0=y^2+12y+15

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Solution for 0=y^2+12y+15 equation: 


Simplifying
0 = y2 + 12y + 15

Reorder the terms:
0 = 15 + 12y + y2

Solving
0 = 15 + 12y + y2

Solving for variable 'y'.

Combine like terms: 0 + -15 = -15
-15 + -12y + -1y2 = 15 + 12y + y2 + -15 + -12y + -1y2

Reorder the terms:
-15 + -12y + -1y2 = 15 + -15 + 12y + -12y + y2 + -1y2

Combine like terms: 15 + -15 = 0
-15 + -12y + -1y2 = 0 + 12y + -12y + y2 + -1y2
-15 + -12y + -1y2 = 12y + -12y + y2 + -1y2

Combine like terms: 12y + -12y = 0
-15 + -12y + -1y2 = 0 + y2 + -1y2
-15 + -12y + -1y2 = y2 + -1y2

Combine like terms: y2 + -1y2 = 0
-15 + -12y + -1y2 = 0

Factor out the Greatest Common Factor (GCF), '-1'.
-1(15 + 12y + y2) = 0

Ignore the factor -1.

Subproblem 1
Set the factor '(15 + 12y + y2)' equal to zero and attempt to solve:

Simplifying
15 + 12y + y2 = 0

Solving
15 + 12y + y2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-15' to each side of the equation.
15 + 12y + -15 + y2 = 0 + -15

Reorder the terms:
15 + -15 + 12y + y2 = 0 + -15

Combine like terms: 15 + -15 = 0
0 + 12y + y2 = 0 + -15
12y + y2 = 0 + -15

Combine like terms: 0 + -15 = -15
12y + y2 = -15

The y term is 12y.  Take half its coefficient (6).
Square it (36) and add it to both sides.

Add '36' to each side of the equation.
12y + 36 + y2 = -15 + 36

Reorder the terms:
36 + 12y + y2 = -15 + 36

Combine like terms: -15 + 36 = 21
36 + 12y + y2 = 21

Factor a perfect square on the left side:
(y + 6)(y + 6) = 21

Calculate the square root of the right side: 4.582575695

Break this problem into two subproblems by setting 
(y + 6) equal to 4.582575695 and -4.582575695.


Subproblem 1
y + 6 = 4.582575695

Simplifying
y + 6 = 4.582575695

Reorder the terms:
6 + y = 4.582575695

Solving
6 + y = 4.582575695

Solving for variable 'y'.

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

Add '-6' to each side of the equation.
6 + -6 + y = 4.582575695 + -6

Combine like terms: 6 + -6 = 0
0 + y = 4.582575695 + -6
y = 4.582575695 + -6

Combine like terms: 4.582575695 + -6 = -1.417424305
y = -1.417424305

Simplifying
y = -1.417424305


Subproblem 2
y + 6 = -4.582575695

Simplifying
y + 6 = -4.582575695

Reorder the terms:
6 + y = -4.582575695

Solving
6 + y = -4.582575695

Solving for variable 'y'.

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

Add '-6' to each side of the equation.
6 + -6 + y = -4.582575695 + -6

Combine like terms: 6 + -6 = 0
0 + y = -4.582575695 + -6
y = -4.582575695 + -6

Combine like terms: -4.582575695 + -6 = -10.582575695
y = -10.582575695

Simplifying
y = -10.582575695


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {-1.417424305, -10.582575695}
Solution
y = {-1.417424305, -10.582575695}

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