Algebra Examples

$\frac{3 a^{2} + 6 a + 3}{a^{2} - 3 a - 10} \div \frac{12 a^{2} - 12}{a^{2} - 4}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{3 a^{2} + 6 a + 3}{a^{2} - 3 a - 10} \cdot \frac{a^{2} - 4}{12 a^{2} - 12}$

Step 2

Simplify the numerator.

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Step 2.1

Factor $3$ out of $3 a^{2} + 6 a + 3$ .

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Step 2.1.1

Factor $3$ out of $3 a^{2}$ .

$\frac{3 (a^{2}) + 6 a + 3}{a^{2} - 3 a - 10} \cdot \frac{a^{2} - 4}{12 a^{2} - 12}$

Step 2.1.2

Factor $3$ out of $6 a$ .

$\frac{3 (a^{2}) + 3 (2 a) + 3}{a^{2} - 3 a - 10} \cdot \frac{a^{2} - 4}{12 a^{2} - 12}$

Step 2.1.3

Factor $3$ out of $3$ .

$\frac{3 (a^{2}) + 3 (2 a) + 3 (1)}{a^{2} - 3 a - 10} \cdot \frac{a^{2} - 4}{12 a^{2} - 12}$

Step 2.1.4

Factor $3$ out of $3 (a^{2}) + 3 (2 a)$ .

$\frac{3 (a^{2} + 2 a) + 3 (1)}{a^{2} - 3 a - 10} \cdot \frac{a^{2} - 4}{12 a^{2} - 12}$

Step 2.1.5

Factor $3$ out of $3 (a^{2} + 2 a) + 3 (1)$ .

$\frac{3 (a^{2} + 2 a + 1)}{a^{2} - 3 a - 10} \cdot \frac{a^{2} - 4}{12 a^{2} - 12}$

Step 2.2

Factor using the perfect square rule.

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Step 2.2.1

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

$\frac{3 (a^{2} + 2 a + 1^{2})}{a^{2} - 3 a - 10} \cdot \frac{a^{2} - 4}{12 a^{2} - 12}$

Step 2.2.2

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

$2 a = 2 \cdot a \cdot 1$

Step 2.2.3

Rewrite the polynomial.

$\frac{3 (a^{2} + 2 \cdot a \cdot 1 + 1^{2})}{a^{2} - 3 a - 10} \cdot \frac{a^{2} - 4}{12 a^{2} - 12}$

Step 2.2.4

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

$\frac{3 {(a + 1)}^{2}}{a^{2} - 3 a - 10} \cdot \frac{a^{2} - 4}{12 a^{2} - 12}$

Step 3

Factor $a^{2} - 3 a - 10$ using the AC method.

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Step 3.1

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 $- 10$ and whose sum is $- 3$ .

$- 5, 2$

Step 3.2

Write the factored form using these integers.

$\frac{3 {(a + 1)}^{2}}{(a - 5) (a + 2)} \cdot \frac{a^{2} - 4}{12 a^{2} - 12}$

Step 4

Simplify the numerator.

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Step 4.1

Rewrite $4$ as $2^{2}$ .

$\frac{3 {(a + 1)}^{2}}{(a - 5) (a + 2)} \cdot \frac{a^{2} - 2^{2}}{12 a^{2} - 12}$

Step 4.2

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 $b = 2$ .

$\frac{3 {(a + 1)}^{2}}{(a - 5) (a + 2)} \cdot \frac{(a + 2) (a - 2)}{12 a^{2} - 12}$

Step 5

Simplify the denominator.

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Step 5.1

Factor $12$ out of $12 a^{2} - 12$ .

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Step 5.1.1

Factor $12$ out of $12 a^{2}$ .

$\frac{3 {(a + 1)}^{2}}{(a - 5) (a + 2)} \cdot \frac{(a + 2) (a - 2)}{12 (a^{2}) - 12}$

Step 5.1.2

Factor $12$ out of $- 12$ .

$\frac{3 {(a + 1)}^{2}}{(a - 5) (a + 2)} \cdot \frac{(a + 2) (a - 2)}{12 (a^{2}) + 12 (- 1)}$

Step 5.1.3

Factor $12$ out of $12 (a^{2}) + 12 (- 1)$ .

$\frac{3 {(a + 1)}^{2}}{(a - 5) (a + 2)} \cdot \frac{(a + 2) (a - 2)}{12 (a^{2} - 1)}$

Step 5.2

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

$\frac{3 {(a + 1)}^{2}}{(a - 5) (a + 2)} \cdot \frac{(a + 2) (a - 2)}{12 (a^{2} - 1^{2})}$

Step 5.3

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 $b = 1$ .

$\frac{3 {(a + 1)}^{2}}{(a - 5) (a + 2)} \cdot \frac{(a + 2) (a - 2)}{12 (a + 1) (a - 1)}$

Step 6

Simplify terms.

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Step 6.1

Combine.

$\frac{3 {(a + 1)}^{2} ((a + 2) (a - 2))}{(a - 5) (a + 2) (12 (a + 1) (a - 1))}$

Step 6.2

Cancel the common factor of $3$ and $12$ .

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Step 6.2.1

Factor $3$ out of $3 {(a + 1)}^{2} ((a + 2) (a - 2))$ .

$\frac{3 ({(a + 1)}^{2} ((a + 2) (a - 2)))}{(a - 5) (a + 2) (12 (a + 1) (a - 1))}$

Step 6.2.2

Cancel the common factors.

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Step 6.2.2.1

Factor $3$ out of $(a - 5) (a + 2) (12 (a + 1) (a - 1))$ .

$\frac{3 ({(a + 1)}^{2} ((a + 2) (a - 2)))}{3 ((a - 5) (a + 2) (4 (a + 1) (a - 1)))}$

Step 6.2.2.2

Cancel the common factor.

$\frac{3 ({(a + 1)}^{2} ((a + 2) (a - 2)))}{3 ((a - 5) (a + 2) (4 (a + 1) (a - 1)))}$

Step 6.2.2.3

Rewrite the expression.

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

Step 6.3

Cancel the common factor of ${(a + 1)}^{2}$ and $a + 1$ .

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Step 6.3.1

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

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

Step 6.3.2

Cancel the common factors.

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Step 6.3.2.1

Factor $a + 1$ out of $(a - 5) (a + 2) (4 (a + 1) (a - 1))$ .

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

Step 6.3.2.2

Cancel the common factor.

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

Step 6.3.2.3

Rewrite the expression.

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

Step 6.4

Cancel the common factor of $a + 2$ .

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Step 6.4.1

Cancel the common factor.

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

Step 6.4.2

Rewrite the expression.

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

Step 6.5

Move $4$ to the left of $a - 5$ .

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