3y^2+32y+12=0

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Solution for 3y^2+32y+12=0 equation: 


Simplifying
3y2 + 32y + 12 = 0

Reorder the terms:
12 + 32y + 3y2 = 0

Solving
12 + 32y + 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'.
4 + 10.66666667y + y2 = 0

Move the constant term to the right:

Add '-4' to each side of the equation.
4 + 10.66666667y + -4 + y2 = 0 + -4

Reorder the terms:
4 + -4 + 10.66666667y + y2 = 0 + -4

Combine like terms: 4 + -4 = 0
0 + 10.66666667y + y2 = 0 + -4
10.66666667y + y2 = 0 + -4

Combine like terms: 0 + -4 = -4
10.66666667y + y2 = -4

The y term is 10.66666667y.  Take half its coefficient (5.333333335).
Square it (28.44444446) and add it to both sides.

Add '28.44444446' to each side of the equation.
10.66666667y + 28.44444446 + y2 = -4 + 28.44444446

Reorder the terms:
28.44444446 + 10.66666667y + y2 = -4 + 28.44444446

Combine like terms: -4 + 28.44444446 = 24.44444446
28.44444446 + 10.66666667y + y2 = 24.44444446

Factor a perfect square on the left side:
(y + 5.333333335)(y + 5.333333335) = 24.44444446

Calculate the square root of the right side: 4.944132326

Break this problem into two subproblems by setting 
(y + 5.333333335) equal to 4.944132326 and -4.944132326.


Subproblem 1
y + 5.333333335 = 4.944132326

Simplifying
y + 5.333333335 = 4.944132326

Reorder the terms:
5.333333335 + y = 4.944132326

Solving
5.333333335 + y = 4.944132326

Solving for variable 'y'.

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

Add '-5.333333335' to each side of the equation.
5.333333335 + -5.333333335 + y = 4.944132326 + -5.333333335

Combine like terms: 5.333333335 + -5.333333335 = 0.000000000
0.000000000 + y = 4.944132326 + -5.333333335
y = 4.944132326 + -5.333333335

Combine like terms: 4.944132326 + -5.333333335 = -0.389201009
y = -0.389201009

Simplifying
y = -0.389201009


Subproblem 2
y + 5.333333335 = -4.944132326

Simplifying
y + 5.333333335 = -4.944132326

Reorder the terms:
5.333333335 + y = -4.944132326

Solving
5.333333335 + y = -4.944132326

Solving for variable 'y'.

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

Add '-5.333333335' to each side of the equation.
5.333333335 + -5.333333335 + y = -4.944132326 + -5.333333335

Combine like terms: 5.333333335 + -5.333333335 = 0.000000000
0.000000000 + y = -4.944132326 + -5.333333335
y = -4.944132326 + -5.333333335

Combine like terms: -4.944132326 + -5.333333335 = -10.277465661
y = -10.277465661

Simplifying
y = -10.277465661


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {-0.389201009, -10.277465661}

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