Basic Math Examples

$\frac{1}{a^{2} - 25} - \frac{a - 5}{a^{2} - 4 a - 5}$

Step 1

Factor $a^{2} - 4 a - 5$ 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 $- 5$ and whose sum is $- 4$ .

$- 5, 1$

Step 1.2

Write the factored form using these integers.

$\frac{1}{a^{2} - 25} - \frac{a - 5}{(a - 5) (a + 1)}$

Step 2

Cancel the common factor of $a - 5$ .

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Step 2.1

Cancel the common factor.

$\frac{1}{a^{2} - 25} - \frac{a - 5}{(a - 5) (a + 1)}$

Step 2.2

Rewrite the expression.

$\frac{1}{a^{2} - 25} - \frac{1}{a + 1}$

Step 3

To write $\frac{1}{a^{2} - 25}$ as a fraction with a common denominator, multiply by $\frac{a + 1}{a + 1}$ .

$\frac{1}{a^{2} - 25} \cdot \frac{a + 1}{a + 1} - \frac{1}{a + 1}$

Step 4

To write $- \frac{1}{a + 1}$ as a fraction with a common denominator, multiply by $\frac{a^{2} - 25}{a^{2} - 25}$ .

$\frac{1}{a^{2} - 25} \cdot \frac{a + 1}{a + 1} - \frac{1}{a + 1} \cdot \frac{a^{2} - 25}{a^{2} - 25}$

Step 5

Write each expression with a common denominator of $(a^{2} - 25) (a + 1)$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.1

Multiply $\frac{1}{a^{2} - 25}$ by $\frac{a + 1}{a + 1}$ .

$\frac{a + 1}{(a^{2} - 25) (a + 1)} - \frac{1}{a + 1} \cdot \frac{a^{2} - 25}{a^{2} - 25}$

Step 5.2

Multiply $\frac{1}{a + 1}$ by $\frac{a^{2} - 25}{a^{2} - 25}$ .

$\frac{a + 1}{(a^{2} - 25) (a + 1)} - \frac{a^{2} - 25}{(a + 1) (a^{2} - 25)}$

Step 5.3

Reorder the factors of $(a^{2} - 25) (a + 1)$ .

$\frac{a + 1}{(a + 1) (a^{2} - 25)} - \frac{a^{2} - 25}{(a + 1) (a^{2} - 25)}$

Step 6

Combine the numerators over the common denominator.

$\frac{a + 1 - (a^{2} - 25)}{(a + 1) (a^{2} - 25)}$

Step 7

Rewrite $\frac{a + 1 - (a^{2} - 25)}{(a + 1) (a^{2} - 25)}$ in a factored form.

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Step 7.1

Apply the distributive property.

$\frac{a + 1 - a^{2} - - 25}{(a + 1) (a^{2} - 25)}$

Step 7.2

Multiply $- 1$ by $- 25$ .

$\frac{a + 1 - a^{2} + 25}{(a + 1) (a^{2} - 25)}$

Step 7.3

Add $1$ and $25$ .

$\frac{a - a^{2} + 26}{(a + 1) (a^{2} - 25)}$

Step 7.4

Reorder terms.

$\frac{- a^{2} + a + 26}{(a + 1) (a^{2} - 25)}$

Step 7.5

Factor $- 1$ out of $- a^{2} + a + 26$ .

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Step 7.5.1

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

$\frac{- (a^{2}) + a + 26}{(a + 1) (a^{2} - 25)}$

Step 7.5.2

Factor $- 1$ out of $a$ .

$\frac{- (a^{2}) - 1 (- a) + 26}{(a + 1) (a^{2} - 25)}$

Step 7.5.3

Rewrite $26$ as $- 1 (- 26)$ .

$\frac{- (a^{2}) - 1 (- a) - 1 (- 26)}{(a + 1) (a^{2} - 25)}$

Step 7.5.4

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

$\frac{- (a^{2} - a) - 1 (- 26)}{(a + 1) (a^{2} - 25)}$

Step 7.5.5

Factor $- 1$ out of $- (a^{2} - a) - 1 (- 26)$ .

$\frac{- (a^{2} - a - 26)}{(a + 1) (a^{2} - 25)}$

Step 7.6

Rewrite $25$ as $5^{2}$ .

$\frac{- (a^{2} - a - 26)}{(a + 1) (a^{2} - 5^{2})}$

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 $b = 5$ .

$\frac{- (a^{2} - a - 26)}{(a + 1) ((a + 5) (a - 5))}$

Step 7.7.2

Remove unnecessary parentheses.

$\frac{- (a^{2} - a - 26)}{(a + 1) (a + 5) (a - 5)}$