z^4+2z^2-1=0

Solution for z^4+2z^2-1=0 equation:

Simplifying
z4 + 2z2 + -1 = 0

Reorder the terms:
-1 + 2z2 + z4 = 0

Solving
-1 + 2z2 + z4 = 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 + 2z2 + 1 + z4 = 0 + 1

Reorder the terms:
-1 + 1 + 2z2 + z4 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + 2z2 + z4 = 0 + 1
2z2 + z4 = 0 + 1

Combine like terms: 0 + 1 = 1
2z2 + z4 = 1

The z term is 2z2.  Take half its coefficient (1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
2z2 + 1 + z4 = 1 + 1

Reorder the terms:
1 + 2z2 + z4 = 1 + 1

Combine like terms: 1 + 1 = 2
1 + 2z2 + z4 = 2

Factor a perfect square on the left side:
(z2 + 1)(z2 + 1) = 2

Calculate the square root of the right side: 1.414213562

Break this problem into two subproblems by setting 
(z2 + 1) equal to 1.414213562 and -1.414213562.

Subproblem 1
z2 + 1 = 1.414213562

Simplifying
z2 + 1 = 1.414213562

Reorder the terms:
1 + z2 = 1.414213562

Solving
1 + z2 = 1.414213562

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 + z2 = 1.414213562 + -1

Combine like terms: 1 + -1 = 0
0 + z2 = 1.414213562 + -1
z2 = 1.414213562 + -1

Combine like terms: 1.414213562 + -1 = 0.414213562
z2 = 0.414213562

Simplifying
z2 = 0.414213562

Take the square root of each side:
z = {-0.643594253, 0.643594253}

Subproblem 2
z2 + 1 = -1.414213562

Simplifying
z2 + 1 = -1.414213562

Reorder the terms:
1 + z2 = -1.414213562

Solving
1 + z2 = -1.414213562

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 + z2 = -1.414213562 + -1

Combine like terms: 1 + -1 = 0
0 + z2 = -1.414213562 + -1
z2 = -1.414213562 + -1

Combine like terms: -1.414213562 + -1 = -2.414213562
z2 = -2.414213562

Simplifying
z2 = -2.414213562

Reorder the terms:
2.414213562 + z2 = -2.414213562 + 2.414213562

Combine like terms: -2.414213562 + 2.414213562 = 0.000000000
2.414213562 + z2 = 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.