12z^5+14z^4+3z^3=0

Solution for 12z^5+14z^4+3z^3=0 equation:

Simplifying
12z5 + 14z4 + 3z3 = 0

Reorder the terms:
3z3 + 14z4 + 12z5 = 0

Solving
3z3 + 14z4 + 12z5 = 0

Solving for variable 'z'.

Factor out the Greatest Common Factor (GCF), 'z3'.
z3(3 + 14z + 12z2) = 0

Subproblem 1
Set the factor 'z3' equal to zero and attempt to solve:

Simplifying
z3 = 0

Solving
z3 = 0

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

Simplifying
z3 = 0

The solution to this equation could not be determined.

This subproblem is being ignored because a solution could not be determined.
Subproblem 2
Set the factor '(3 + 14z + 12z2)' equal to zero and attempt to solve:

Simplifying
3 + 14z + 12z2 = 0

Solving
3 + 14z + 12z2 = 0

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

Divide each side by '12'.
0.25 + 1.166666667z + z2 = 0

Move the constant term to the right:

Add '-0.25' to each side of the equation.
0.25 + 1.166666667z + -0.25 + z2 = 0 + -0.25

Reorder the terms:
0.25 + -0.25 + 1.166666667z + z2 = 0 + -0.25

Combine like terms: 0.25 + -0.25 = 0.00
0.00 + 1.166666667z + z2 = 0 + -0.25
1.166666667z + z2 = 0 + -0.25

Combine like terms: 0 + -0.25 = -0.25
1.166666667z + z2 = -0.25

The z term is 1.166666667z.  Take half its coefficient (0.5833333335).
Square it (0.3402777780) and add it to both sides.

Add '0.3402777780' to each side of the equation.
1.166666667z + 0.3402777780 + z2 = -0.25 + 0.3402777780

Reorder the terms:
0.3402777780 + 1.166666667z + z2 = -0.25 + 0.3402777780

Combine like terms: -0.25 + 0.3402777780 = 0.090277778
0.3402777780 + 1.166666667z + z2 = 0.090277778

Factor a perfect square on the left side:
(z + 0.5833333335)(z + 0.5833333335) = 0.090277778

Calculate the square root of the right side: 0.300462607

Break this problem into two subproblems by setting 
(z + 0.5833333335) equal to 0.300462607 and -0.300462607.

Subproblem 1
z + 0.5833333335 = 0.300462607

Simplifying
z + 0.5833333335 = 0.300462607

Reorder the terms:
0.5833333335 + z = 0.300462607

Solving
0.5833333335 + z = 0.300462607

Solving for variable 'z'.

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

Add '-0.5833333335' to each side of the equation.
0.5833333335 + -0.5833333335 + z = 0.300462607 + -0.5833333335

Combine like terms: 0.5833333335 + -0.5833333335 = 0.0000000000
0.0000000000 + z = 0.300462607 + -0.5833333335
z = 0.300462607 + -0.5833333335

Combine like terms: 0.300462607 + -0.5833333335 = -0.2828707265
z = -0.2828707265

Simplifying
z = -0.2828707265

Subproblem 2
z + 0.5833333335 = -0.300462607

Simplifying
z + 0.5833333335 = -0.300462607

Reorder the terms:
0.5833333335 + z = -0.300462607

Solving
0.5833333335 + z = -0.300462607

Solving for variable 'z'.

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

Add '-0.5833333335' to each side of the equation.
0.5833333335 + -0.5833333335 + z = -0.300462607 + -0.5833333335

Combine like terms: 0.5833333335 + -0.5833333335 = 0.0000000000
0.0000000000 + z = -0.300462607 + -0.5833333335
z = -0.300462607 + -0.5833333335

Combine like terms: -0.300462607 + -0.5833333335 = -0.8837959405
z = -0.8837959405

Simplifying
z = -0.8837959405

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {-0.2828707265, -0.8837959405}
Solution
z = {-0.2828707265, -0.8837959405}