5y(y-1)=40

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Solution for 5y(y-1)=40 equation: 


Simplifying
5y(y + -1) = 40

Reorder the terms:
5y(-1 + y) = 40
(-1 * 5y + y * 5y) = 40
(-5y + 5y2) = 40

Solving
-5y + 5y2 = 40

Solving for variable 'y'.

Reorder the terms:
-40 + -5y + 5y2 = 40 + -40

Combine like terms: 40 + -40 = 0
-40 + -5y + 5y2 = 0

Factor out the Greatest Common Factor (GCF), '5'.
5(-8 + -1y + y2) = 0

Ignore the factor 5.

Subproblem 1
Set the factor '(-8 + -1y + y2)' equal to zero and attempt to solve:

Simplifying
-8 + -1y + y2 = 0

Solving
-8 + -1y + y2 = 0

Begin completing the square.

Move the constant term to the right:

Add '8' to each side of the equation.
-8 + -1y + 8 + y2 = 0 + 8

Reorder the terms:
-8 + 8 + -1y + y2 = 0 + 8

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

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

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 = 8 + 0.25

Reorder the terms:
0.25 + -1y + y2 = 8 + 0.25

Combine like terms: 8 + 0.25 = 8.25
0.25 + -1y + y2 = 8.25

Factor a perfect square on the left side:
(y + -0.5)(y + -0.5) = 8.25

Calculate the square root of the right side: 2.872281323

Break this problem into two subproblems by setting 
(y + -0.5) equal to 2.872281323 and -2.872281323.


Subproblem 1
y + -0.5 = 2.872281323

Simplifying
y + -0.5 = 2.872281323

Reorder the terms:
-0.5 + y = 2.872281323

Solving
-0.5 + y = 2.872281323

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 = 2.872281323 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + y = 2.872281323 + 0.5
y = 2.872281323 + 0.5

Combine like terms: 2.872281323 + 0.5 = 3.372281323
y = 3.372281323

Simplifying
y = 3.372281323


Subproblem 2
y + -0.5 = -2.872281323

Simplifying
y + -0.5 = -2.872281323

Reorder the terms:
-0.5 + y = -2.872281323

Solving
-0.5 + y = -2.872281323

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 = -2.872281323 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + y = -2.872281323 + 0.5
y = -2.872281323 + 0.5

Combine like terms: -2.872281323 + 0.5 = -2.372281323
y = -2.372281323

Simplifying
y = -2.372281323


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {3.372281323, -2.372281323}
Solution
y = {3.372281323, -2.372281323}

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