Algebra Examples

$\frac{x^{2} + 4 x - 12}{x^{2} - 25} \div \frac{x + 6}{x^{2} - 5 x}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{2} + 4 x - 12}{x^{2} - 25} \cdot \frac{x^{2} - 5 x}{x + 6}$

Step 2

Factor $x^{2} + 4 x - 12$ 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 $- 12$ and whose sum is $4$ .

$- 2, 6$

Step 2.2

Write the factored form using these integers.

$\frac{(x - 2) (x + 6)}{x^{2} - 25} \cdot \frac{x^{2} - 5 x}{x + 6}$

Step 3

Simplify the denominator.

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Step 3.1

Rewrite $25$ as $5^{2}$ .

$\frac{(x - 2) (x + 6)}{x^{2} - 5^{2}} \cdot \frac{x^{2} - 5 x}{x + 6}$

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 $b = 5$ .

$\frac{(x - 2) (x + 6)}{(x + 5) (x - 5)} \cdot \frac{x^{2} - 5 x}{x + 6}$

Step 4

Simplify terms.

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Step 4.1

Cancel the common factor of $x + 6$ .

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Step 4.1.1

Factor $x + 6$ out of $(x - 2) (x + 6)$ .

$\frac{(x + 6) (x - 2)}{(x + 5) (x - 5)} \cdot \frac{x^{2} - 5 x}{x + 6}$

Step 4.1.2

Cancel the common factor.

$\frac{(x + 6) (x - 2)}{(x + 5) (x - 5)} \cdot \frac{x^{2} - 5 x}{x + 6}$

Step 4.1.3

Rewrite the expression.

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

Step 4.2

Multiply $\frac{x - 2}{(x + 5) (x - 5)}$ by $x^{2} - 5 x$ .

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

Step 5

Factor $x$ out of $x^{2} - 5 x$ .

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Step 5.1

Factor $x$ out of $x^{2}$ .

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

Step 5.2

Factor $x$ out of $- 5 x$ .

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

Step 5.3

Factor $x$ out of $x \cdot x + x \cdot - 5$ .

$\frac{(x - 2) (x (x - 5))}{(x + 5) (x - 5)}$

$\frac{(x - 2) x (x - 5)}{(x + 5) (x - 5)}$

Step 6

Cancel the common factor of $x - 5$ .

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Step 6.1

Cancel the common factor.

$\frac{(x - 2) x (x - 5)}{(x + 5) (x - 5)}$

Step 6.2

Rewrite the expression.

$\frac{(x - 2) x}{x + 5}$

Step 7

Reorder factors in $\frac{(x - 2) x}{x + 5}$ .

$\frac{x (x - 2)}{x + 5}$