Algebra Examples

$\frac{x^{2} - 4}{x^{2} - 11 x + 18} \div \frac{6 x}{x - 9}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{2} - 4}{x^{2} - 11 x + 18} \cdot \frac{x - 9}{6 x}$

Step 2

Simplify the numerator.

Tap for more steps...

Step 2.1

Rewrite $4$ as $2^{2}$ .

$\frac{x^{2} - 2^{2}}{x^{2} - 11 x + 18} \cdot \frac{x - 9}{6 x}$

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 $b = 2$ .

$\frac{(x + 2) (x - 2)}{x^{2} - 11 x + 18} \cdot \frac{x - 9}{6 x}$

Step 3

Factor $x^{2} - 11 x + 18$ using the AC method.

Tap for more steps...

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 $18$ and whose sum is $- 11$ .

$- 9, - 2$

Step 3.2

Write the factored form using these integers.

$\frac{(x + 2) (x - 2)}{(x - 9) (x - 2)} \cdot \frac{x - 9}{6 x}$

Step 4

Cancel the common factor of $x - 9$ .

Tap for more steps...

Step 4.1

Cancel the common factor.

$\frac{(x + 2) (x - 2)}{(x - 9) (x - 2)} \cdot \frac{x - 9}{6 x}$

Step 4.2

Rewrite the expression.

$\frac{(x + 2) (x - 2)}{x - 2} \cdot \frac{1}{6 x}$

Step 5

Multiply $\frac{(x + 2) (x - 2)}{x - 2}$ by $\frac{1}{6 x}$ .

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

Step 6

Cancel the common factor of $x - 2$ .

Tap for more steps...

Step 6.1

Cancel the common factor.

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

Step 6.2

Rewrite the expression.

$\frac{x + 2}{6 x}$

Step 7

Split the fraction $\frac{x + 2}{6 x}$ into two fractions.

$\frac{x}{6 x} + \frac{2}{6 x}$

Step 8

Cancel the common factor of $x$ .

Tap for more steps...

Step 8.1

Cancel the common factor.

$\frac{x}{6 x} + \frac{2}{6 x}$

Step 8.2

Rewrite the expression.

$\frac{1}{6} + \frac{2}{6 x}$

Step 9

Cancel the common factor of $2$ and $6$ .

Tap for more steps...

Step 9.1

Factor $2$ out of $2$ .

$\frac{1}{6} + \frac{2 (1)}{6 x}$

Step 9.2

Cancel the common factors.

Tap for more steps...

Step 9.2.1

Factor $2$ out of $6 x$ .

$\frac{1}{6} + \frac{2 (1)}{2 (3 x)}$

Step 9.2.2

Cancel the common factor.

$\frac{1}{6} + \frac{2 \cdot 1}{2 (3 x)}$

Step 9.2.3

Rewrite the expression.

$\frac{1}{6} + \frac{1}{3 x}$