Algebra Examples

$\frac{7 - x}{7 - x} \cdot \frac{x^{2} - 1}{64 - x^{2}} \div \frac{x^{2} + 3 x - 4}{x^{2} + 12 x + 32}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{7 - x}{7 - x} \cdot \frac{x^{2} - 1}{64 - x^{2}} \frac{x^{2} + 12 x + 32}{x^{2} + 3 x - 4}$

Step 2

Reduce the expression by cancelling the common factors.

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Step 2.1

Cancel the common factor of $7 - x$ .

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Step 2.1.1

Cancel the common factor.

$\frac{7 - x}{7 - x} \cdot \frac{x^{2} - 1}{64 - x^{2}} \frac{x^{2} + 12 x + 32}{x^{2} + 3 x - 4}$

Step 2.1.2

Rewrite the expression.

$1 \cdot \frac{x^{2} - 1}{64 - x^{2}} \frac{x^{2} + 12 x + 32}{x^{2} + 3 x - 4}$

Step 2.2

Multiply $\frac{x^{2} - 1}{64 - x^{2}}$ by $1$ .

$\frac{x^{2} - 1}{64 - x^{2}} \cdot \frac{x^{2} + 12 x + 32}{x^{2} + 3 x - 4}$

Step 3

Simplify the numerator.

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Step 3.1

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

$\frac{x^{2} - 1^{2}}{64 - x^{2}} \cdot \frac{x^{2} + 12 x + 32}{x^{2} + 3 x - 4}$

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 $a = x$ and $b = 1$ .

$\frac{(x + 1) (x - 1)}{64 - x^{2}} \cdot \frac{x^{2} + 12 x + 32}{x^{2} + 3 x - 4}$

Step 4

Simplify the denominator.

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Step 4.1

Rewrite $64$ as $8^{2}$ .

$\frac{(x + 1) (x - 1)}{8^{2} - x^{2}} \cdot \frac{x^{2} + 12 x + 32}{x^{2} + 3 x - 4}$

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 $a = 8$ and $b = x$ .

$\frac{(x + 1) (x - 1)}{(8 + x) (8 - x)} \cdot \frac{x^{2} + 12 x + 32}{x^{2} + 3 x - 4}$

Step 5

Factor $x^{2} + 12 x + 32$ 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 $12$ .

$4, 8$

Step 5.2

Write the factored form using these integers.

$\frac{(x + 1) (x - 1)}{(8 + x) (8 - x)} \cdot \frac{(x + 4) (x + 8)}{x^{2} + 3 x - 4}$

Step 6

Factor $x^{2} + 3 x - 4$ 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 $- 4$ and whose sum is $3$ .

$- 1, 4$

Step 6.2

Write the factored form using these integers.

$\frac{(x + 1) (x - 1)}{(8 + x) (8 - x)} \cdot \frac{(x + 4) (x + 8)}{(x - 1) (x + 4)}$

Step 7

Simplify terms.

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Step 7.1

Cancel the common factor of $x - 1$ .

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Step 7.1.1

Factor $x - 1$ out of $(x + 1) (x - 1)$ .

$\frac{(x - 1) (x + 1)}{(8 + x) (8 - x)} \cdot \frac{(x + 4) (x + 8)}{(x - 1) (x + 4)}$

Step 7.1.2

Cancel the common factor.

$\frac{(x - 1) (x + 1)}{(8 + x) (8 - x)} \cdot \frac{(x + 4) (x + 8)}{(x - 1) (x + 4)}$

Step 7.1.3

Rewrite the expression.

$\frac{x + 1}{(8 + x) (8 - x)} \cdot \frac{(x + 4) (x + 8)}{x + 4}$

Step 7.2

Multiply $\frac{x + 1}{(8 + x) (8 - x)}$ by $\frac{(x + 4) (x + 8)}{x + 4}$ .

$\frac{(x + 1) ((x + 4) (x + 8))}{(8 + x) (8 - x) (x + 4)}$

Step 7.3

Cancel the common factor of $x + 4$ .

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Step 7.3.1

Cancel the common factor.

$\frac{(x + 1) ((x + 4) (x + 8))}{(8 + x) (8 - x) (x + 4)}$

Step 7.3.2

Rewrite the expression.

$\frac{(x + 1) (x + 8)}{(8 + x) (8 - x)}$

Step 7.4

Cancel the common factor of $x + 8$ and $8 + x$ .

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Step 7.4.1

Reorder terms.

$\frac{(x + 1) (x + 8)}{(x + 8) (8 - x)}$

Step 7.4.2

Cancel the common factor.

$\frac{(x + 1) (x + 8)}{(x + 8) (8 - x)}$

Step 7.4.3

Rewrite the expression.

$\frac{x + 1}{8 - x}$