12c^4+24c^3+c^2=0

Solution for 12c^4+24c^3+c^2=0 equation:

Simplifying
12c4 + 24c3 + c2 = 0

Reorder the terms:
c2 + 24c3 + 12c4 = 0

Solving
c2 + 24c3 + 12c4 = 0

Solving for variable 'c'.

Factor out the Greatest Common Factor (GCF), 'c2'.
c2(1 + 24c + 12c2) = 0

Subproblem 1
Set the factor 'c2' equal to zero and attempt to solve:

Simplifying
c2 = 0

Solving
c2 = 0

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

Simplifying
c2 = 0

Take the square root of each side:
c = {0}
Subproblem 2
Set the factor '(1 + 24c + 12c2)' equal to zero and attempt to solve:

Simplifying
1 + 24c + 12c2 = 0

Solving
1 + 24c + 12c2 = 0

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

Divide each side by '12'.
0.08333333333 + 2c + c2 = 0

Move the constant term to the right:

Add '-0.08333333333' to each side of the equation.
0.08333333333 + 2c + -0.08333333333 + c2 = 0 + -0.08333333333

Reorder the terms:
0.08333333333 + -0.08333333333 + 2c + c2 = 0 + -0.08333333333

Combine like terms: 0.08333333333 + -0.08333333333 = 0.00000000000
0.00000000000 + 2c + c2 = 0 + -0.08333333333
2c + c2 = 0 + -0.08333333333

Combine like terms: 0 + -0.08333333333 = -0.08333333333
2c + c2 = -0.08333333333

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

Add '1' to each side of the equation.
2c + 1 + c2 = -0.08333333333 + 1

Reorder the terms:
1 + 2c + c2 = -0.08333333333 + 1

Combine like terms: -0.08333333333 + 1 = 0.91666666667
1 + 2c + c2 = 0.91666666667

Factor a perfect square on the left side:
(c + 1)(c + 1) = 0.91666666667

Calculate the square root of the right side: 0.957427108

Break this problem into two subproblems by setting 
(c + 1) equal to 0.957427108 and -0.957427108.

Subproblem 1
c + 1 = 0.957427108

Simplifying
c + 1 = 0.957427108

Reorder the terms:
1 + c = 0.957427108

Solving
1 + c = 0.957427108

Solving for variable 'c'.

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

Add '-1' to each side of the equation.
1 + -1 + c = 0.957427108 + -1

Combine like terms: 1 + -1 = 0
0 + c = 0.957427108 + -1
c = 0.957427108 + -1

Combine like terms: 0.957427108 + -1 = -0.042572892
c = -0.042572892

Simplifying
c = -0.042572892

Subproblem 2
c + 1 = -0.957427108

Simplifying
c + 1 = -0.957427108

Reorder the terms:
1 + c = -0.957427108

Solving
1 + c = -0.957427108

Solving for variable 'c'.

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

Add '-1' to each side of the equation.
1 + -1 + c = -0.957427108 + -1

Combine like terms: 1 + -1 = 0
0 + c = -0.957427108 + -1
c = -0.957427108 + -1

Combine like terms: -0.957427108 + -1 = -1.957427108
c = -1.957427108

Simplifying
c = -1.957427108

Solution
The solution to the problem is based on the solutions
from the subproblems.
c = {-0.042572892, -1.957427108}
Solution
c = {0, -0.042572892, -1.957427108}