Algebra Examples

$\frac{2 x^{2} - 2 x}{4 x^{4} - 4 x^{2}}$

Step 1

Factor $2 x$ out of $2 x^{2} - 2 x$ .

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Step 1.1

Factor $2 x$ out of $2 x^{2}$ .

$\frac{2 x (x) - 2 x}{4 x^{4} - 4 x^{2}}$

Step 1.2

Factor $2 x$ out of $- 2 x$ .

$\frac{2 x (x) + 2 x (- 1)}{4 x^{4} - 4 x^{2}}$

Step 1.3

Factor $2 x$ out of $2 x (x) + 2 x (- 1)$ .

$\frac{2 x (x - 1)}{4 x^{4} - 4 x^{2}}$

Step 2

Simplify the denominator.

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Step 2.1

Factor $4 x^{2}$ out of $4 x^{4} - 4 x^{2}$ .

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Step 2.1.1

Factor $4 x^{2}$ out of $4 x^{4}$ .

$\frac{2 x (x - 1)}{4 x^{2} (x^{2}) - 4 x^{2}}$

Step 2.1.2

Factor $4 x^{2}$ out of $- 4 x^{2}$ .

$\frac{2 x (x - 1)}{4 x^{2} (x^{2}) + 4 x^{2} (- 1)}$

Step 2.1.3

Factor $4 x^{2}$ out of $4 x^{2} (x^{2}) + 4 x^{2} (- 1)$ .

$\frac{2 x (x - 1)}{4 x^{2} (x^{2} - 1)}$

Step 2.2

Rewrite $1$ as $1^{2}$ .

$\frac{2 x (x - 1)}{4 x^{2} (x^{2} - 1^{2})}$

Step 2.3

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 $b = 1$ .

$\frac{2 x (x - 1)}{4 x^{2} (x + 1) (x - 1)}$

Step 3

Cancel the common factor of $2$ and $4$ .

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Step 3.1

Factor $2$ out of $2 x (x - 1)$ .

$\frac{2 (x (x - 1))}{4 x^{2} (x + 1) (x - 1)}$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $2$ out of $4 x^{2} (x + 1) (x - 1)$ .

$\frac{2 (x (x - 1))}{2 (2 x^{2} (x + 1) (x - 1))}$

Step 3.2.2

Cancel the common factor.

$\frac{2 (x (x - 1))}{2 (2 x^{2} (x + 1) (x - 1))}$

Step 3.2.3

Rewrite the expression.

$\frac{x (x - 1)}{2 x^{2} (x + 1) (x - 1)}$

Step 4

Cancel the common factors.

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Step 4.1

Factor $x$ out of $2 x^{2} (x + 1) (x - 1)$ .

$\frac{x (x - 1)}{x (2 x (x + 1) (x - 1))}$

Step 4.2

Cancel the common factor.

$\frac{x (x - 1)}{x (2 x (x + 1) (x - 1))}$

Step 4.3

Rewrite the expression.

$\frac{x - 1}{2 x (x + 1) (x - 1)}$

Step 5

Cancel the common factor of $x - 1$ .

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Step 5.1

Cancel the common factor.

$\frac{x - 1}{2 x (x + 1) (x - 1)}$

Step 5.2

Rewrite the expression.

$\frac{1}{2 x (x + 1)}$