171-8Y=5Y^2-108

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Solution for 171-8Y=5Y^2-108 equation: 


Simplifying
171 + -8Y = 5Y2 + -108

Reorder the terms:
171 + -8Y = -108 + 5Y2

Solving
171 + -8Y = -108 + 5Y2

Solving for variable 'Y'.

Reorder the terms:
171 + 108 + -8Y + -5Y2 = -108 + 5Y2 + 108 + -5Y2

Combine like terms: 171 + 108 = 279
279 + -8Y + -5Y2 = -108 + 5Y2 + 108 + -5Y2

Reorder the terms:
279 + -8Y + -5Y2 = -108 + 108 + 5Y2 + -5Y2

Combine like terms: -108 + 108 = 0
279 + -8Y + -5Y2 = 0 + 5Y2 + -5Y2
279 + -8Y + -5Y2 = 5Y2 + -5Y2

Combine like terms: 5Y2 + -5Y2 = 0
279 + -8Y + -5Y2 = 0

Begin completing the square.  Divide all terms by
-5 the coefficient of the squared term: 

Divide each side by '-5'.
-55.8 + 1.6Y + Y2 = 0

Move the constant term to the right:

Add '55.8' to each side of the equation.
-55.8 + 1.6Y + 55.8 + Y2 = 0 + 55.8

Reorder the terms:
-55.8 + 55.8 + 1.6Y + Y2 = 0 + 55.8

Combine like terms: -55.8 + 55.8 = 0.0
0.0 + 1.6Y + Y2 = 0 + 55.8
1.6Y + Y2 = 0 + 55.8

Combine like terms: 0 + 55.8 = 55.8
1.6Y + Y2 = 55.8

The Y term is 1.6Y.  Take half its coefficient (0.8).
Square it (0.64) and add it to both sides.

Add '0.64' to each side of the equation.
1.6Y + 0.64 + Y2 = 55.8 + 0.64

Reorder the terms:
0.64 + 1.6Y + Y2 = 55.8 + 0.64

Combine like terms: 55.8 + 0.64 = 56.44
0.64 + 1.6Y + Y2 = 56.44

Factor a perfect square on the left side:
(Y + 0.8)(Y + 0.8) = 56.44

Calculate the square root of the right side: 7.512655988

Break this problem into two subproblems by setting 
(Y + 0.8) equal to 7.512655988 and -7.512655988.


Subproblem 1
Y + 0.8 = 7.512655988

Simplifying
Y + 0.8 = 7.512655988

Reorder the terms:
0.8 + Y = 7.512655988

Solving
0.8 + Y = 7.512655988

Solving for variable 'Y'.

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

Add '-0.8' to each side of the equation.
0.8 + -0.8 + Y = 7.512655988 + -0.8

Combine like terms: 0.8 + -0.8 = 0.0
0.0 + Y = 7.512655988 + -0.8
Y = 7.512655988 + -0.8

Combine like terms: 7.512655988 + -0.8 = 6.712655988
Y = 6.712655988

Simplifying
Y = 6.712655988


Subproblem 2
Y + 0.8 = -7.512655988

Simplifying
Y + 0.8 = -7.512655988

Reorder the terms:
0.8 + Y = -7.512655988

Solving
0.8 + Y = -7.512655988

Solving for variable 'Y'.

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

Add '-0.8' to each side of the equation.
0.8 + -0.8 + Y = -7.512655988 + -0.8

Combine like terms: 0.8 + -0.8 = 0.0
0.0 + Y = -7.512655988 + -0.8
Y = -7.512655988 + -0.8

Combine like terms: -7.512655988 + -0.8 = -8.312655988
Y = -8.312655988

Simplifying
Y = -8.312655988


Solution
The solution to the problem is based on the solutions
from the subproblems.
Y = {6.712655988, -8.312655988}

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