z^6+z^3+1=0

Solution for z^6+z^3+1=0 equation:

Simplifying
z6 + z3 + 1 = 0

Reorder the terms:
1 + z3 + z6 = 0

Solving
1 + z3 + z6 = 0

Solving for variable 'z'.

Begin completing the square.

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + z3 + -1 + z6 = 0 + -1

Reorder the terms:
1 + -1 + z3 + z6 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + z3 + z6 = 0 + -1
z3 + z6 = 0 + -1

Combine like terms: 0 + -1 = -1
z3 + z6 = -1

The z term is z3.  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.
z3 + 0.25 + z6 = -1 + 0.25

Reorder the terms:
0.25 + z3 + z6 = -1 + 0.25

Combine like terms: -1 + 0.25 = -0.75
0.25 + z3 + z6 = -0.75

Factor a perfect square on the left side:
(z3 + 0.5)(z3 + 0.5) = -0.75

Can't calculate square root of the right side.

The solution to this equation could not be determined.