12y^3+33y^2-6y=

Solution for 12y^3+33y^2-6y= equation:

Simplifying
12y3 + 33y2 + -6y = 0

Reorder the terms:
-6y + 33y2 + 12y3 = 0

Solving
-6y + 33y2 + 12y3 = 0

Solving for variable 'y'.

Factor out the Greatest Common Factor (GCF), '3y'.
3y(-2 + 11y + 4y2) = 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 '(-2 + 11y + 4y2)' equal to zero and attempt to solve:

Simplifying
-2 + 11y + 4y2 = 0

Solving
-2 + 11y + 4y2 = 0

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

Divide each side by '4'.
-0.5 + 2.75y + y2 = 0

Move the constant term to the right:

Add '0.5' to each side of the equation.
-0.5 + 2.75y + 0.5 + y2 = 0 + 0.5

Reorder the terms:
-0.5 + 0.5 + 2.75y + y2 = 0 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + 2.75y + y2 = 0 + 0.5
2.75y + y2 = 0 + 0.5

Combine like terms: 0 + 0.5 = 0.5
2.75y + y2 = 0.5

The y term is 2.75y.  Take half its coefficient (1.375).
Square it (1.890625) and add it to both sides.

Add '1.890625' to each side of the equation.
2.75y + 1.890625 + y2 = 0.5 + 1.890625

Reorder the terms:
1.890625 + 2.75y + y2 = 0.5 + 1.890625

Combine like terms: 0.5 + 1.890625 = 2.390625
1.890625 + 2.75y + y2 = 2.390625

Factor a perfect square on the left side:
(y + 1.375)(y + 1.375) = 2.390625

Calculate the square root of the right side: 1.54616461

Break this problem into two subproblems by setting 
(y + 1.375) equal to 1.54616461 and -1.54616461.

Subproblem 1
y + 1.375 = 1.54616461

Simplifying
y + 1.375 = 1.54616461

Reorder the terms:
1.375 + y = 1.54616461

Solving
1.375 + y = 1.54616461

Solving for variable 'y'.

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

Add '-1.375' to each side of the equation.
1.375 + -1.375 + y = 1.54616461 + -1.375

Combine like terms: 1.375 + -1.375 = 0.000
0.000 + y = 1.54616461 + -1.375
y = 1.54616461 + -1.375

Combine like terms: 1.54616461 + -1.375 = 0.17116461
y = 0.17116461

Simplifying
y = 0.17116461

Subproblem 2
y + 1.375 = -1.54616461

Simplifying
y + 1.375 = -1.54616461

Reorder the terms:
1.375 + y = -1.54616461

Solving
1.375 + y = -1.54616461

Solving for variable 'y'.

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

Add '-1.375' to each side of the equation.
1.375 + -1.375 + y = -1.54616461 + -1.375

Combine like terms: 1.375 + -1.375 = 0.000
0.000 + y = -1.54616461 + -1.375
y = -1.54616461 + -1.375

Combine like terms: -1.54616461 + -1.375 = -2.92116461
y = -2.92116461

Simplifying
y = -2.92116461

Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {0.17116461, -2.92116461}
Solution
y = {0, 0.17116461, -2.92116461}