6a^3+12a^2+9a=0

Solution for 6a^3+12a^2+9a=0 equation:

Simplifying
6a3 + 12a2 + 9a = 0

Reorder the terms:
9a + 12a2 + 6a3 = 0

Solving
9a + 12a2 + 6a3 = 0

Solving for variable 'a'.

Factor out the Greatest Common Factor (GCF), '3a'.
3a(3 + 4a + 2a2) = 0

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

Simplifying
a = 0

Solving
a = 0

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

Simplifying
a = 0
Subproblem 2
Set the factor '(3 + 4a + 2a2)' equal to zero and attempt to solve:

Simplifying
3 + 4a + 2a2 = 0

Solving
3 + 4a + 2a2 = 0

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

Divide each side by '2'.
1.5 + 2a + a2 = 0

Move the constant term to the right:

Add '-1.5' to each side of the equation.
1.5 + 2a + -1.5 + a2 = 0 + -1.5

Reorder the terms:
1.5 + -1.5 + 2a + a2 = 0 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + 2a + a2 = 0 + -1.5
2a + a2 = 0 + -1.5

Combine like terms: 0 + -1.5 = -1.5
2a + a2 = -1.5

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

Add '1' to each side of the equation.
2a + 1 + a2 = -1.5 + 1

Reorder the terms:
1 + 2a + a2 = -1.5 + 1

Combine like terms: -1.5 + 1 = -0.5
1 + 2a + a2 = -0.5

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

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
a = {0}