z^4+3z^2-1=0

Solution for z^4+3z^2-1=0 equation:

Simplifying
z4 + 3z2 + -1 = 0

Reorder the terms:
-1 + 3z2 + z4 = 0

Solving
-1 + 3z2 + 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 + 3z2 + 1 + z4 = 0 + 1

Reorder the terms:
-1 + 1 + 3z2 + z4 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + 3z2 + z4 = 0 + 1
3z2 + z4 = 0 + 1

Combine like terms: 0 + 1 = 1
3z2 + z4 = 1

The z term is 3z2.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3z2 + 2.25 + z4 = 1 + 2.25

Reorder the terms:
2.25 + 3z2 + z4 = 1 + 2.25

Combine like terms: 1 + 2.25 = 3.25
2.25 + 3z2 + z4 = 3.25

Factor a perfect square on the left side:
(z2 + 1.5)(z2 + 1.5) = 3.25

Calculate the square root of the right side: 1.802775638

Break this problem into two subproblems by setting 
(z2 + 1.5) equal to 1.802775638 and -1.802775638.

Subproblem 1
z2 + 1.5 = 1.802775638

Simplifying
z2 + 1.5 = 1.802775638

Reorder the terms:
1.5 + z2 = 1.802775638

Solving
1.5 + z2 = 1.802775638

Solving for variable 'z'.

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

Add '-1.5' to each side of the equation.
1.5 + -1.5 + z2 = 1.802775638 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + z2 = 1.802775638 + -1.5
z2 = 1.802775638 + -1.5

Combine like terms: 1.802775638 + -1.5 = 0.302775638
z2 = 0.302775638

Simplifying
z2 = 0.302775638

Take the square root of each side:
z = {-0.550250523, 0.550250523}

Subproblem 2
z2 + 1.5 = -1.802775638

Simplifying
z2 + 1.5 = -1.802775638

Reorder the terms:
1.5 + z2 = -1.802775638

Solving
1.5 + z2 = -1.802775638

Solving for variable 'z'.

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

Add '-1.5' to each side of the equation.
1.5 + -1.5 + z2 = -1.802775638 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + z2 = -1.802775638 + -1.5
z2 = -1.802775638 + -1.5

Combine like terms: -1.802775638 + -1.5 = -3.302775638
z2 = -3.302775638

Simplifying
z2 = -3.302775638

Reorder the terms:
3.302775638 + z2 = -3.302775638 + 3.302775638

Combine like terms: -3.302775638 + 3.302775638 = 0.000000000
3.302775638 + 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.