9y^2+36y+37=7

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Solution for 9y^2+36y+37=7 equation: 


Simplifying
9y2 + 36y + 37 = 7

Reorder the terms:
37 + 36y + 9y2 = 7

Solving
37 + 36y + 9y2 = 7

Solving for variable 'y'.

Reorder the terms:
37 + -7 + 36y + 9y2 = 7 + -7

Combine like terms: 37 + -7 = 30
30 + 36y + 9y2 = 7 + -7

Combine like terms: 7 + -7 = 0
30 + 36y + 9y2 = 0

Factor out the Greatest Common Factor (GCF), '3'.
3(10 + 12y + 3y2) = 0

Ignore the factor 3.

Subproblem 1
Set the factor '(10 + 12y + 3y2)' equal to zero and attempt to solve:

Simplifying
10 + 12y + 3y2 = 0

Solving
10 + 12y + 3y2 = 0

Begin completing the square.  Divide all terms by
3 the coefficient of the squared term: 

Divide each side by '3'.
3.333333333 + 4y + y2 = 0

Move the constant term to the right:

Add '-3.333333333' to each side of the equation.
3.333333333 + 4y + -3.333333333 + y2 = 0 + -3.333333333

Reorder the terms:
3.333333333 + -3.333333333 + 4y + y2 = 0 + -3.333333333

Combine like terms: 3.333333333 + -3.333333333 = 0.000000000
0.000000000 + 4y + y2 = 0 + -3.333333333
4y + y2 = 0 + -3.333333333

Combine like terms: 0 + -3.333333333 = -3.333333333
4y + y2 = -3.333333333

The y term is 4y.  Take half its coefficient (2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
4y + 4 + y2 = -3.333333333 + 4

Reorder the terms:
4 + 4y + y2 = -3.333333333 + 4

Combine like terms: -3.333333333 + 4 = 0.666666667
4 + 4y + y2 = 0.666666667

Factor a perfect square on the left side:
(y + 2)(y + 2) = 0.666666667

Calculate the square root of the right side: 0.816496581

Break this problem into two subproblems by setting 
(y + 2) equal to 0.816496581 and -0.816496581.


Subproblem 1
y + 2 = 0.816496581

Simplifying
y + 2 = 0.816496581

Reorder the terms:
2 + y = 0.816496581

Solving
2 + y = 0.816496581

Solving for variable 'y'.

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

Add '-2' to each side of the equation.
2 + -2 + y = 0.816496581 + -2

Combine like terms: 2 + -2 = 0
0 + y = 0.816496581 + -2
y = 0.816496581 + -2

Combine like terms: 0.816496581 + -2 = -1.183503419
y = -1.183503419

Simplifying
y = -1.183503419


Subproblem 2
y + 2 = -0.816496581

Simplifying
y + 2 = -0.816496581

Reorder the terms:
2 + y = -0.816496581

Solving
2 + y = -0.816496581

Solving for variable 'y'.

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

Add '-2' to each side of the equation.
2 + -2 + y = -0.816496581 + -2

Combine like terms: 2 + -2 = 0
0 + y = -0.816496581 + -2
y = -0.816496581 + -2

Combine like terms: -0.816496581 + -2 = -2.816496581
y = -2.816496581

Simplifying
y = -2.816496581


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {-1.183503419, -2.816496581}
Solution
y = {-1.183503419, -2.816496581}

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