Pre-Algebra Examples

$\frac{b^{2} - 49}{b^{2} + 3 b - 28} \div \frac{7 - b}{b^{2} + 4 b - 32}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{b^{2} - 49}{b^{2} + 3 b - 28} \cdot \frac{b^{2} + 4 b - 32}{7 - b}$

Step 2

Simplify the numerator.

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Step 2.1

Rewrite $49$ as $7^{2}$ .

$\frac{b^{2} - 7^{2}}{b^{2} + 3 b - 28} \cdot \frac{b^{2} + 4 b - 32}{7 - b}$

Step 2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = b$ and $b = 7$ .

$\frac{(b + 7) (b - 7)}{b^{2} + 3 b - 28} \cdot \frac{b^{2} + 4 b - 32}{7 - b}$

Step 3

Factor $b^{2} + 3 b - 28$ 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 $- 28$ and whose sum is $3$ .

$- 4, 7$

Step 3.2

Write the factored form using these integers.

$\frac{(b + 7) (b - 7)}{(b - 4) (b + 7)} \cdot \frac{b^{2} + 4 b - 32}{7 - b}$

Step 4

Cancel the common factor of $b + 7$ .

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Step 4.1

Cancel the common factor.

$\frac{(b + 7) (b - 7)}{(b - 4) (b + 7)} \cdot \frac{b^{2} + 4 b - 32}{7 - b}$

Step 4.2

Rewrite the expression.

$\frac{b - 7}{b - 4} \cdot \frac{b^{2} + 4 b - 32}{7 - b}$

Step 5

Factor $b^{2} + 4 b - 32$ using the AC method.

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Step 5.1

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 $- 32$ and whose sum is $4$ .

$- 4, 8$

Step 5.2

Write the factored form using these integers.

$\frac{b - 7}{b - 4} \cdot \frac{(b - 4) (b + 8)}{7 - b}$

Step 6

Cancel the common factor of $b - 4$ .

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Step 6.1

Cancel the common factor.

$\frac{b - 7}{b - 4} \cdot \frac{(b - 4) (b + 8)}{7 - b}$

Step 6.2

Rewrite the expression.

$(b - 7) \frac{b + 8}{7 - b}$

Step 7

Multiply $b - 7$ by $\frac{b + 8}{7 - b}$ .

$\frac{(b - 7) (b + 8)}{7 - b}$

Step 8

Cancel the common factor of $b - 7$ and $7 - b$ .

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Step 8.1

Factor $- 1$ out of $b$ .

$\frac{(- 1 (- b) - 7) (b + 8)}{7 - b}$

Step 8.2

Rewrite $- 7$ as $- 1 (7)$ .

$\frac{(- 1 (- b) - 1 (7)) (b + 8)}{7 - b}$

Step 8.3

Factor $- 1$ out of $- 1 (- b) - 1 (7)$ .

$\frac{- 1 (- b + 7) (b + 8)}{7 - b}$

Step 8.4

Reorder terms.

$\frac{- 1 (- b + 7) (b + 8)}{- b + 7}$

Step 8.5

Cancel the common factor.

$\frac{- 1 (- b + 7) (b + 8)}{- b + 7}$

Step 8.6

Divide $- 1 (b + 8)$ by $1$ .

$- 1 (b + 8)$

Step 9

Rewrite $- 1 (b + 8)$ as $- (b + 8)$ .

$- (b + 8)$

Step 10

Apply the distributive property.

$- b - 1 \cdot 8$

Step 11

Multiply $- 1$ by $8$ .

$- b - 8$