y + 6y2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-1 + -3y + 3y2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(-1 + -3y + 3y2)' equal to zero and attempt to solve:

Simplifying
-1 + -3y + 3y2 = 0

Solving
-1 + -3y + 3y2 = 0

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

Divide each side by '3'.
-0.3333333333 + -1y + y2 = 0

Move the constant term to the right:

Add '0.3333333333' to each side of the equation.
-0.3333333333 + -1y + 0.3333333333 + y2 = 0 + 0.3333333333

Reorder the terms:
-0.3333333333 + 0.3333333333 + -1y + y2 = 0 + 0.3333333333

Combine like terms: -0.3333333333 + 0.3333333333 = 0.0000000000
0.0000000000 + -1y + y2 = 0 + 0.3333333333
-1y + y2 = 0 + 0.3333333333

Combine like terms: 0 + 0.3333333333 = 0.3333333333
-1y + y2 = 0.3333333333

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 = 0.3333333333 + 0.25

Reorder the terms:
0.25 + -1y + y2 = 0.3333333333 + 0.25

Combine like terms: 0.3333333333 + 0.25 = 0.5833333333
0.25 + -1y + y2 = 0.5833333333

Factor a perfect square on the left side:
(y + -0.5)(y + -0.5) = 0.5833333333

Calculate the square root of the right side: 0.763762616

Break this problem into two subproblems by setting 
(y + -0.5) equal to 0.763762616 and -0.763762616.


Subproblem 1
y + -0.5 = 0.763762616

Simplifying
y + -0.5 = 0.763762616

Reorder the terms:
-0.5 + y = 0.763762616

Solving
-0.5 + y = 0.763762616

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 = 0.763762616 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + y = 0.763762616 + 0.5
y = 0.763762616 + 0.5

Combine like terms: 0.763762616 + 0.5 = 1.263762616
y = 1.263762616

Simplifying
y = 1.263762616


Subproblem 2
y + -0.5 = -0.763762616

Simplifying
y + -0.5 = -0.763762616

Reorder the terms:
-0.5 + y = -0.763762616

Solving
-0.5 + y = -0.763762616

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 = -0.763762616 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + y = -0.763762616 + 0.5
y = -0.763762616 + 0.5

Combine like terms: -0.763762616 + 0.5 = -0.263762616
y = -0.263762616

Simplifying
y = -0.263762616


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {1.263762616, -0.263762616}
Solution
y = {1.263762616, -0.263762616}

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