(4y+16)(4y+16)=8

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Solution for (4y+16)(4y+16)=8 equation: 


Simplifying
(4y + 16)(4y + 16) = 8

Reorder the terms:
(16 + 4y)(4y + 16) = 8

Reorder the terms:
(16 + 4y)(16 + 4y) = 8

Multiply (16 + 4y) * (16 + 4y)
(16(16 + 4y) + 4y * (16 + 4y)) = 8
((16 * 16 + 4y * 16) + 4y * (16 + 4y)) = 8
((256 + 64y) + 4y * (16 + 4y)) = 8
(256 + 64y + (16 * 4y + 4y * 4y)) = 8
(256 + 64y + (64y + 16y2)) = 8

Combine like terms: 64y + 64y = 128y
(256 + 128y + 16y2) = 8

Solving
256 + 128y + 16y2 = 8

Solving for variable 'y'.

Reorder the terms:
256 + -8 + 128y + 16y2 = 8 + -8

Combine like terms: 256 + -8 = 248
248 + 128y + 16y2 = 8 + -8

Combine like terms: 8 + -8 = 0
248 + 128y + 16y2 = 0

Factor out the Greatest Common Factor (GCF), '8'.
8(31 + 16y + 2y2) = 0

Ignore the factor 8.

Subproblem 1
Set the factor '(31 + 16y + 2y2)' equal to zero and attempt to solve:

Simplifying
31 + 16y + 2y2 = 0

Solving
31 + 16y + 2y2 = 0

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

Divide each side by '2'.
15.5 + 8y + y2 = 0

Move the constant term to the right:

Add '-15.5' to each side of the equation.
15.5 + 8y + -15.5 + y2 = 0 + -15.5

Reorder the terms:
15.5 + -15.5 + 8y + y2 = 0 + -15.5

Combine like terms: 15.5 + -15.5 = 0.0
0.0 + 8y + y2 = 0 + -15.5
8y + y2 = 0 + -15.5

Combine like terms: 0 + -15.5 = -15.5
8y + y2 = -15.5

The y term is 8y.  Take half its coefficient (4).
Square it (16) and add it to both sides.

Add '16' to each side of the equation.
8y + 16 + y2 = -15.5 + 16

Reorder the terms:
16 + 8y + y2 = -15.5 + 16

Combine like terms: -15.5 + 16 = 0.5
16 + 8y + y2 = 0.5

Factor a perfect square on the left side:
(y + 4)(y + 4) = 0.5

Calculate the square root of the right side: 0.707106781

Break this problem into two subproblems by setting 
(y + 4) equal to 0.707106781 and -0.707106781.


Subproblem 1
y + 4 = 0.707106781

Simplifying
y + 4 = 0.707106781

Reorder the terms:
4 + y = 0.707106781

Solving
4 + y = 0.707106781

Solving for variable 'y'.

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

Add '-4' to each side of the equation.
4 + -4 + y = 0.707106781 + -4

Combine like terms: 4 + -4 = 0
0 + y = 0.707106781 + -4
y = 0.707106781 + -4

Combine like terms: 0.707106781 + -4 = -3.292893219
y = -3.292893219

Simplifying
y = -3.292893219


Subproblem 2
y + 4 = -0.707106781

Simplifying
y + 4 = -0.707106781

Reorder the terms:
4 + y = -0.707106781

Solving
4 + y = -0.707106781

Solving for variable 'y'.

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

Add '-4' to each side of the equation.
4 + -4 + y = -0.707106781 + -4

Combine like terms: 4 + -4 = 0
0 + y = -0.707106781 + -4
y = -0.707106781 + -4

Combine like terms: -0.707106781 + -4 = -4.707106781
y = -4.707106781

Simplifying
y = -4.707106781


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {-3.292893219, -4.707106781}
Solution
y = {-3.292893219, -4.707106781}

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