0=x^3-x

Solution for 0=x^3-x equation:

Simplifying
0 = x3 + -1x

Reorder the terms:
0 = -1x + x3

Solving
0 = -1x + x3

Solving for variable 'x'.
Remove the zero:
x + -1x3 = -1x + x3 + x + -1x3

Reorder the terms:
x + -1x3 = -1x + x + x3 + -1x3

Combine like terms: -1x + x = 0
x + -1x3 = 0 + x3 + -1x3
x + -1x3 = x3 + -1x3

Combine like terms: x3 + -1x3 = 0
x + -1x3 = 0

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

Factor a difference between two squares.
x((1 + x)(1 + -1x)) = 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 + x)' equal to zero and attempt to solve:

Simplifying
1 + x = 0

Solving
1 + x = 0

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

Add '-1' to each side of the equation.
1 + -1 + x = 0 + -1

Combine like terms: 1 + -1 = 0
0 + x = 0 + -1
x = 0 + -1

Combine like terms: 0 + -1 = -1
x = -1

Simplifying
x = -1
Subproblem 3
Set the factor '(1 + -1x)' equal to zero and attempt to solve:

Simplifying
1 + -1x = 0

Solving
1 + -1x = 0

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

Add '-1' to each side of the equation.
1 + -1 + -1x = 0 + -1

Combine like terms: 1 + -1 = 0
0 + -1x = 0 + -1
-1x = 0 + -1

Combine like terms: 0 + -1 = -1
-1x = -1

Divide each side by '-1'.
x = 1

Simplifying
x = 1
Solution
x = {0, -1, 1}