6z^3+3z^2+9z=0

Solution for 6z^3+3z^2+9z=0 equation:

Simplifying
6z3 + 3z2 + 9z = 0

Reorder the terms:
9z + 3z2 + 6z3 = 0

Solving
9z + 3z2 + 6z3 = 0

Solving for variable 'z'.

Factor out the Greatest Common Factor (GCF), '3z'.
3z(3 + z + 2z2) = 0

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

Simplifying
z = 0

Solving
z = 0

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

Simplifying
z = 0
Subproblem 2
Set the factor '(3 + z + 2z2)' equal to zero and attempt to solve:

Simplifying
3 + z + 2z2 = 0

Solving
3 + z + 2z2 = 0

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

Divide each side by '2'.
1.5 + 0.5z + z2 = 0

Move the constant term to the right:

Add '-1.5' to each side of the equation.
1.5 + 0.5z + -1.5 + z2 = 0 + -1.5

Reorder the terms:
1.5 + -1.5 + 0.5z + z2 = 0 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + 0.5z + z2 = 0 + -1.5
0.5z + z2 = 0 + -1.5

Combine like terms: 0 + -1.5 = -1.5
0.5z + z2 = -1.5

The z term is z.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
0.5z + 0.25 + z2 = -1.5 + 0.25

Reorder the terms:
0.25 + 0.5z + z2 = -1.5 + 0.25

Combine like terms: -1.5 + 0.25 = -1.25
0.25 + 0.5z + z2 = -1.25

Factor a perfect square on the left side:
(z + 0.5)(z + 0.5) = -1.25

Can't calculate square root of the right side.

The solution to this equation could not be determined.

This subproblem is being ignored because a solution could not be determined.
Solution
z = {0}