x^5-2x^3-x=0

Solution for x^5-2x^3-x=0 equation:

Simplifying
x5 + -2x3 + -1x = 0

Reorder the terms:
-1x + -2x3 + x5 = 0

Solving
-1x + -2x3 + x5 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), 'x'.
x(-1 + -2x2 + x4) = 0

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

Simplifying
x = 0

Solving
x = 0

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

Simplifying
x = 0
Subproblem 2
Set the factor '(-1 + -2x2 + x4)' equal to zero and attempt to solve:

Simplifying
-1 + -2x2 + x4 = 0

Solving
-1 + -2x2 + x4 = 0

Begin completing the square.

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + -2x2 + 1 + x4 = 0 + 1

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

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

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

The x term is -2x2.  Take half its coefficient (-1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
-2x2 + 1 + x4 = 1 + 1

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

Combine like terms: 1 + 1 = 2
1 + -2x2 + x4 = 2

Factor a perfect square on the left side:
(x2 + -1)(x2 + -1) = 2

Calculate the square root of the right side: 1.414213562

Break this problem into two subproblems by setting 
(x2 + -1) equal to 1.414213562 and -1.414213562.

Subproblem 1
x2 + -1 = 1.414213562

Simplifying
x2 + -1 = 1.414213562

Reorder the terms:
-1 + x2 = 1.414213562

Solving
-1 + x2 = 1.414213562

Solving for variable 'x'.

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

Add '1' to each side of the equation.
-1 + 1 + x2 = 1.414213562 + 1

Combine like terms: -1 + 1 = 0
0 + x2 = 1.414213562 + 1
x2 = 1.414213562 + 1

Combine like terms: 1.414213562 + 1 = 2.414213562
x2 = 2.414213562

Simplifying
x2 = 2.414213562

Take the square root of each side:
x = {-1.553773974, 1.553773974}

Subproblem 2
x2 + -1 = -1.414213562

Simplifying
x2 + -1 = -1.414213562

Reorder the terms:
-1 + x2 = -1.414213562

Solving
-1 + x2 = -1.414213562

Solving for variable 'x'.

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

Add '1' to each side of the equation.
-1 + 1 + x2 = -1.414213562 + 1

Combine like terms: -1 + 1 = 0
0 + x2 = -1.414213562 + 1
x2 = -1.414213562 + 1

Combine like terms: -1.414213562 + 1 = -0.414213562
x2 = -0.414213562

Simplifying
x2 = -0.414213562

Reorder the terms:
0.414213562 + x2 = -0.414213562 + 0.414213562

Combine like terms: -0.414213562 + 0.414213562 = 0.000000000
0.414213562 + 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}