2[2+3y(y-1)]=22

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Solution for 2[2+3y(y-1)]=22 equation: 


Simplifying
2[2 + 3y(y + -1)] = 22

Reorder the terms:
2[2 + 3y(-1 + y)] = 22
2[2 + (-1 * 3y + y * 3y)] = 22
2[2 + (-3y + 3y2)] = 22
[2 * 2 + -3y * 2 + 3y2 * 2] = 22
[4 + -6y + 6y2] = 22

Solving
4 + -6y + 6y2 = 22

Solving for variable 'y'.

Reorder the terms:
4 + -22 + -6y + 6y2 = 22 + -22

Combine like terms: 4 + -22 = -18
-18 + -6y + 6y2 = 22 + -22

Combine like terms: 22 + -22 = 0
-18 + -6y + 6y2 = 0

Factor out the Greatest Common Factor (GCF), '6'.
6(-3 + -1y + y2) = 0

Ignore the factor 6.

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

Simplifying
-3 + -1y + y2 = 0

Solving
-3 + -1y + y2 = 0

Begin completing the square.

Move the constant term to the right:

Add '3' to each side of the equation.
-3 + -1y + 3 + y2 = 0 + 3

Reorder the terms:
-3 + 3 + -1y + y2 = 0 + 3

Combine like terms: -3 + 3 = 0
0 + -1y + y2 = 0 + 3
-1y + y2 = 0 + 3

Combine like terms: 0 + 3 = 3
-1y + y2 = 3

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 = 3 + 0.25

Reorder the terms:
0.25 + -1y + y2 = 3 + 0.25

Combine like terms: 3 + 0.25 = 3.25
0.25 + -1y + y2 = 3.25

Factor a perfect square on the left side:
(y + -0.5)(y + -0.5) = 3.25

Calculate the square root of the right side: 1.802775638

Break this problem into two subproblems by setting 
(y + -0.5) equal to 1.802775638 and -1.802775638.


Subproblem 1
y + -0.5 = 1.802775638

Simplifying
y + -0.5 = 1.802775638

Reorder the terms:
-0.5 + y = 1.802775638

Solving
-0.5 + y = 1.802775638

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 = 1.802775638 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + y = 1.802775638 + 0.5
y = 1.802775638 + 0.5

Combine like terms: 1.802775638 + 0.5 = 2.302775638
y = 2.302775638

Simplifying
y = 2.302775638


Subproblem 2
y + -0.5 = -1.802775638

Simplifying
y + -0.5 = -1.802775638

Reorder the terms:
-0.5 + y = -1.802775638

Solving
-0.5 + y = -1.802775638

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 = -1.802775638 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + y = -1.802775638 + 0.5
y = -1.802775638 + 0.5

Combine like terms: -1.802775638 + 0.5 = -1.302775638
y = -1.302775638

Simplifying
y = -1.302775638


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {2.302775638, -1.302775638}
Solution
y = {2.302775638, -1.302775638}

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