Algebra Examples

$\frac{x^{2} - 5 x - 6}{5 x + 15} \div \frac{x^{2} - 3 x - 4}{7 x + 21}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{2} - 5 x - 6}{5 x + 15} \cdot \frac{7 x + 21}{x^{2} - 3 x - 4}$

Step 2

Factor $x^{2} - 5 x - 6$ 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 $- 6$ and whose sum is $- 5$ .

$- 6, 1$

Step 2.2

Write the factored form using these integers.

$\frac{(x - 6) (x + 1)}{5 x + 15} \cdot \frac{7 x + 21}{x^{2} - 3 x - 4}$

Step 3

Factor $5$ out of $5 x + 15$ .

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Step 3.1

Factor $5$ out of $5 x$ .

$\frac{(x - 6) (x + 1)}{5 (x) + 15} \cdot \frac{7 x + 21}{x^{2} - 3 x - 4}$

Step 3.2

Factor $5$ out of $15$ .

$\frac{(x - 6) (x + 1)}{5 x + 5 \cdot 3} \cdot \frac{7 x + 21}{x^{2} - 3 x - 4}$

Step 3.3

Factor $5$ out of $5 x + 5 \cdot 3$ .

$\frac{(x - 6) (x + 1)}{5 (x + 3)} \cdot \frac{7 x + 21}{x^{2} - 3 x - 4}$

Step 4

Factor $7$ out of $7 x + 21$ .

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Step 4.1

Factor $7$ out of $7 x$ .

$\frac{(x - 6) (x + 1)}{5 (x + 3)} \cdot \frac{7 (x) + 21}{x^{2} - 3 x - 4}$

Step 4.2

Factor $7$ out of $21$ .

$\frac{(x - 6) (x + 1)}{5 (x + 3)} \cdot \frac{7 x + 7 \cdot 3}{x^{2} - 3 x - 4}$

Step 4.3

Factor $7$ out of $7 x + 7 \cdot 3$ .

$\frac{(x - 6) (x + 1)}{5 (x + 3)} \cdot \frac{7 (x + 3)}{x^{2} - 3 x - 4}$

Step 5

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

$- 4, 1$

Step 5.2

Write the factored form using these integers.

$\frac{(x - 6) (x + 1)}{5 (x + 3)} \cdot \frac{7 (x + 3)}{(x - 4) (x + 1)}$

Step 6

Cancel the common factor of $x + 1$ .

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Step 6.1

Factor $x + 1$ out of $(x - 6) (x + 1)$ .

$\frac{(x + 1) (x - 6)}{5 (x + 3)} \cdot \frac{7 (x + 3)}{(x - 4) (x + 1)}$

Step 6.2

Factor $x + 1$ out of $(x - 4) (x + 1)$ .

$\frac{(x + 1) (x - 6)}{5 (x + 3)} \cdot \frac{7 (x + 3)}{(x + 1) (x - 4)}$

Step 6.3

Cancel the common factor.

$\frac{(x + 1) (x - 6)}{5 (x + 3)} \cdot \frac{7 (x + 3)}{(x + 1) (x - 4)}$

Step 6.4

Rewrite the expression.

$\frac{x - 6}{5 (x + 3)} \cdot \frac{7 (x + 3)}{x - 4}$

Step 7

Cancel the common factor of $x + 3$ .

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Step 7.1

Factor $x + 3$ out of $5 (x + 3)$ .

$\frac{x - 6}{(x + 3) \cdot 5} \cdot \frac{7 (x + 3)}{x - 4}$

Step 7.2

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

$\frac{x - 6}{(x + 3) \cdot 5} \cdot \frac{(x + 3) \cdot 7}{x - 4}$

Step 7.3

Cancel the common factor.

$\frac{x - 6}{(x + 3) \cdot 5} \cdot \frac{(x + 3) \cdot 7}{x - 4}$

Step 7.4

Rewrite the expression.

$\frac{x - 6}{5} \cdot \frac{7}{x - 4}$

Step 8

Multiply $\frac{x - 6}{5}$ by $\frac{7}{x - 4}$ .

$\frac{(x - 6) \cdot 7}{5 (x - 4)}$

Step 9

Move $7$ to the left of $x - 6$ .

$\frac{7 (x - 6)}{5 (x - 4)}$