2y(y-1)=5y+2

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Solution for 2y(y-1)=5y+2 equation: 


Simplifying
2y(y + -1) = 5y + 2

Reorder the terms:
2y(-1 + y) = 5y + 2
(-1 * 2y + y * 2y) = 5y + 2
(-2y + 2y2) = 5y + 2

Reorder the terms:
-2y + 2y2 = 2 + 5y

Solving
-2y + 2y2 = 2 + 5y

Solving for variable 'y'.

Reorder the terms:
-2 + -2y + -5y + 2y2 = 2 + 5y + -2 + -5y

Combine like terms: -2y + -5y = -7y
-2 + -7y + 2y2 = 2 + 5y + -2 + -5y

Reorder the terms:
-2 + -7y + 2y2 = 2 + -2 + 5y + -5y

Combine like terms: 2 + -2 = 0
-2 + -7y + 2y2 = 0 + 5y + -5y
-2 + -7y + 2y2 = 5y + -5y

Combine like terms: 5y + -5y = 0
-2 + -7y + 2y2 = 0

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

Divide each side by '2'.
-1 + -3.5y + y2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + -3.5y + 1 + y2 = 0 + 1

Reorder the terms:
-1 + 1 + -3.5y + y2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + -3.5y + y2 = 0 + 1
-3.5y + y2 = 0 + 1

Combine like terms: 0 + 1 = 1
-3.5y + y2 = 1

The y term is -3.5y.  Take half its coefficient (-1.75).
Square it (3.0625) and add it to both sides.

Add '3.0625' to each side of the equation.
-3.5y + 3.0625 + y2 = 1 + 3.0625

Reorder the terms:
3.0625 + -3.5y + y2 = 1 + 3.0625

Combine like terms: 1 + 3.0625 = 4.0625
3.0625 + -3.5y + y2 = 4.0625

Factor a perfect square on the left side:
(y + -1.75)(y + -1.75) = 4.0625

Calculate the square root of the right side: 2.015564437

Break this problem into two subproblems by setting 
(y + -1.75) equal to 2.015564437 and -2.015564437.


Subproblem 1
y + -1.75 = 2.015564437

Simplifying
y + -1.75 = 2.015564437

Reorder the terms:
-1.75 + y = 2.015564437

Solving
-1.75 + y = 2.015564437

Solving for variable 'y'.

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

Add '1.75' to each side of the equation.
-1.75 + 1.75 + y = 2.015564437 + 1.75

Combine like terms: -1.75 + 1.75 = 0.00
0.00 + y = 2.015564437 + 1.75
y = 2.015564437 + 1.75

Combine like terms: 2.015564437 + 1.75 = 3.765564437
y = 3.765564437

Simplifying
y = 3.765564437


Subproblem 2
y + -1.75 = -2.015564437

Simplifying
y + -1.75 = -2.015564437

Reorder the terms:
-1.75 + y = -2.015564437

Solving
-1.75 + y = -2.015564437

Solving for variable 'y'.

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

Add '1.75' to each side of the equation.
-1.75 + 1.75 + y = -2.015564437 + 1.75

Combine like terms: -1.75 + 1.75 = 0.00
0.00 + y = -2.015564437 + 1.75
y = -2.015564437 + 1.75

Combine like terms: -2.015564437 + 1.75 = -0.265564437
y = -0.265564437

Simplifying
y = -0.265564437


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {3.765564437, -0.265564437}

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