Simplify (8/(9-a^2))÷((4a^2-4a-24)/(a^2-6a+9))

$\frac{8}{9 - a^{2}} \div \frac{4 a^{2} - 4 a - 24}{a^{2} - 6 a + 9}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{8}{9 - a^{2}} \cdot \frac{a^{2} - 6 a + 9}{4 a^{2} - 4 a - 24}$

Step 2

Simplify the denominator.

Rewrite $9$ as $3^{2}$ .

$\frac{8}{3^{2} - a^{2}} \cdot \frac{a^{2} - 6 a + 9}{4 a^{2} - 4 a - 24}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 3$ and $b = a$ .

$\frac{8}{(3 + a) (3 - a)} \cdot \frac{a^{2} - 6 a + 9}{4 a^{2} - 4 a - 24}$

Step 3

Factor using the perfect square rule.

Rewrite $9$ as $3^{2}$ .

$\frac{8}{(3 + a) (3 - a)} \cdot \frac{a^{2} - 6 a + 3^{2}}{4 a^{2} - 4 a - 24}$

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

$6 a = 2 \cdot a \cdot 3$

Rewrite the polynomial.

$\frac{8}{(3 + a) (3 - a)} \cdot \frac{a^{2} - 2 \cdot a \cdot 3 + 3^{2}}{4 a^{2} - 4 a - 24}$

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = a$ and $b = 3$ .

$\frac{8}{(3 + a) (3 - a)} \cdot \frac{{(a - 3)}^{2}}{4 a^{2} - 4 a - 24}$

Step 4

Simplify the denominator.

Factor $4$ out of $4 a^{2} - 4 a - 24$ .

Factor $4$ out of $4 a^{2}$ .

$\frac{8}{(3 + a) (3 - a)} \cdot \frac{{(a - 3)}^{2}}{4 (a^{2}) - 4 a - 24}$

Factor $4$ out of $- 4 a$ .

$\frac{8}{(3 + a) (3 - a)} \cdot \frac{{(a - 3)}^{2}}{4 (a^{2}) + 4 (- a) - 24}$

Factor $4$ out of $- 24$ .

$\frac{8}{(3 + a) (3 - a)} \cdot \frac{{(a - 3)}^{2}}{4 (a^{2}) + 4 (- a) + 4 (- 6)}$

Factor $4$ out of $4 (a^{2}) + 4 (- a)$ .

$\frac{8}{(3 + a) (3 - a)} \cdot \frac{{(a - 3)}^{2}}{4 (a^{2} - a) + 4 (- 6)}$

Factor $4$ out of $4 (a^{2} - a) + 4 (- 6)$ .

$\frac{8}{(3 + a) (3 - a)} \cdot \frac{{(a - 3)}^{2}}{4 (a^{2} - a - 6)}$

Factor $a^{2} - a - 6$ using the AC method.

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 $- 6$ and whose sum is $- 1$ .

$- 3, 2$

Write the factored form using these integers.

$\frac{8}{(3 + a) (3 - a)} \cdot \frac{{(a - 3)}^{2}}{4 ((a - 3) (a + 2))}$

$\frac{8}{(3 + a) (3 - a)} \cdot \frac{{(a - 3)}^{2}}{4 (a - 3) (a + 2)}$

Step 5

Simplify terms.

Combine.

$\frac{8 {(a - 3)}^{2}}{(3 + a) (3 - a) (4 (a - 3) (a + 2))}$

Cancel the common factor of $8$ and $4$ .

Factor $4$ out of $8 {(a - 3)}^{2}$ .

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

Cancel the common factors.

Factor $4$ out of $(3 + a) (3 - a) (4 (a - 3) (a + 2))$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of ${(a - 3)}^{2}$ and $a - 3$ .

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

$\frac{(a - 3) (2 (a - 3))}{(3 + a) (3 - a) ((a - 3) (a + 2))}$

Cancel the common factors.

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

$\frac{(a - 3) (2 (a - 3))}{(a - 3) ((3 + a) (3 - a) (a + 2))}$

Cancel the common factor.

$\frac{(a - 3) (2 (a - 3))}{(a - 3) ((3 + a) (3 - a) (a + 2))}$

Rewrite the expression.

$\frac{2 (a - 3)}{(3 + a) (3 - a) (a + 2)}$

Cancel the common factor of $a - 3$ and $3 - a$ .

Factor $- 1$ out of $a$ .

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

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

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

Factor $- 1$ out of $- 1 (- a) - 1 (3)$ .

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

Reorder terms.

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 6

Multiply $2$ by $- 1$ .

$\frac{- 2}{(3 + a) (a + 2)}$

Step 7

Move the negative in front of the fraction.

$- \frac{2}{(3 + a) (a + 2)}$

Simplify (8/(9-a^2))÷((4a^2-4a-24)/(a^2-6a+9))

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