21z^5-42z^4+6z^3=0

Solution for 21z^5-42z^4+6z^3=0 equation:

Simplifying
21z5 + -42z4 + 6z3 = 0

Reorder the terms:
6z3 + -42z4 + 21z5 = 0

Solving
6z3 + -42z4 + 21z5 = 0

Solving for variable 'z'.

Factor out the Greatest Common Factor (GCF), '3z3'.
3z3(2 + -14z + 7z2) = 0

Ignore the factor 3.
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 '(2 + -14z + 7z2)' equal to zero and attempt to solve:

Simplifying
2 + -14z + 7z2 = 0

Solving
2 + -14z + 7z2 = 0

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

Divide each side by '7'.
0.2857142857 + -2z + z2 = 0

Move the constant term to the right:

Add '-0.2857142857' to each side of the equation.
0.2857142857 + -2z + -0.2857142857 + z2 = 0 + -0.2857142857

Reorder the terms:
0.2857142857 + -0.2857142857 + -2z + z2 = 0 + -0.2857142857

Combine like terms: 0.2857142857 + -0.2857142857 = 0.0000000000
0.0000000000 + -2z + z2 = 0 + -0.2857142857
-2z + z2 = 0 + -0.2857142857

Combine like terms: 0 + -0.2857142857 = -0.2857142857
-2z + z2 = -0.2857142857

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

Add '1' to each side of the equation.
-2z + 1 + z2 = -0.2857142857 + 1

Reorder the terms:
1 + -2z + z2 = -0.2857142857 + 1

Combine like terms: -0.2857142857 + 1 = 0.7142857143
1 + -2z + z2 = 0.7142857143

Factor a perfect square on the left side:
(z + -1)(z + -1) = 0.7142857143

Calculate the square root of the right side: 0.845154255

Break this problem into two subproblems by setting 
(z + -1) equal to 0.845154255 and -0.845154255.

Subproblem 1
z + -1 = 0.845154255

Simplifying
z + -1 = 0.845154255

Reorder the terms:
-1 + z = 0.845154255

Solving
-1 + z = 0.845154255

Solving for variable 'z'.

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

Add '1' to each side of the equation.
-1 + 1 + z = 0.845154255 + 1

Combine like terms: -1 + 1 = 0
0 + z = 0.845154255 + 1
z = 0.845154255 + 1

Combine like terms: 0.845154255 + 1 = 1.845154255
z = 1.845154255

Simplifying
z = 1.845154255

Subproblem 2
z + -1 = -0.845154255

Simplifying
z + -1 = -0.845154255

Reorder the terms:
-1 + z = -0.845154255

Solving
-1 + z = -0.845154255

Solving for variable 'z'.

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

Add '1' to each side of the equation.
-1 + 1 + z = -0.845154255 + 1

Combine like terms: -1 + 1 = 0
0 + z = -0.845154255 + 1
z = -0.845154255 + 1

Combine like terms: -0.845154255 + 1 = 0.154845745
z = 0.154845745

Simplifying
z = 0.154845745

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {1.845154255, 0.154845745}
Solution
z = {1.845154255, 0.154845745}