Algebra Examples

$\frac{x - 5}{x - 7} \div \frac{x^{2} + 2 x - 35}{x^{2} - 49}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{x - 5}{x - 7} \cdot \frac{x^{2} - 49}{x^{2} + 2 x - 35}$

Step 2

Simplify the numerator.

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Step 2.1

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

$\frac{x - 5}{x - 7} \cdot \frac{x^{2} - 7^{2}}{x^{2} + 2 x - 35}$

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 $a = x$ and $b = 7$ .

$\frac{x - 5}{x - 7} \cdot \frac{(x + 7) (x - 7)}{x^{2} + 2 x - 35}$

Step 3

Factor $x^{2} + 2 x - 35$ 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 $- 35$ and whose sum is $2$ .

$- 5, 7$

Step 3.2

Write the factored form using these integers.

$\frac{x - 5}{x - 7} \cdot \frac{(x + 7) (x - 7)}{(x - 5) (x + 7)}$

Step 4

Reduce the expression by cancelling the common factors.

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Step 4.1

Cancel the common factor of $x - 5$ .

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Step 4.1.1

Cancel the common factor.

$\frac{x - 5}{x - 7} \cdot \frac{(x + 7) (x - 7)}{(x - 5) (x + 7)}$

Step 4.1.2

Rewrite the expression.

$\frac{1}{x - 7} \cdot \frac{(x + 7) (x - 7)}{x + 7}$

Step 4.2

Cancel the common factor of $x - 7$ .

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Step 4.2.1

Factor $x - 7$ out of $(x + 7) (x - 7)$ .

$\frac{1}{x - 7} \cdot \frac{(x - 7) (x + 7)}{x + 7}$

Step 4.2.2

Cancel the common factor.

$\frac{1}{x - 7} \cdot \frac{(x - 7) (x + 7)}{x + 7}$

Step 4.2.3

Rewrite the expression.

$\frac{x + 7}{x + 7}$

Step 4.3

Cancel the common factor of $x + 7$ .

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Step 4.3.1

Cancel the common factor.

$\frac{x + 7}{x + 7}$

Step 4.3.2

Rewrite the expression.

$1$