z^8+z^4=1

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Solution for z^8+z^4=1 equation:


Simplifying
z8 + z4 = 1

Reorder the terms:
z4 + z8 = 1

Solving
z4 + z8 = 1

Solving for variable 'z'.

Reorder the terms:
-1 + z4 + z8 = 1 + -1

Combine like terms: 1 + -1 = 0
-1 + z4 + z8 = 0

Begin completing the square.

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + z4 + 1 + z8 = 0 + 1

Reorder the terms:
-1 + 1 + z4 + z8 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + z4 + z8 = 0 + 1
z4 + z8 = 0 + 1

Combine like terms: 0 + 1 = 1
z4 + z8 = 1

The z term is z4.  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.
z4 + 0.25 + z8 = 1 + 0.25

Reorder the terms:
0.25 + z4 + z8 = 1 + 0.25

Combine like terms: 1 + 0.25 = 1.25
0.25 + z4 + z8 = 1.25

Factor a perfect square on the left side:
(z4 + 0.5)(z4 + 0.5) = 1.25

Calculate the square root of the right side: 1.118033989

Break this problem into two subproblems by setting 
(z4 + 0.5) equal to 1.118033989 and -1.118033989.

Subproblem 1

z4 + 0.5 = 1.118033989 Simplifying z4 + 0.5 = 1.118033989 Reorder the terms: 0.5 + z4 = 1.118033989 Solving 0.5 + z4 = 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 + z4 = 1.118033989 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + z4 = 1.118033989 + -0.5 z4 = 1.118033989 + -0.5 Combine like terms: 1.118033989 + -0.5 = 0.618033989 z4 = 0.618033989 Simplifying z4 = 0.618033989 Reorder the terms: -0.618033989 + z4 = 0.618033989 + -0.618033989 Combine like terms: 0.618033989 + -0.618033989 = 0.000000000 -0.618033989 + z4 = 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

z4 + 0.5 = -1.118033989 Simplifying z4 + 0.5 = -1.118033989 Reorder the terms: 0.5 + z4 = -1.118033989 Solving 0.5 + z4 = -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 + z4 = -1.118033989 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + z4 = -1.118033989 + -0.5 z4 = -1.118033989 + -0.5 Combine like terms: -1.118033989 + -0.5 = -1.618033989 z4 = -1.618033989 Simplifying z4 = -1.618033989 Reorder the terms: 1.618033989 + z4 = -1.618033989 + 1.618033989 Combine like terms: -1.618033989 + 1.618033989 = 0.000000000 1.618033989 + z4 = 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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