Algebra Examples

$\frac{x}{x^{2} - 1} - \frac{x + 3}{x^{2} - x - 2}$

Step 1

Simplify each term.

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Step 1.1

Simplify the denominator.

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Step 1.1.1

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

$\frac{x}{x^{2} - 1^{2}} - \frac{x + 3}{x^{2} - x - 2}$

Step 1.1.2

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 $b = 1$ .

$\frac{x}{(x + 1) (x - 1)} - \frac{x + 3}{x^{2} - x - 2}$

Step 1.2

Factor $x^{2} - x - 2$ using the AC method.

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Step 1.2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 2$ and whose sum is $- 1$ .

$- 2, 1$

Step 1.2.2

Write the factored form using these integers.

$\frac{x}{(x + 1) (x - 1)} - \frac{x + 3}{(x - 2) (x + 1)}$

Step 2

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

$\frac{x}{(x + 1) (x - 1)} \cdot \frac{x - 2}{x - 2} - \frac{x + 3}{(x - 2) (x + 1)}$

Step 3

To write $- \frac{x + 3}{(x - 2) (x + 1)}$ as a fraction with a common denominator, multiply by $\frac{x - 1}{x - 1}$ .

$\frac{x}{(x + 1) (x - 1)} \cdot \frac{x - 2}{x - 2} - \frac{x + 3}{(x - 2) (x + 1)} \cdot \frac{x - 1}{x - 1}$

Step 4

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

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Step 4.1

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

$\frac{x (x - 2)}{(x + 1) (x - 1) (x - 2)} - \frac{x + 3}{(x - 2) (x + 1)} \cdot \frac{x - 1}{x - 1}$

Step 4.2

Multiply $\frac{x + 3}{(x - 2) (x + 1)}$ by $\frac{x - 1}{x - 1}$ .

$\frac{x (x - 2)}{(x + 1) (x - 1) (x - 2)} - \frac{(x + 3) (x - 1)}{(x - 2) (x + 1) (x - 1)}$

Step 4.3

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

$\frac{x (x - 2)}{(x + 1) (x - 1) (x - 2)} - \frac{(x + 3) (x - 1)}{(x + 1) (x - 1) (x - 2)}$

Step 5

Combine the numerators over the common denominator.

$\frac{x (x - 2) - (x + 3) (x - 1)}{(x + 1) (x - 1) (x - 2)}$

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

$\frac{x \cdot x + x \cdot - 2 - (x + 3) (x - 1)}{(x + 1) (x - 1) (x - 2)}$

Step 6.2

Multiply $x$ by $x$ .

$\frac{x^{2} + x \cdot - 2 - (x + 3) (x - 1)}{(x + 1) (x - 1) (x - 2)}$

Step 6.3

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

$\frac{x^{2} - 2 \cdot x - (x + 3) (x - 1)}{(x + 1) (x - 1) (x - 2)}$

Step 6.4

Apply the distributive property.

$\frac{x^{2} - 2 x + (- x - 1 \cdot 3) (x - 1)}{(x + 1) (x - 1) (x - 2)}$

Step 6.5

Multiply $- 1$ by $3$ .

$\frac{x^{2} - 2 x + (- x - 3) (x - 1)}{(x + 1) (x - 1) (x - 2)}$

Step 6.6

Expand $(- x - 3) (x - 1)$ using the FOIL Method.

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Step 6.6.1

Apply the distributive property.

$\frac{x^{2} - 2 x - x (x - 1) - 3 (x - 1)}{(x + 1) (x - 1) (x - 2)}$

Step 6.6.2

Apply the distributive property.

$\frac{x^{2} - 2 x - x \cdot x - x \cdot - 1 - 3 (x - 1)}{(x + 1) (x - 1) (x - 2)}$

Step 6.6.3

Apply the distributive property.

$\frac{x^{2} - 2 x - x \cdot x - x \cdot - 1 - 3 x - 3 \cdot - 1}{(x + 1) (x - 1) (x - 2)}$

Step 6.7

Simplify and combine like terms.

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Step 6.7.1

Simplify each term.

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Step 6.7.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 6.7.1.1.1

Move $x$ .

$\frac{x^{2} - 2 x - (x \cdot x) - x \cdot - 1 - 3 x - 3 \cdot - 1}{(x + 1) (x - 1) (x - 2)}$

Step 6.7.1.1.2

Multiply $x$ by $x$ .

$\frac{x^{2} - 2 x - x^{2} - x \cdot - 1 - 3 x - 3 \cdot - 1}{(x + 1) (x - 1) (x - 2)}$

Step 6.7.1.2

Multiply $- x \cdot - 1$ .

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Step 6.7.1.2.1

Multiply $- 1$ by $- 1$ .

$\frac{x^{2} - 2 x - x^{2} + 1 x - 3 x - 3 \cdot - 1}{(x + 1) (x - 1) (x - 2)}$

Step 6.7.1.2.2

Multiply $x$ by $1$ .

$\frac{x^{2} - 2 x - x^{2} + x - 3 x - 3 \cdot - 1}{(x + 1) (x - 1) (x - 2)}$

Step 6.7.1.3

Multiply $- 3$ by $- 1$ .

$\frac{x^{2} - 2 x - x^{2} + x - 3 x + 3}{(x + 1) (x - 1) (x - 2)}$

Step 6.7.2

Subtract $3 x$ from $x$ .

$\frac{x^{2} - 2 x - x^{2} - 2 x + 3}{(x + 1) (x - 1) (x - 2)}$

Step 6.8

Subtract $x^{2}$ from $x^{2}$ .

$\frac{- 2 x + 0 - 2 x + 3}{(x + 1) (x - 1) (x - 2)}$

Step 6.9

Add $- 2 x$ and $0$ .

$\frac{- 2 x - 2 x + 3}{(x + 1) (x - 1) (x - 2)}$

Step 6.10

Subtract $2 x$ from $- 2 x$ .

$\frac{- 4 x + 3}{(x + 1) (x - 1) (x - 2)}$

Step 7

Simplify with factoring out.

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Step 7.1

Factor $- 1$ out of $- 4 x$ .

$\frac{- (4 x) + 3}{(x + 1) (x - 1) (x - 2)}$

Step 7.2

Rewrite $3$ as $- 1 (- 3)$ .

$\frac{- (4 x) - 1 (- 3)}{(x + 1) (x - 1) (x - 2)}$

Step 7.3

Factor $- 1$ out of $- (4 x) - 1 (- 3)$ .

$\frac{- (4 x - 3)}{(x + 1) (x - 1) (x - 2)}$

Step 7.4

Simplify the expression.

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Step 7.4.1

Rewrite $- (4 x - 3)$ as $- 1 (4 x - 3)$ .

$\frac{- 1 (4 x - 3)}{(x + 1) (x - 1) (x - 2)}$

Step 7.4.2

Move the negative in front of the fraction.

$- \frac{4 x - 3}{(x + 1) (x - 1) (x - 2)}$