3y+y^2=47.5

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Solution for 3y+y^2=47.5 equation:


Simplifying
3y + y2 = 47.5

Solving
3y + y2 = 47.5

Solving for variable 'y'.

Reorder the terms:
-47.5 + 3y + y2 = 47.5 + -47.5

Combine like terms: 47.5 + -47.5 = 0.0
-47.5 + 3y + y2 = 0.0

Begin completing the square.

Move the constant term to the right:

Add '47.5' to each side of the equation.
-47.5 + 3y + 47.5 + y2 = 0.0 + 47.5

Reorder the terms:
-47.5 + 47.5 + 3y + y2 = 0.0 + 47.5

Combine like terms: -47.5 + 47.5 = 0.0
0.0 + 3y + y2 = 0.0 + 47.5
3y + y2 = 0.0 + 47.5

Combine like terms: 0.0 + 47.5 = 47.5
3y + y2 = 47.5

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 = 47.5 + 2.25

Reorder the terms:
2.25 + 3y + y2 = 47.5 + 2.25

Combine like terms: 47.5 + 2.25 = 49.75
2.25 + 3y + y2 = 49.75

Factor a perfect square on the left side:
(y + 1.5)(y + 1.5) = 49.75

Calculate the square root of the right side: 7.05336799

Break this problem into two subproblems by setting 
(y + 1.5) equal to 7.05336799 and -7.05336799.


Subproblem 1
y + 1.5 = 7.05336799

Simplifying
y + 1.5 = 7.05336799

Reorder the terms:
1.5 + y = 7.05336799

Solving
1.5 + y = 7.05336799

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 = 7.05336799 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + y = 7.05336799 + -1.5
y = 7.05336799 + -1.5

Combine like terms: 7.05336799 + -1.5 = 5.55336799
y = 5.55336799

Simplifying
y = 5.55336799


Subproblem 2
y + 1.5 = -7.05336799

Simplifying
y + 1.5 = -7.05336799

Reorder the terms:
1.5 + y = -7.05336799

Solving
1.5 + y = -7.05336799

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 = -7.05336799 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + y = -7.05336799 + -1.5
y = -7.05336799 + -1.5

Combine like terms: -7.05336799 + -1.5 = -8.55336799
y = -8.55336799

Simplifying
y = -8.55336799


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {5.55336799, -8.55336799}

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