3x^6+x^4-x^2=0

Solution for 3x^6+x^4-x^2=0 equation:

Simplifying
3x6 + x4 + -1x2 = 0

Reorder the terms:
-1x2 + x4 + 3x6 = 0

Solving
-1x2 + x4 + 3x6 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), 'x2'.
x2(-1 + x2 + 3x4) = 0

Subproblem 1
Set the factor 'x2' equal to zero and attempt to solve:

Simplifying
x2 = 0

Solving
x2 = 0

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

Simplifying
x2 = 0

Take the square root of each side:
x = {0}
Subproblem 2
Set the factor '(-1 + x2 + 3x4)' equal to zero and attempt to solve:

Simplifying
-1 + x2 + 3x4 = 0

Solving
-1 + x2 + 3x4 = 0

Begin completing the square.  Divide all terms by
3 the coefficient of the squared term: 

Divide each side by '3'.
-0.3333333333 + 0.3333333333x2 + x4 = 0

Move the constant term to the right:

Add '0.3333333333' to each side of the equation.
-0.3333333333 + 0.3333333333x2 + 0.3333333333 + x4 = 0 + 0.3333333333

Reorder the terms:
-0.3333333333 + 0.3333333333 + 0.3333333333x2 + x4 = 0 + 0.3333333333

Combine like terms: -0.3333333333 + 0.3333333333 = 0.0000000000
0.0000000000 + 0.3333333333x2 + x4 = 0 + 0.3333333333
0.3333333333x2 + x4 = 0 + 0.3333333333

Combine like terms: 0 + 0.3333333333 = 0.3333333333
0.3333333333x2 + x4 = 0.3333333333

The x term is x2.  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.
0.3333333333x2 + 0.25 + x4 = 0.3333333333 + 0.25

Reorder the terms:
0.25 + 0.3333333333x2 + x4 = 0.3333333333 + 0.25

Combine like terms: 0.3333333333 + 0.25 = 0.5833333333
0.25 + 0.3333333333x2 + x4 = 0.5833333333

Factor a perfect square on the left side:
(x2 + 0.5)(x2 + 0.5) = 0.5833333333

Calculate the square root of the right side: 0.763762616

Break this problem into two subproblems by setting 
(x2 + 0.5) equal to 0.763762616 and -0.763762616.

Subproblem 1
x2 + 0.5 = 0.763762616

Simplifying
x2 + 0.5 = 0.763762616

Reorder the terms:
0.5 + x2 = 0.763762616

Solving
0.5 + x2 = 0.763762616

Solving for variable 'x'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + x2 = 0.763762616 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x2 = 0.763762616 + -0.5
x2 = 0.763762616 + -0.5

Combine like terms: 0.763762616 + -0.5 = 0.263762616
x2 = 0.263762616

Simplifying
x2 = 0.263762616

Take the square root of each side:
x = {-0.513578247, 0.513578247}

Subproblem 2
x2 + 0.5 = -0.763762616

Simplifying
x2 + 0.5 = -0.763762616

Reorder the terms:
0.5 + x2 = -0.763762616

Solving
0.5 + x2 = -0.763762616

Solving for variable 'x'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + x2 = -0.763762616 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x2 = -0.763762616 + -0.5
x2 = -0.763762616 + -0.5

Combine like terms: -0.763762616 + -0.5 = -1.263762616
x2 = -1.263762616

Simplifying
x2 = -1.263762616

Reorder the terms:
1.263762616 + x2 = -1.263762616 + 1.263762616

Combine like terms: -1.263762616 + 1.263762616 = 0.000000000
1.263762616 + x2 = 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.

This subproblem is being ignored because a solution could not be determined.
Solution
x = {0}