y^2+5y+7y=-2

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Solution for y^2+5y+7y=-2 equation: 


Simplifying
y2 + 5y + 7y = -2

Reorder the terms:
5y + 7y + y2 = -2

Combine like terms: 5y + 7y = 12y
12y + y2 = -2

Solving
12y + y2 = -2

Solving for variable 'y'.

Reorder the terms:
2 + 12y + y2 = -2 + 2

Combine like terms: -2 + 2 = 0
2 + 12y + y2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-2' to each side of the equation.
2 + 12y + -2 + y2 = 0 + -2

Reorder the terms:
2 + -2 + 12y + y2 = 0 + -2

Combine like terms: 2 + -2 = 0
0 + 12y + y2 = 0 + -2
12y + y2 = 0 + -2

Combine like terms: 0 + -2 = -2
12y + y2 = -2

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 = -2 + 36

Reorder the terms:
36 + 12y + y2 = -2 + 36

Combine like terms: -2 + 36 = 34
36 + 12y + y2 = 34

Factor a perfect square on the left side:
(y + 6)(y + 6) = 34

Calculate the square root of the right side: 5.830951895

Break this problem into two subproblems by setting 
(y + 6) equal to 5.830951895 and -5.830951895.


Subproblem 1
y + 6 = 5.830951895

Simplifying
y + 6 = 5.830951895

Reorder the terms:
6 + y = 5.830951895

Solving
6 + y = 5.830951895

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 = 5.830951895 + -6

Combine like terms: 6 + -6 = 0
0 + y = 5.830951895 + -6
y = 5.830951895 + -6

Combine like terms: 5.830951895 + -6 = -0.169048105
y = -0.169048105

Simplifying
y = -0.169048105


Subproblem 2
y + 6 = -5.830951895

Simplifying
y + 6 = -5.830951895

Reorder the terms:
6 + y = -5.830951895

Solving
6 + y = -5.830951895

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 = -5.830951895 + -6

Combine like terms: 6 + -6 = 0
0 + y = -5.830951895 + -6
y = -5.830951895 + -6

Combine like terms: -5.830951895 + -6 = -11.830951895
y = -11.830951895

Simplifying
y = -11.830951895


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {-0.169048105, -11.830951895}

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