3y+y^2=80

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Solution for 3y+y^2=80 equation: 


Simplifying
3y + y2 = 80

Solving
3y + y2 = 80

Solving for variable 'y'.

Reorder the terms:
-80 + 3y + y2 = 80 + -80

Combine like terms: 80 + -80 = 0
-80 + 3y + y2 = 0

Begin completing the square.

Move the constant term to the right:

Add '80' to each side of the equation.
-80 + 3y + 80 + y2 = 0 + 80

Reorder the terms:
-80 + 80 + 3y + y2 = 0 + 80

Combine like terms: -80 + 80 = 0
0 + 3y + y2 = 0 + 80
3y + y2 = 0 + 80

Combine like terms: 0 + 80 = 80
3y + y2 = 80

The y term is 3y.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3y + 2.25 + y2 = 80 + 2.25

Reorder the terms:
2.25 + 3y + y2 = 80 + 2.25

Combine like terms: 80 + 2.25 = 82.25
2.25 + 3y + y2 = 82.25

Factor a perfect square on the left side:
(y + 1.5)(y + 1.5) = 82.25

Calculate the square root of the right side: 9.069178574

Break this problem into two subproblems by setting 
(y + 1.5) equal to 9.069178574 and -9.069178574.


Subproblem 1
y + 1.5 = 9.069178574

Simplifying
y + 1.5 = 9.069178574

Reorder the terms:
1.5 + y = 9.069178574

Solving
1.5 + y = 9.069178574

Solving for variable 'y'.

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

Add '-1.5' to each side of the equation.
1.5 + -1.5 + y = 9.069178574 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + y = 9.069178574 + -1.5
y = 9.069178574 + -1.5

Combine like terms: 9.069178574 + -1.5 = 7.569178574
y = 7.569178574

Simplifying
y = 7.569178574


Subproblem 2
y + 1.5 = -9.069178574

Simplifying
y + 1.5 = -9.069178574

Reorder the terms:
1.5 + y = -9.069178574

Solving
1.5 + y = -9.069178574

Solving for variable 'y'.

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

Add '-1.5' to each side of the equation.
1.5 + -1.5 + y = -9.069178574 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + y = -9.069178574 + -1.5
y = -9.069178574 + -1.5

Combine like terms: -9.069178574 + -1.5 = -10.569178574
y = -10.569178574

Simplifying
y = -10.569178574


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {7.569178574, -10.569178574}

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