z^6+4z^3-16=0

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Solution for z^6+4z^3-16=0 equation: 


Simplifying
z6 + 4z3 + -16 = 0

Reorder the terms:
-16 + 4z3 + z6 = 0

Solving
-16 + 4z3 + z6 = 0

Solving for variable 'z'.

Begin completing the square.

Move the constant term to the right:

Add '16' to each side of the equation.
-16 + 4z3 + 16 + z6 = 0 + 16

Reorder the terms:
-16 + 16 + 4z3 + z6 = 0 + 16

Combine like terms: -16 + 16 = 0
0 + 4z3 + z6 = 0 + 16
4z3 + z6 = 0 + 16

Combine like terms: 0 + 16 = 16
4z3 + z6 = 16

The z term is 4z3.  Take half its coefficient (2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
4z3 + 4 + z6 = 16 + 4

Reorder the terms:
4 + 4z3 + z6 = 16 + 4

Combine like terms: 16 + 4 = 20
4 + 4z3 + z6 = 20

Factor a perfect square on the left side:
(z3 + 2)(z3 + 2) = 20

Calculate the square root of the right side: 4.472135955

Break this problem into two subproblems by setting 
(z3 + 2) equal to 4.472135955 and -4.472135955.


Subproblem 1
z3 + 2 = 4.472135955

Simplifying
z3 + 2 = 4.472135955

Reorder the terms:
2 + z3 = 4.472135955

Solving
2 + z3 = 4.472135955

Solving for variable 'z'.

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

Add '-2' to each side of the equation.
2 + -2 + z3 = 4.472135955 + -2

Combine like terms: 2 + -2 = 0
0 + z3 = 4.472135955 + -2
z3 = 4.472135955 + -2

Combine like terms: 4.472135955 + -2 = 2.472135955
z3 = 2.472135955

Simplifying
z3 = 2.472135955

Reorder the terms:
-2.472135955 + z3 = 2.472135955 + -2.472135955

Combine like terms: 2.472135955 + -2.472135955 = 0.000000000
-2.472135955 + 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 + 2 = -4.472135955

Simplifying
z3 + 2 = -4.472135955

Reorder the terms:
2 + z3 = -4.472135955

Solving
2 + z3 = -4.472135955

Solving for variable 'z'.

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

Add '-2' to each side of the equation.
2 + -2 + z3 = -4.472135955 + -2

Combine like terms: 2 + -2 = 0
0 + z3 = -4.472135955 + -2
z3 = -4.472135955 + -2

Combine like terms: -4.472135955 + -2 = -6.472135955
z3 = -6.472135955

Simplifying
z3 = -6.472135955

Reorder the terms:
6.472135955 + z3 = -6.472135955 + 6.472135955

Combine like terms: -6.472135955 + 6.472135955 = 0.000000000
6.472135955 + 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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