(7y+6)(4y+116)=180

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Solution for (7y+6)(4y+116)=180 equation:


Simplifying
(7y + 6)(4y + 116) = 180

Reorder the terms:
(6 + 7y)(4y + 116) = 180

Reorder the terms:
(6 + 7y)(116 + 4y) = 180

Multiply (6 + 7y) * (116 + 4y)
(6(116 + 4y) + 7y * (116 + 4y)) = 180
((116 * 6 + 4y * 6) + 7y * (116 + 4y)) = 180
((696 + 24y) + 7y * (116 + 4y)) = 180
(696 + 24y + (116 * 7y + 4y * 7y)) = 180
(696 + 24y + (812y + 28y2)) = 180

Combine like terms: 24y + 812y = 836y
(696 + 836y + 28y2) = 180

Solving
696 + 836y + 28y2 = 180

Solving for variable 'y'.

Reorder the terms:
696 + -180 + 836y + 28y2 = 180 + -180

Combine like terms: 696 + -180 = 516
516 + 836y + 28y2 = 180 + -180

Combine like terms: 180 + -180 = 0
516 + 836y + 28y2 = 0

Factor out the Greatest Common Factor (GCF), '4'.
4(129 + 209y + 7y2) = 0

Ignore the factor 4.

Subproblem 1
Set the factor '(129 + 209y + 7y2)' equal to zero and attempt to solve:

Simplifying
129 + 209y + 7y2 = 0

Solving
129 + 209y + 7y2 = 0

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

Divide each side by '7'.
18.42857143 + 29.85714286y + y2 = 0

Move the constant term to the right:

Add '-18.42857143' to each side of the equation.
18.42857143 + 29.85714286y + -18.42857143 + y2 = 0 + -18.42857143

Reorder the terms:
18.42857143 + -18.42857143 + 29.85714286y + y2 = 0 + -18.42857143

Combine like terms: 18.42857143 + -18.42857143 = 0.00000000
0.00000000 + 29.85714286y + y2 = 0 + -18.42857143
29.85714286y + y2 = 0 + -18.42857143

Combine like terms: 0 + -18.42857143 = -18.42857143
29.85714286y + y2 = -18.42857143

The y term is 29.85714286y.  Take half its coefficient (14.92857143).
Square it (222.8622449) and add it to both sides.

Add '222.8622449' to each side of the equation.
29.85714286y + 222.8622449 + y2 = -18.42857143 + 222.8622449

Reorder the terms:
222.8622449 + 29.85714286y + y2 = -18.42857143 + 222.8622449

Combine like terms: -18.42857143 + 222.8622449 = 204.43367347
222.8622449 + 29.85714286y + y2 = 204.43367347

Factor a perfect square on the left side:
(y + 14.92857143)(y + 14.92857143) = 204.43367347

Calculate the square root of the right side: 14.298030405

Break this problem into two subproblems by setting 
(y + 14.92857143) equal to 14.298030405 and -14.298030405.


Subproblem 1
y + 14.92857143 = 14.298030405

Simplifying
y + 14.92857143 = 14.298030405

Reorder the terms:
14.92857143 + y = 14.298030405

Solving
14.92857143 + y = 14.298030405

Solving for variable 'y'.

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

Add '-14.92857143' to each side of the equation.
14.92857143 + -14.92857143 + y = 14.298030405 + -14.92857143

Combine like terms: 14.92857143 + -14.92857143 = 0.00000000
0.00000000 + y = 14.298030405 + -14.92857143
y = 14.298030405 + -14.92857143

Combine like terms: 14.298030405 + -14.92857143 = -0.630541025
y = -0.630541025

Simplifying
y = -0.630541025


Subproblem 2
y + 14.92857143 = -14.298030405

Simplifying
y + 14.92857143 = -14.298030405

Reorder the terms:
14.92857143 + y = -14.298030405

Solving
14.92857143 + y = -14.298030405

Solving for variable 'y'.

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

Add '-14.92857143' to each side of the equation.
14.92857143 + -14.92857143 + y = -14.298030405 + -14.92857143

Combine like terms: 14.92857143 + -14.92857143 = 0.00000000
0.00000000 + y = -14.298030405 + -14.92857143
y = -14.298030405 + -14.92857143

Combine like terms: -14.298030405 + -14.92857143 = -29.226601835
y = -29.226601835

Simplifying
y = -29.226601835


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {-0.630541025, -29.226601835}
Solution
y = {-0.630541025, -29.226601835}

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