n^3+12n^2+3n=0

Solution for n^3+12n^2+3n=0 equation:

Simplifying
n3 + 12n2 + 3n = 0

Reorder the terms:
3n + 12n2 + n3 = 0

Solving
3n + 12n2 + n3 = 0

Solving for variable 'n'.

Factor out the Greatest Common Factor (GCF), 'n'.
n(3 + 12n + n2) = 0

Subproblem 1
Set the factor 'n' equal to zero and attempt to solve:

Simplifying
n = 0

Solving
n = 0

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

Simplifying
n = 0
Subproblem 2
Set the factor '(3 + 12n + n2)' equal to zero and attempt to solve:

Simplifying
3 + 12n + n2 = 0

Solving
3 + 12n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-3' to each side of the equation.
3 + 12n + -3 + n2 = 0 + -3

Reorder the terms:
3 + -3 + 12n + n2 = 0 + -3

Combine like terms: 3 + -3 = 0
0 + 12n + n2 = 0 + -3
12n + n2 = 0 + -3

Combine like terms: 0 + -3 = -3
12n + n2 = -3

The n term is 12n.  Take half its coefficient (6).
Square it (36) and add it to both sides.

Add '36' to each side of the equation.
12n + 36 + n2 = -3 + 36

Reorder the terms:
36 + 12n + n2 = -3 + 36

Combine like terms: -3 + 36 = 33
36 + 12n + n2 = 33

Factor a perfect square on the left side:
(n + 6)(n + 6) = 33

Calculate the square root of the right side: 5.744562647

Break this problem into two subproblems by setting 
(n + 6) equal to 5.744562647 and -5.744562647.

Subproblem 1
n + 6 = 5.744562647

Simplifying
n + 6 = 5.744562647

Reorder the terms:
6 + n = 5.744562647

Solving
6 + n = 5.744562647

Solving for variable 'n'.

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

Add '-6' to each side of the equation.
6 + -6 + n = 5.744562647 + -6

Combine like terms: 6 + -6 = 0
0 + n = 5.744562647 + -6
n = 5.744562647 + -6

Combine like terms: 5.744562647 + -6 = -0.255437353
n = -0.255437353

Simplifying
n = -0.255437353

Subproblem 2
n + 6 = -5.744562647

Simplifying
n + 6 = -5.744562647

Reorder the terms:
6 + n = -5.744562647

Solving
6 + n = -5.744562647

Solving for variable 'n'.

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

Add '-6' to each side of the equation.
6 + -6 + n = -5.744562647 + -6

Combine like terms: 6 + -6 = 0
0 + n = -5.744562647 + -6
n = -5.744562647 + -6

Combine like terms: -5.744562647 + -6 = -11.744562647
n = -11.744562647

Simplifying
n = -11.744562647

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {-0.255437353, -11.744562647}
Solution
n = {0, -0.255437353, -11.744562647}