Pre-Algebra Examples

$\frac{a^{2} - a - 20}{a^{2} - 25} \div \frac{a^{2} - 13 a + 36}{45 - 5 a}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{a^{2} - a - 20}{a^{2} - 25} \cdot \frac{45 - 5 a}{a^{2} - 13 a + 36}$

Step 2

Factor $a^{2} - a - 20$ using the AC method.

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Step 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 $- 20$ and whose sum is $- 1$ .

$- 5, 4$

Step 2.2

Write the factored form using these integers.

$\frac{(a - 5) (a + 4)}{a^{2} - 25} \cdot \frac{45 - 5 a}{a^{2} - 13 a + 36}$

Step 3

Simplify the denominator.

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Step 3.1

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

$\frac{(a - 5) (a + 4)}{a^{2} - 5^{2}} \cdot \frac{45 - 5 a}{a^{2} - 13 a + 36}$

Step 3.2

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 $b = 5$ .

$\frac{(a - 5) (a + 4)}{(a + 5) (a - 5)} \cdot \frac{45 - 5 a}{a^{2} - 13 a + 36}$

Step 4

Cancel the common factor of $a - 5$ .

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Step 4.1

Cancel the common factor.

$\frac{(a - 5) (a + 4)}{(a + 5) (a - 5)} \cdot \frac{45 - 5 a}{a^{2} - 13 a + 36}$

Step 4.2

Rewrite the expression.

$\frac{a + 4}{a + 5} \cdot \frac{45 - 5 a}{a^{2} - 13 a + 36}$

Step 5

Factor $5$ out of $45 - 5 a$ .

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Step 5.1

Factor $5$ out of $45$ .

$\frac{a + 4}{a + 5} \cdot \frac{5 (9) - 5 a}{a^{2} - 13 a + 36}$

Step 5.2

Factor $5$ out of $- 5 a$ .

$\frac{a + 4}{a + 5} \cdot \frac{5 (9) + 5 (- a)}{a^{2} - 13 a + 36}$

Step 5.3

Factor $5$ out of $5 (9) + 5 (- a)$ .

$\frac{a + 4}{a + 5} \cdot \frac{5 (9 - a)}{a^{2} - 13 a + 36}$

Step 6

Factor $a^{2} - 13 a + 36$ using the AC method.

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Step 6.1

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 $36$ and whose sum is $- 13$ .

$- 9, - 4$

Step 6.2

Write the factored form using these integers.

$\frac{a + 4}{a + 5} \cdot \frac{5 (9 - a)}{(a - 9) (a - 4)}$

Step 7

Cancel the common factor of $9 - a$ and $a - 9$ .

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Step 7.1

Rewrite $9$ as $- 1 (- 9)$ .

$\frac{a + 4}{a + 5} \cdot \frac{5 (- 1 (- 9) - a)}{(a - 9) (a - 4)}$

Step 7.2

Factor $- 1$ out of $- a$ .

$\frac{a + 4}{a + 5} \cdot \frac{5 (- 1 (- 9) - (a))}{(a - 9) (a - 4)}$

Step 7.3

Factor $- 1$ out of $- 1 (- 9) - (a)$ .

$\frac{a + 4}{a + 5} \cdot \frac{5 (- 1 (- 9 + a))}{(a - 9) (a - 4)}$

Step 7.4

Reorder terms.

$\frac{a + 4}{a + 5} \cdot \frac{5 (- 1 (a - 9))}{(a - 9) (a - 4)}$

Step 7.5

Cancel the common factor.

$\frac{a + 4}{a + 5} \cdot \frac{5 (- 1 (a - 9))}{(a - 9) (a - 4)}$

Step 7.6

Rewrite the expression.

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

Step 8

Multiply $5$ by $- 1$ .

$\frac{a + 4}{a + 5} \cdot \frac{- 5}{a - 4}$

Step 9

Move the negative in front of the fraction.

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

Step 10

Rewrite using the commutative property of multiplication.

$- \frac{a + 4}{a + 5} \cdot \frac{5}{a - 4}$

Step 11

Multiply $\frac{5}{a - 4}$ by $\frac{a + 4}{a + 5}$ .

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

Step 12

Apply the distributive property.

$- \frac{5 a + 5 \cdot 4}{(a - 4) (a + 5)}$

Step 13

Multiply $5$ by $4$ .

$- \frac{5 a + 20}{(a - 4) (a + 5)}$

Step 14

Split the fraction $\frac{5 a + 20}{(a - 4) (a + 5)}$ into two fractions.

$- (\frac{5 a}{(a - 4) (a + 5)} + \frac{20}{(a - 4) (a + 5)})$