y^2+y=84

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Solution for y^2+y=84 equation: 


Simplifying
y2 + y = 84

Reorder the terms:
y + y2 = 84

Solving
y + y2 = 84

Solving for variable 'y'.

Reorder the terms:
-84 + y + y2 = 84 + -84

Combine like terms: 84 + -84 = 0
-84 + y + y2 = 0

Begin completing the square.

Move the constant term to the right:

Add '84' to each side of the equation.
-84 + y + 84 + y2 = 0 + 84

Reorder the terms:
-84 + 84 + y + y2 = 0 + 84

Combine like terms: -84 + 84 = 0
0 + y + y2 = 0 + 84
y + y2 = 0 + 84

Combine like terms: 0 + 84 = 84
y + y2 = 84

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 = 84 + 0.25

Reorder the terms:
0.25 + y + y2 = 84 + 0.25

Combine like terms: 84 + 0.25 = 84.25
0.25 + y + y2 = 84.25

Factor a perfect square on the left side:
(y + 0.5)(y + 0.5) = 84.25

Calculate the square root of the right side: 9.178779875

Break this problem into two subproblems by setting 
(y + 0.5) equal to 9.178779875 and -9.178779875.


Subproblem 1
y + 0.5 = 9.178779875

Simplifying
y + 0.5 = 9.178779875

Reorder the terms:
0.5 + y = 9.178779875

Solving
0.5 + y = 9.178779875

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 = 9.178779875 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + y = 9.178779875 + -0.5
y = 9.178779875 + -0.5

Combine like terms: 9.178779875 + -0.5 = 8.678779875
y = 8.678779875

Simplifying
y = 8.678779875


Subproblem 2
y + 0.5 = -9.178779875

Simplifying
y + 0.5 = -9.178779875

Reorder the terms:
0.5 + y = -9.178779875

Solving
0.5 + y = -9.178779875

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 = -9.178779875 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + y = -9.178779875 + -0.5
y = -9.178779875 + -0.5

Combine like terms: -9.178779875 + -0.5 = -9.678779875
y = -9.678779875

Simplifying
y = -9.678779875


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {8.678779875, -9.678779875}

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