0=2x^6+5x^4-x^2

Solution for 0=2x^6+5x^4-x^2 equation:

Simplifying
0 = 2x6 + 5x4 + -1x2

Reorder the terms:
0 = -1x2 + 5x4 + 2x6

Solving
0 = -1x2 + 5x4 + 2x6

Solving for variable 'x'.
Remove the zero:
x2 + -5x4 + -2x6 = -1x2 + 5x4 + 2x6 + x2 + -5x4 + -2x6

Reorder the terms:
x2 + -5x4 + -2x6 = -1x2 + x2 + 5x4 + -5x4 + 2x6 + -2x6

Combine like terms: -1x2 + x2 = 0
x2 + -5x4 + -2x6 = 0 + 5x4 + -5x4 + 2x6 + -2x6
x2 + -5x4 + -2x6 = 5x4 + -5x4 + 2x6 + -2x6

Combine like terms: 5x4 + -5x4 = 0
x2 + -5x4 + -2x6 = 0 + 2x6 + -2x6
x2 + -5x4 + -2x6 = 2x6 + -2x6

Combine like terms: 2x6 + -2x6 = 0
x2 + -5x4 + -2x6 = 0

Factor out the Greatest Common Factor (GCF), 'x2'.
x2(1 + -5x2 + -2x4) = 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 + -5x2 + -2x4)' equal to zero and attempt to solve:

Simplifying
1 + -5x2 + -2x4 = 0

Solving
1 + -5x2 + -2x4 = 0

Begin completing the square.  Divide all terms by
-2 the coefficient of the squared term: 

Divide each side by '-2'.
-0.5 + 2.5x2 + x4 = 0

Move the constant term to the right:

Add '0.5' to each side of the equation.
-0.5 + 2.5x2 + 0.5 + x4 = 0 + 0.5

Reorder the terms:
-0.5 + 0.5 + 2.5x2 + x4 = 0 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + 2.5x2 + x4 = 0 + 0.5
2.5x2 + x4 = 0 + 0.5

Combine like terms: 0 + 0.5 = 0.5
2.5x2 + x4 = 0.5

The x term is 2.5x2.  Take half its coefficient (1.25).
Square it (1.5625) and add it to both sides.

Add '1.5625' to each side of the equation.
2.5x2 + 1.5625 + x4 = 0.5 + 1.5625

Reorder the terms:
1.5625 + 2.5x2 + x4 = 0.5 + 1.5625

Combine like terms: 0.5 + 1.5625 = 2.0625
1.5625 + 2.5x2 + x4 = 2.0625

Factor a perfect square on the left side:
(x2 + 1.25)(x2 + 1.25) = 2.0625

Calculate the square root of the right side: 1.436140662

Break this problem into two subproblems by setting 
(x2 + 1.25) equal to 1.436140662 and -1.436140662.

Subproblem 1
x2 + 1.25 = 1.436140662

Simplifying
x2 + 1.25 = 1.436140662

Reorder the terms:
1.25 + x2 = 1.436140662

Solving
1.25 + x2 = 1.436140662

Solving for variable 'x'.

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

Add '-1.25' to each side of the equation.
1.25 + -1.25 + x2 = 1.436140662 + -1.25

Combine like terms: 1.25 + -1.25 = 0.00
0.00 + x2 = 1.436140662 + -1.25
x2 = 1.436140662 + -1.25

Combine like terms: 1.436140662 + -1.25 = 0.186140662
x2 = 0.186140662

Simplifying
x2 = 0.186140662

Take the square root of each side:
x = {-0.431440218, 0.431440218}

Subproblem 2
x2 + 1.25 = -1.436140662

Simplifying
x2 + 1.25 = -1.436140662

Reorder the terms:
1.25 + x2 = -1.436140662

Solving
1.25 + x2 = -1.436140662

Solving for variable 'x'.

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

Add '-1.25' to each side of the equation.
1.25 + -1.25 + x2 = -1.436140662 + -1.25

Combine like terms: 1.25 + -1.25 = 0.00
0.00 + x2 = -1.436140662 + -1.25
x2 = -1.436140662 + -1.25

Combine like terms: -1.436140662 + -1.25 = -2.686140662
x2 = -2.686140662

Simplifying
x2 = -2.686140662

Reorder the terms:
2.686140662 + x2 = -2.686140662 + 2.686140662

Combine like terms: -2.686140662 + 2.686140662 = 0.000000000
2.686140662 + 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}