Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

Factor using the perfect square rule.

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Step 1.1.1

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

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

Step 1.1.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 x = 2 \cdot x \cdot 1$

Step 1.1.3

Rewrite the polynomial.

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

Step 1.1.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 1$ .

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

Step 1.2

Simplify the denominator.

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Step 1.2.1

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

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

Step 1.2.2

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 $b = 1$ .

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

Step 2

To write $\frac{x}{{(x - 1)}^{2}}$ as a fraction with a common denominator, multiply by $\frac{x + 1}{x + 1}$ .

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

Step 3

To write $- \frac{x}{(x + 1) (x - 1)}$ as a fraction with a common denominator, multiply by $\frac{x - 1}{x - 1}$ .

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

Step 4

Write each expression with a common denominator of $(x + 1) {(x - 1)}^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

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

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

Step 4.2

Multiply $\frac{x}{(x + 1) (x - 1)}$ by $\frac{x - 1}{x - 1}$ .

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

Step 4.3

Raise $x - 1$ to the power of $1$ .

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

Step 4.4

Raise $x - 1$ to the power of $1$ .

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

Step 4.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 4.6

Add $1$ and $1$ .

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

Step 4.7

Reorder the factors of $(x + 1) {(x - 1)}^{2}$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

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

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Step 6.1.1

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

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

Step 6.1.2

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

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

Step 6.2

Apply the distributive property.

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

Step 6.3

Multiply $- 1$ by $- 1$ .

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

Step 6.4

Subtract $x$ from $x$ .

$\frac{x (0 + 1 + 1)}{{(x - 1)}^{2} (x + 1)}$

Step 6.5

Add $0$ and $1$ .

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

Step 6.6

Add $1$ and $1$ .

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

Step 7

Move $2$ to the left of $x$ .

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