y^2-83=5y

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Solution for y^2-83=5y equation: 


Simplifying
y2 + -83 = 5y

Reorder the terms:
-83 + y2 = 5y

Solving
-83 + y2 = 5y

Solving for variable 'y'.

Reorder the terms:
-83 + -5y + y2 = 5y + -5y

Combine like terms: 5y + -5y = 0
-83 + -5y + y2 = 0

Begin completing the square.

Move the constant term to the right:

Add '83' to each side of the equation.
-83 + -5y + 83 + y2 = 0 + 83

Reorder the terms:
-83 + 83 + -5y + y2 = 0 + 83

Combine like terms: -83 + 83 = 0
0 + -5y + y2 = 0 + 83
-5y + y2 = 0 + 83

Combine like terms: 0 + 83 = 83
-5y + y2 = 83

The y term is -5y.  Take half its coefficient (-2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
-5y + 6.25 + y2 = 83 + 6.25

Reorder the terms:
6.25 + -5y + y2 = 83 + 6.25

Combine like terms: 83 + 6.25 = 89.25
6.25 + -5y + y2 = 89.25

Factor a perfect square on the left side:
(y + -2.5)(y + -2.5) = 89.25

Calculate the square root of the right side: 9.447221814

Break this problem into two subproblems by setting 
(y + -2.5) equal to 9.447221814 and -9.447221814.


Subproblem 1
y + -2.5 = 9.447221814

Simplifying
y + -2.5 = 9.447221814

Reorder the terms:
-2.5 + y = 9.447221814

Solving
-2.5 + y = 9.447221814

Solving for variable 'y'.

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

Add '2.5' to each side of the equation.
-2.5 + 2.5 + y = 9.447221814 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + y = 9.447221814 + 2.5
y = 9.447221814 + 2.5

Combine like terms: 9.447221814 + 2.5 = 11.947221814
y = 11.947221814

Simplifying
y = 11.947221814


Subproblem 2
y + -2.5 = -9.447221814

Simplifying
y + -2.5 = -9.447221814

Reorder the terms:
-2.5 + y = -9.447221814

Solving
-2.5 + y = -9.447221814

Solving for variable 'y'.

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

Add '2.5' to each side of the equation.
-2.5 + 2.5 + y = -9.447221814 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + y = -9.447221814 + 2.5
y = -9.447221814 + 2.5

Combine like terms: -9.447221814 + 2.5 = -6.947221814
y = -6.947221814

Simplifying
y = -6.947221814


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {11.947221814, -6.947221814}

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