y^2+y=8.5

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Solution for y^2+y=8.5 equation: 


Simplifying
y2 + y = 8.5

Reorder the terms:
y + y2 = 8.5

Solving
y + y2 = 8.5

Solving for variable 'y'.

Reorder the terms:
-8.5 + y + y2 = 8.5 + -8.5

Combine like terms: 8.5 + -8.5 = 0.0
-8.5 + y + y2 = 0.0

Begin completing the square.

Move the constant term to the right:

Add '8.5' to each side of the equation.
-8.5 + y + 8.5 + y2 = 0.0 + 8.5

Reorder the terms:
-8.5 + 8.5 + y + y2 = 0.0 + 8.5

Combine like terms: -8.5 + 8.5 = 0.0
0.0 + y + y2 = 0.0 + 8.5
y + y2 = 0.0 + 8.5

Combine like terms: 0.0 + 8.5 = 8.5
y + y2 = 8.5

The y term is y.  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.
y + 0.25 + y2 = 8.5 + 0.25

Reorder the terms:
0.25 + y + y2 = 8.5 + 0.25

Combine like terms: 8.5 + 0.25 = 8.75
0.25 + y + y2 = 8.75

Factor a perfect square on the left side:
(y + 0.5)(y + 0.5) = 8.75

Calculate the square root of the right side: 2.958039892

Break this problem into two subproblems by setting 
(y + 0.5) equal to 2.958039892 and -2.958039892.


Subproblem 1
y + 0.5 = 2.958039892

Simplifying
y + 0.5 = 2.958039892

Reorder the terms:
0.5 + y = 2.958039892

Solving
0.5 + y = 2.958039892

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 = 2.958039892 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + y = 2.958039892 + -0.5
y = 2.958039892 + -0.5

Combine like terms: 2.958039892 + -0.5 = 2.458039892
y = 2.458039892

Simplifying
y = 2.458039892


Subproblem 2
y + 0.5 = -2.958039892

Simplifying
y + 0.5 = -2.958039892

Reorder the terms:
0.5 + y = -2.958039892

Solving
0.5 + y = -2.958039892

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 = -2.958039892 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + y = -2.958039892 + -0.5
y = -2.958039892 + -0.5

Combine like terms: -2.958039892 + -0.5 = -3.458039892
y = -3.458039892

Simplifying
y = -3.458039892


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {2.458039892, -3.458039892}

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