Algebra Examples

x^{3} - x = 0

$x^{3} - x = 0$

Step 1

Factor

$x$ out of

$x^{3} - x$ .

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Step 1.1

Factor

$x$ out of

$x^{3}$ .

$x \cdot x^{2} - x = 0$

Step 1.2

Factor

$x$ out of

$- x$ .

$x \cdot x^{2} + x \cdot - 1 = 0$

Step 1.3

Factor

$x$ out of

$x \cdot x^{2} + x \cdot - 1$ .

$x (x^{2} - 1) = 0$

Step 2

Rewrite

$1$ as

$1^{2}$ .

$x (x^{2} - 1^{2}) = 0$

Step 3

Factor.

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Step 3.1

Since both terms are perfect squares, factor using the difference of squares formula,

$a^{2} - b^{2} = (a + b) (a - b)$ where

$a = x$ and

$b = 1$ .

$x ((x + 1) (x - 1)) = 0$

Step 3.2

Remove unnecessary parentheses.

$x (x + 1) (x - 1) = 0$

Step 4

If any individual factor on the left side of the equation is equal to

$0$ , the entire expression will be equal to

$0$ .

$x = 0$

$x + 1 = 0$

$x - 1 = 0$

Step 5

Set

$x$ equal to

$0$ .

$x = 0$

Step 6

Set

$x + 1$ equal to

$0$ and solve for

$x$ .

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Step 6.1

Set

$x + 1$ equal to

$0$ .

$x + 1 = 0$

Step 6.2

Subtract

$1$ from both sides of the equation.

$x = - 1$

Step 7

Set

$x - 1$ equal to

$0$ and solve for

$x$ .

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Step 7.1

Set

$x - 1$ equal to

$0$ .

$x - 1 = 0$

Step 7.2

Add

$1$ to both sides of the equation.

$x = 1$

Step 8

The final solution is all the values that make

$x (x + 1) (x - 1) = 0$ true.

$x = 0, - 1, 1$

$x^{3} - x = 0$

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