z^6+2z^3-2=0

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Solution for z^6+2z^3-2=0 equation: 


Simplifying
z6 + 2z3 + -2 = 0

Reorder the terms:
-2 + 2z3 + z6 = 0

Solving
-2 + 2z3 + z6 = 0

Solving for variable 'z'.

Begin completing the square.

Move the constant term to the right:

Add '2' to each side of the equation.
-2 + 2z3 + 2 + z6 = 0 + 2

Reorder the terms:
-2 + 2 + 2z3 + z6 = 0 + 2

Combine like terms: -2 + 2 = 0
0 + 2z3 + z6 = 0 + 2
2z3 + z6 = 0 + 2

Combine like terms: 0 + 2 = 2
2z3 + z6 = 2

The z term is 2z3.  Take half its coefficient (1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
2z3 + 1 + z6 = 2 + 1

Reorder the terms:
1 + 2z3 + z6 = 2 + 1

Combine like terms: 2 + 1 = 3
1 + 2z3 + z6 = 3

Factor a perfect square on the left side:
(z3 + 1)(z3 + 1) = 3

Calculate the square root of the right side: 1.732050808

Break this problem into two subproblems by setting 
(z3 + 1) equal to 1.732050808 and -1.732050808.


Subproblem 1
z3 + 1 = 1.732050808

Simplifying
z3 + 1 = 1.732050808

Reorder the terms:
1 + z3 = 1.732050808

Solving
1 + z3 = 1.732050808

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 + z3 = 1.732050808 + -1

Combine like terms: 1 + -1 = 0
0 + z3 = 1.732050808 + -1
z3 = 1.732050808 + -1

Combine like terms: 1.732050808 + -1 = 0.732050808
z3 = 0.732050808

Simplifying
z3 = 0.732050808

Reorder the terms:
-0.732050808 + z3 = 0.732050808 + -0.732050808

Combine like terms: 0.732050808 + -0.732050808 = 0.000000000
-0.732050808 + 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 + 1 = -1.732050808

Simplifying
z3 + 1 = -1.732050808

Reorder the terms:
1 + z3 = -1.732050808

Solving
1 + z3 = -1.732050808

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 + z3 = -1.732050808 + -1

Combine like terms: 1 + -1 = 0
0 + z3 = -1.732050808 + -1
z3 = -1.732050808 + -1

Combine like terms: -1.732050808 + -1 = -2.732050808
z3 = -2.732050808

Simplifying
z3 = -2.732050808

Reorder the terms:
2.732050808 + z3 = -2.732050808 + 2.732050808

Combine like terms: -2.732050808 + 2.732050808 = 0.000000000
2.732050808 + 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.

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