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

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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 = 1.25
2.25 + 3z2 + z4 = 1.25

Factor a perfect square on the left side:
(z2 + 1.5)(z2 + 1.5) = 1.25

Calculate the square root of the right side: 1.118033989

Break this problem into two subproblems by setting 
(z2 + 1.5) equal to 1.118033989 and -1.118033989.

Subproblem 1

z2 + 1.5 = 1.118033989 Simplifying z2 + 1.5 = 1.118033989 Reorder the terms: 1.5 + z2 = 1.118033989 Solving 1.5 + z2 = 1.118033989 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 = 1.118033989 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + z2 = 1.118033989 + -1.5 z2 = 1.118033989 + -1.5 Combine like terms: 1.118033989 + -1.5 = -0.381966011 z2 = -0.381966011 Simplifying z2 = -0.381966011 Reorder the terms: 0.381966011 + z2 = -0.381966011 + 0.381966011 Combine like terms: -0.381966011 + 0.381966011 = 0.000000000 0.381966011 + z2 = 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

z2 + 1.5 = -1.118033989 Simplifying z2 + 1.5 = -1.118033989 Reorder the terms: 1.5 + z2 = -1.118033989 Solving 1.5 + z2 = -1.118033989 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 = -1.118033989 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + z2 = -1.118033989 + -1.5 z2 = -1.118033989 + -1.5 Combine like terms: -1.118033989 + -1.5 = -2.618033989 z2 = -2.618033989 Simplifying z2 = -2.618033989 Reorder the terms: 2.618033989 + z2 = -2.618033989 + 2.618033989 Combine like terms: -2.618033989 + 2.618033989 = 0.000000000 2.618033989 + 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.

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