6y^3+12y^2+3y=0

Solution for 6y^3+12y^2+3y=0 equation:

Simplifying
6y3 + 12y2 + 3y = 0

Reorder the terms:
3y + 12y2 + 6y3 = 0

Solving
3y + 12y2 + 6y3 = 0

Solving for variable 'y'.

Factor out the Greatest Common Factor (GCF), '3y'.
3y(1 + 4y + 2y2) = 0

Ignore the factor 3.
Subproblem 1
Set the factor 'y' equal to zero and attempt to solve:

Simplifying
y = 0

Solving
y = 0

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

Simplifying
y = 0
Subproblem 2
Set the factor '(1 + 4y + 2y2)' equal to zero and attempt to solve:

Simplifying
1 + 4y + 2y2 = 0

Solving
1 + 4y + 2y2 = 0

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

Divide each side by '2'.
0.5 + 2y + y2 = 0

Move the constant term to the right:

Add '-0.5' to each side of the equation.
0.5 + 2y + -0.5 + y2 = 0 + -0.5

Reorder the terms:
0.5 + -0.5 + 2y + y2 = 0 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + 2y + y2 = 0 + -0.5
2y + y2 = 0 + -0.5

Combine like terms: 0 + -0.5 = -0.5
2y + y2 = -0.5

The y term is 2y.  Take half its coefficient (1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
2y + 1 + y2 = -0.5 + 1

Reorder the terms:
1 + 2y + y2 = -0.5 + 1

Combine like terms: -0.5 + 1 = 0.5
1 + 2y + y2 = 0.5

Factor a perfect square on the left side:
(y + 1)(y + 1) = 0.5

Calculate the square root of the right side: 0.707106781

Break this problem into two subproblems by setting 
(y + 1) equal to 0.707106781 and -0.707106781.

Subproblem 1
y + 1 = 0.707106781

Simplifying
y + 1 = 0.707106781

Reorder the terms:
1 + y = 0.707106781

Solving
1 + y = 0.707106781

Solving for variable 'y'.

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

Add '-1' to each side of the equation.
1 + -1 + y = 0.707106781 + -1

Combine like terms: 1 + -1 = 0
0 + y = 0.707106781 + -1
y = 0.707106781 + -1

Combine like terms: 0.707106781 + -1 = -0.292893219
y = -0.292893219

Simplifying
y = -0.292893219

Subproblem 2
y + 1 = -0.707106781

Simplifying
y + 1 = -0.707106781

Reorder the terms:
1 + y = -0.707106781

Solving
1 + y = -0.707106781

Solving for variable 'y'.

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

Add '-1' to each side of the equation.
1 + -1 + y = -0.707106781 + -1

Combine like terms: 1 + -1 = 0
0 + y = -0.707106781 + -1
y = -0.707106781 + -1

Combine like terms: -0.707106781 + -1 = -1.707106781
y = -1.707106781

Simplifying
y = -1.707106781

Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {-0.292893219, -1.707106781}
Solution
y = {0, -0.292893219, -1.707106781}