Algebra Examples

$\frac{x^{2} - 49}{x^{2} - 4 x - 21} \cdot \frac{x + 3}{x}$

Step 1

Simplify the numerator.

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Step 1.1

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

$\frac{x^{2} - 7^{2}}{x^{2} - 4 x - 21} \cdot \frac{x + 3}{x}$

Step 1.2

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 $b = 7$ .

$\frac{(x + 7) (x - 7)}{x^{2} - 4 x - 21} \cdot \frac{x + 3}{x}$

Step 2

Factor $x^{2} - 4 x - 21$ using the AC method.

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Step 2.1

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 $- 21$ and whose sum is $- 4$ .

$- 7, 3$

Step 2.2

Write the factored form using these integers.

$\frac{(x + 7) (x - 7)}{(x - 7) (x + 3)} \cdot \frac{x + 3}{x}$

Step 3

Simplify terms.

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Step 3.1

Cancel the common factor of $x + 3$ .

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Step 3.1.1

Factor $x + 3$ out of $(x - 7) (x + 3)$ .

$\frac{(x + 7) (x - 7)}{(x + 3) (x - 7)} \cdot \frac{x + 3}{x}$

Step 3.1.2

Cancel the common factor.

$\frac{(x + 7) (x - 7)}{(x + 3) (x - 7)} \cdot \frac{x + 3}{x}$

Step 3.1.3

Rewrite the expression.

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

Step 3.2

Multiply $\frac{(x + 7) (x - 7)}{x - 7}$ by $\frac{1}{x}$ .

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

Step 3.3

Cancel the common factor of $x - 7$ .

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Step 3.3.1

Cancel the common factor.

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

Step 3.3.2

Rewrite the expression.

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