(y+8)(2y-5)=1

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Solution for (y+8)(2y-5)=1 equation: 


Simplifying
(y + 8)(2y + -5) = 1

Reorder the terms:
(8 + y)(2y + -5) = 1

Reorder the terms:
(8 + y)(-5 + 2y) = 1

Multiply (8 + y) * (-5 + 2y)
(8(-5 + 2y) + y(-5 + 2y)) = 1
((-5 * 8 + 2y * 8) + y(-5 + 2y)) = 1
((-40 + 16y) + y(-5 + 2y)) = 1
(-40 + 16y + (-5 * y + 2y * y)) = 1
(-40 + 16y + (-5y + 2y2)) = 1

Combine like terms: 16y + -5y = 11y
(-40 + 11y + 2y2) = 1

Solving
-40 + 11y + 2y2 = 1

Solving for variable 'y'.

Reorder the terms:
-40 + -1 + 11y + 2y2 = 1 + -1

Combine like terms: -40 + -1 = -41
-41 + 11y + 2y2 = 1 + -1

Combine like terms: 1 + -1 = 0
-41 + 11y + 2y2 = 0

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

Divide each side by '2'.
-20.5 + 5.5y + y2 = 0

Move the constant term to the right:

Add '20.5' to each side of the equation.
-20.5 + 5.5y + 20.5 + y2 = 0 + 20.5

Reorder the terms:
-20.5 + 20.5 + 5.5y + y2 = 0 + 20.5

Combine like terms: -20.5 + 20.5 = 0.0
0.0 + 5.5y + y2 = 0 + 20.5
5.5y + y2 = 0 + 20.5

Combine like terms: 0 + 20.5 = 20.5
5.5y + y2 = 20.5

The y term is 5.5y.  Take half its coefficient (2.75).
Square it (7.5625) and add it to both sides.

Add '7.5625' to each side of the equation.
5.5y + 7.5625 + y2 = 20.5 + 7.5625

Reorder the terms:
7.5625 + 5.5y + y2 = 20.5 + 7.5625

Combine like terms: 20.5 + 7.5625 = 28.0625
7.5625 + 5.5y + y2 = 28.0625

Factor a perfect square on the left side:
(y + 2.75)(y + 2.75) = 28.0625

Calculate the square root of the right side: 5.297405025

Break this problem into two subproblems by setting 
(y + 2.75) equal to 5.297405025 and -5.297405025.


Subproblem 1
y + 2.75 = 5.297405025

Simplifying
y + 2.75 = 5.297405025

Reorder the terms:
2.75 + y = 5.297405025

Solving
2.75 + y = 5.297405025

Solving for variable 'y'.

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

Add '-2.75' to each side of the equation.
2.75 + -2.75 + y = 5.297405025 + -2.75

Combine like terms: 2.75 + -2.75 = 0.00
0.00 + y = 5.297405025 + -2.75
y = 5.297405025 + -2.75

Combine like terms: 5.297405025 + -2.75 = 2.547405025
y = 2.547405025

Simplifying
y = 2.547405025


Subproblem 2
y + 2.75 = -5.297405025

Simplifying
y + 2.75 = -5.297405025

Reorder the terms:
2.75 + y = -5.297405025

Solving
2.75 + y = -5.297405025

Solving for variable 'y'.

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

Add '-2.75' to each side of the equation.
2.75 + -2.75 + y = -5.297405025 + -2.75

Combine like terms: 2.75 + -2.75 = 0.00
0.00 + y = -5.297405025 + -2.75
y = -5.297405025 + -2.75

Combine like terms: -5.297405025 + -2.75 = -8.047405025
y = -8.047405025

Simplifying
y = -8.047405025


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {2.547405025, -8.047405025}

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