3y^2+12y+7=0

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Solution for 3y^2+12y+7=0 equation: 


Simplifying
3y2 + 12y + 7 = 0

Reorder the terms:
7 + 12y + 3y2 = 0

Solving
7 + 12y + 3y2 = 0

Solving for variable 'y'.

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

Divide each side by '3'.
2.333333333 + 4y + y2 = 0

Move the constant term to the right:

Add '-2.333333333' to each side of the equation.
2.333333333 + 4y + -2.333333333 + y2 = 0 + -2.333333333

Reorder the terms:
2.333333333 + -2.333333333 + 4y + y2 = 0 + -2.333333333

Combine like terms: 2.333333333 + -2.333333333 = 0.000000000
0.000000000 + 4y + y2 = 0 + -2.333333333
4y + y2 = 0 + -2.333333333

Combine like terms: 0 + -2.333333333 = -2.333333333
4y + y2 = -2.333333333

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 = -2.333333333 + 4

Reorder the terms:
4 + 4y + y2 = -2.333333333 + 4

Combine like terms: -2.333333333 + 4 = 1.666666667
4 + 4y + y2 = 1.666666667

Factor a perfect square on the left side:
(y + 2)(y + 2) = 1.666666667

Calculate the square root of the right side: 1.290994449

Break this problem into two subproblems by setting 
(y + 2) equal to 1.290994449 and -1.290994449.


Subproblem 1
y + 2 = 1.290994449

Simplifying
y + 2 = 1.290994449

Reorder the terms:
2 + y = 1.290994449

Solving
2 + y = 1.290994449

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 = 1.290994449 + -2

Combine like terms: 2 + -2 = 0
0 + y = 1.290994449 + -2
y = 1.290994449 + -2

Combine like terms: 1.290994449 + -2 = -0.709005551
y = -0.709005551

Simplifying
y = -0.709005551


Subproblem 2
y + 2 = -1.290994449

Simplifying
y + 2 = -1.290994449

Reorder the terms:
2 + y = -1.290994449

Solving
2 + y = -1.290994449

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 = -1.290994449 + -2

Combine like terms: 2 + -2 = 0
0 + y = -1.290994449 + -2
y = -1.290994449 + -2

Combine like terms: -1.290994449 + -2 = -3.290994449
y = -3.290994449

Simplifying
y = -3.290994449


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {-0.709005551, -3.290994449}

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