(y^2+53)+(15y+16)=180

Solution for (y^2+53)+(15y+16)=180 equation:

Simplifying
(y2 + 53) + (15y + 16) = 180

Reorder the terms:
(53 + y2) + (15y + 16) = 180

Remove parenthesis around (53 + y2)
53 + y2 + (15y + 16) = 180

Reorder the terms:
53 + y2 + (16 + 15y) = 180

Remove parenthesis around (16 + 15y)
53 + y2 + 16 + 15y = 180

Reorder the terms:
53 + 16 + 15y + y2 = 180

Combine like terms: 53 + 16 = 69
69 + 15y + y2 = 180

Solving
69 + 15y + y2 = 180

Solving for variable 'y'.

Reorder the terms:
69 + -180 + 15y + y2 = 180 + -180

Combine like terms: 69 + -180 = -111
-111 + 15y + y2 = 180 + -180

Combine like terms: 180 + -180 = 0
-111 + 15y + y2 = 0

Begin completing the square.

Move the constant term to the right:

Add '111' to each side of the equation.
-111 + 15y + 111 + y2 = 0 + 111

Reorder the terms:
-111 + 111 + 15y + y2 = 0 + 111

Combine like terms: -111 + 111 = 0
0 + 15y + y2 = 0 + 111
15y + y2 = 0 + 111

Combine like terms: 0 + 111 = 111
15y + y2 = 111

The y term is 15y.  Take half its coefficient (7.5).
Square it (56.25) and add it to both sides.

Add '56.25' to each side of the equation.
15y + 56.25 + y2 = 111 + 56.25

Reorder the terms:
56.25 + 15y + y2 = 111 + 56.25

Combine like terms: 111 + 56.25 = 167.25
56.25 + 15y + y2 = 167.25

Factor a perfect square on the left side:
(y + 7.5)(y + 7.5) = 167.25

Calculate the square root of the right side: 12.932517156

Break this problem into two subproblems by setting 
(y + 7.5) equal to 12.932517156 and -12.932517156.

Subproblem 1
y + 7.5 = 12.932517156

Simplifying
y + 7.5 = 12.932517156

Reorder the terms:
7.5 + y = 12.932517156

Solving
7.5 + y = 12.932517156

Solving for variable 'y'.

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

Add '-7.5' to each side of the equation.
7.5 + -7.5 + y = 12.932517156 + -7.5

Combine like terms: 7.5 + -7.5 = 0.0
0.0 + y = 12.932517156 + -7.5
y = 12.932517156 + -7.5

Combine like terms: 12.932517156 + -7.5 = 5.432517156
y = 5.432517156

Simplifying
y = 5.432517156

Subproblem 2
y + 7.5 = -12.932517156

Simplifying
y + 7.5 = -12.932517156

Reorder the terms:
7.5 + y = -12.932517156

Solving
7.5 + y = -12.932517156

Solving for variable 'y'.

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

Add '-7.5' to each side of the equation.
7.5 + -7.5 + y = -12.932517156 + -7.5

Combine like terms: 7.5 + -7.5 = 0.0
0.0 + y = -12.932517156 + -7.5
y = -12.932517156 + -7.5

Combine like terms: -12.932517156 + -7.5 = -20.432517156
y = -20.432517156

Simplifying
y = -20.432517156

Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {5.432517156, -20.432517156}