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 = 1.25
0.25 + z3 + z6 = 1.25

Factor a perfect square on the left side:
(z3 + 0.5)(z3 + 0.5) = 1.25

Calculate the square root of the right side: 1.118033989

Break this problem into two subproblems by setting 
(z3 + 0.5) equal to 1.118033989 and -1.118033989.

Subproblem 1
z3 + 0.5 = 1.118033989

Simplifying
z3 + 0.5 = 1.118033989

Reorder the terms:
0.5 + z3 = 1.118033989

Solving
0.5 + z3 = 1.118033989

Solving for variable 'z'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + z3 = 1.118033989 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + z3 = 1.118033989 + -0.5
z3 = 1.118033989 + -0.5

Combine like terms: 1.118033989 + -0.5 = 0.618033989
z3 = 0.618033989

Simplifying
z3 = 0.618033989

Reorder the terms:
-0.618033989 + z3 = 0.618033989 + -0.618033989

Combine like terms: 0.618033989 + -0.618033989 = 0.000000000
-0.618033989 + z3 = 0.000000000

The solution to this equation could not be determined.This subproblem is being ignored because a solution could not be determined.

Subproblem 2
z3 + 0.5 = -1.118033989

Simplifying
z3 + 0.5 = -1.118033989

Reorder the terms:
0.5 + z3 = -1.118033989

Solving
0.5 + z3 = -1.118033989

Solving for variable 'z'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + z3 = -1.118033989 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + z3 = -1.118033989 + -0.5
z3 = -1.118033989 + -0.5

Combine like terms: -1.118033989 + -0.5 = -1.618033989
z3 = -1.618033989

Simplifying
z3 = -1.618033989

Reorder the terms:
1.618033989 + z3 = -1.618033989 + 1.618033989

Combine like terms: -1.618033989 + 1.618033989 = 0.000000000
1.618033989 + z3 = 0.000000000

The solution to this equation could not be determined.This subproblem is being ignored because a solution could not be determined.

The solution to this equation could not be determined.