Basic Math Examples

$\frac{1}{a^{2} - 1} - \frac{a - 1}{a^{2} + 3 a - 4}$

Step 1

Factor $a^{2} + 3 a - 4$ using the AC method.

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Step 1.1

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 $- 4$ and whose sum is $3$ .

$- 1, 4$

Step 1.2

Write the factored form using these integers.

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

Step 2

Cancel the common factor of $a - 1$ .

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Step 2.1

Cancel the common factor.

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

Step 2.2

Rewrite the expression.

$\frac{1}{a^{2} - 1} - \frac{1}{a + 4}$

Step 3

To write $\frac{1}{a^{2} - 1}$ as a fraction with a common denominator, multiply by $\frac{a + 4}{a + 4}$ .

$\frac{1}{a^{2} - 1} \cdot \frac{a + 4}{a + 4} - \frac{1}{a + 4}$

Step 4

To write $- \frac{1}{a + 4}$ as a fraction with a common denominator, multiply by $\frac{a^{2} - 1}{a^{2} - 1}$ .

$\frac{1}{a^{2} - 1} \cdot \frac{a + 4}{a + 4} - \frac{1}{a + 4} \cdot \frac{a^{2} - 1}{a^{2} - 1}$

Step 5

Write each expression with a common denominator of $(a^{2} - 1) (a + 4)$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.1

Multiply $\frac{1}{a^{2} - 1}$ by $\frac{a + 4}{a + 4}$ .

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

Step 5.2

Multiply $\frac{1}{a + 4}$ by $\frac{a^{2} - 1}{a^{2} - 1}$ .

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

Step 5.3

Reorder the factors of $(a^{2} - 1) (a + 4)$ .

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

Step 6

Combine the numerators over the common denominator.

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

Step 7

Rewrite $\frac{a + 4 - (a^{2} - 1)}{(a + 4) (a^{2} - 1)}$ in a factored form.

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Step 7.1

Apply the distributive property.

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

Step 7.2

Multiply $- 1$ by $- 1$ .

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

Step 7.3

Add $4$ and $1$ .

$\frac{a - a^{2} + 5}{(a + 4) (a^{2} - 1)}$

Step 7.4

Reorder terms.

$\frac{- a^{2} + a + 5}{(a + 4) (a^{2} - 1)}$

Step 7.5

Factor $- 1$ out of $- a^{2} + a + 5$ .

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Step 7.5.1

Factor $- 1$ out of $- a^{2}$ .

$\frac{- (a^{2}) + a + 5}{(a + 4) (a^{2} - 1)}$

Step 7.5.2

Factor $- 1$ out of $a$ .

$\frac{- (a^{2}) - 1 (- a) + 5}{(a + 4) (a^{2} - 1)}$

Step 7.5.3

Rewrite $5$ as $- 1 (- 5)$ .

$\frac{- (a^{2}) - 1 (- a) - 1 (- 5)}{(a + 4) (a^{2} - 1)}$

Step 7.5.4

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

$\frac{- (a^{2} - a) - 1 (- 5)}{(a + 4) (a^{2} - 1)}$

Step 7.5.5

Factor $- 1$ out of $- (a^{2} - a) - 1 (- 5)$ .

$\frac{- (a^{2} - a - 5)}{(a + 4) (a^{2} - 1)}$

Step 7.6

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

$\frac{- (a^{2} - a - 5)}{(a + 4) (a^{2} - 1^{2})}$

Step 7.7

Factor.

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Step 7.7.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = a$ and $b = 1$ .

$\frac{- (a^{2} - a - 5)}{(a + 4) ((a + 1) (a - 1))}$

Step 7.7.2

Remove unnecessary parentheses.

$\frac{- (a^{2} - a - 5)}{(a + 4) (a + 1) (a - 1)}$