z^5+z^3+z=0

Solution for z^5+z^3+z=0 equation:

Simplifying
z5 + z3 + z = 0

Reorder the terms:
z + z3 + z5 = 0

Solving
z + z3 + z5 = 0

Solving for variable 'z'.

Factor out the Greatest Common Factor (GCF), 'z'.
z(1 + z2 + z4) = 0

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

Simplifying
z = 0

Solving
z = 0

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

Simplifying
z = 0
Subproblem 2
Set the factor '(1 + z2 + z4)' equal to zero and attempt to solve:

Simplifying
1 + z2 + z4 = 0

Solving
1 + z2 + z4 = 0

Begin completing the square.

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + z2 + -1 + z4 = 0 + -1

Reorder the terms:
1 + -1 + z2 + z4 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + z2 + z4 = 0 + -1
z2 + z4 = 0 + -1

Combine like terms: 0 + -1 = -1
z2 + z4 = -1

The z term is z2.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
z2 + 0.25 + z4 = -1 + 0.25

Reorder the terms:
0.25 + z2 + z4 = -1 + 0.25

Combine like terms: -1 + 0.25 = -0.75
0.25 + z2 + z4 = -0.75

Factor a perfect square on the left side:
(z2 + 0.5)(z2 + 0.5) = -0.75

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
z = {0}