z^6+24z^3+3=0

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Solution for z^6+24z^3+3=0 equation: 


Simplifying
z6 + 24z3 + 3 = 0

Reorder the terms:
3 + 24z3 + z6 = 0

Solving
3 + 24z3 + z6 = 0

Solving for variable 'z'.

Begin completing the square.

Move the constant term to the right:

Add '-3' to each side of the equation.
3 + 24z3 + -3 + z6 = 0 + -3

Reorder the terms:
3 + -3 + 24z3 + z6 = 0 + -3

Combine like terms: 3 + -3 = 0
0 + 24z3 + z6 = 0 + -3
24z3 + z6 = 0 + -3

Combine like terms: 0 + -3 = -3
24z3 + z6 = -3

The z term is 24z3.  Take half its coefficient (12).
Square it (144) and add it to both sides.

Add '144' to each side of the equation.
24z3 + 144 + z6 = -3 + 144

Reorder the terms:
144 + 24z3 + z6 = -3 + 144

Combine like terms: -3 + 144 = 141
144 + 24z3 + z6 = 141

Factor a perfect square on the left side:
(z3 + 12)(z3 + 12) = 141

Calculate the square root of the right side: 11.874342087

Break this problem into two subproblems by setting 
(z3 + 12) equal to 11.874342087 and -11.874342087.


Subproblem 1
z3 + 12 = 11.874342087

Simplifying
z3 + 12 = 11.874342087

Reorder the terms:
12 + z3 = 11.874342087

Solving
12 + z3 = 11.874342087

Solving for variable 'z'.

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

Add '-12' to each side of the equation.
12 + -12 + z3 = 11.874342087 + -12

Combine like terms: 12 + -12 = 0
0 + z3 = 11.874342087 + -12
z3 = 11.874342087 + -12

Combine like terms: 11.874342087 + -12 = -0.125657913
z3 = -0.125657913

Simplifying
z3 = -0.125657913

Reorder the terms:
0.125657913 + z3 = -0.125657913 + 0.125657913

Combine like terms: -0.125657913 + 0.125657913 = 0.000000000
0.125657913 + 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 + 12 = -11.874342087

Simplifying
z3 + 12 = -11.874342087

Reorder the terms:
12 + z3 = -11.874342087

Solving
12 + z3 = -11.874342087

Solving for variable 'z'.

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

Add '-12' to each side of the equation.
12 + -12 + z3 = -11.874342087 + -12

Combine like terms: 12 + -12 = 0
0 + z3 = -11.874342087 + -12
z3 = -11.874342087 + -12

Combine like terms: -11.874342087 + -12 = -23.874342087
z3 = -23.874342087

Simplifying
z3 = -23.874342087

Reorder the terms:
23.874342087 + z3 = -23.874342087 + 23.874342087

Combine like terms: -23.874342087 + 23.874342087 = 0.000000000
23.874342087 + 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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