6y(y-1)+8=32

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Solution for 6y(y-1)+8=32 equation:


Simplifying
6y(y + -1) + 8 = 32

Reorder the terms:
6y(-1 + y) + 8 = 32
(-1 * 6y + y * 6y) + 8 = 32
(-6y + 6y2) + 8 = 32

Reorder the terms:
8 + -6y + 6y2 = 32

Solving
8 + -6y + 6y2 = 32

Solving for variable 'y'.

Reorder the terms:
8 + -32 + -6y + 6y2 = 32 + -32

Combine like terms: 8 + -32 = -24
-24 + -6y + 6y2 = 32 + -32

Combine like terms: 32 + -32 = 0
-24 + -6y + 6y2 = 0

Factor out the Greatest Common Factor (GCF), '6'.
6(-4 + -1y + y2) = 0

Ignore the factor 6.

Subproblem 1
Set the factor '(-4 + -1y + y2)' equal to zero and attempt to solve:

Simplifying
-4 + -1y + y2 = 0

Solving
-4 + -1y + y2 = 0

Begin completing the square.

Move the constant term to the right:

Add '4' to each side of the equation.
-4 + -1y + 4 + y2 = 0 + 4

Reorder the terms:
-4 + 4 + -1y + y2 = 0 + 4

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

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

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 = 4 + 0.25

Reorder the terms:
0.25 + -1y + y2 = 4 + 0.25

Combine like terms: 4 + 0.25 = 4.25
0.25 + -1y + y2 = 4.25

Factor a perfect square on the left side:
(y + -0.5)(y + -0.5) = 4.25

Calculate the square root of the right side: 2.061552813

Break this problem into two subproblems by setting 
(y + -0.5) equal to 2.061552813 and -2.061552813.


Subproblem 1
y + -0.5 = 2.061552813

Simplifying
y + -0.5 = 2.061552813

Reorder the terms:
-0.5 + y = 2.061552813

Solving
-0.5 + y = 2.061552813

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 = 2.061552813 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + y = 2.061552813 + 0.5
y = 2.061552813 + 0.5

Combine like terms: 2.061552813 + 0.5 = 2.561552813
y = 2.561552813

Simplifying
y = 2.561552813


Subproblem 2
y + -0.5 = -2.061552813

Simplifying
y + -0.5 = -2.061552813

Reorder the terms:
-0.5 + y = -2.061552813

Solving
-0.5 + y = -2.061552813

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 = -2.061552813 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + y = -2.061552813 + 0.5
y = -2.061552813 + 0.5

Combine like terms: -2.061552813 + 0.5 = -1.561552813
y = -1.561552813

Simplifying
y = -1.561552813


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {2.561552813, -1.561552813}
Solution
y = {2.561552813, -1.561552813}

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