Algebra Examples

$\frac{9}{x^{2} - 14 x + 45} \cdot \frac{x - 5}{x + 9} \div \frac{4 x}{9 x^{3} - 729 x}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{9}{x^{2} - 14 x + 45} \cdot \frac{x - 5}{x + 9} \frac{9 x^{3} - 729 x}{4 x}$

Step 2

Factor $x^{2} - 14 x + 45$ using the AC method.

Tap for more steps...

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 $45$ and whose sum is $- 14$ .

$- 9, - 5$

Step 2.2

Write the factored form using these integers.

$\frac{9}{(x - 9) (x - 5)} \cdot \frac{x - 5}{x + 9} \frac{9 x^{3} - 729 x}{4 x}$

Step 3

Simplify terms.

Tap for more steps...

Step 3.1

Cancel the common factor of $x - 5$ .

Tap for more steps...

Step 3.1.1

Factor $x - 5$ out of $(x - 9) (x - 5)$ .

$\frac{9}{(x - 5) (x - 9)} \cdot \frac{x - 5}{x + 9} \frac{9 x^{3} - 729 x}{4 x}$

Step 3.1.2

Cancel the common factor.

$\frac{9}{(x - 5) (x - 9)} \cdot \frac{x - 5}{x + 9} \frac{9 x^{3} - 729 x}{4 x}$

Step 3.1.3

Rewrite the expression.

$\frac{9}{x - 9} \cdot \frac{1}{x + 9} \frac{9 x^{3} - 729 x}{4 x}$

Step 3.2

Multiply $\frac{9}{x - 9}$ by $\frac{1}{x + 9}$ .

$\frac{9}{(x - 9) (x + 9)} \cdot \frac{9 x^{3} - 729 x}{4 x}$

Step 4

Simplify the numerator.

Tap for more steps...

Step 4.1

Factor $9 x$ out of $9 x^{3} - 729 x$ .

Tap for more steps...

Step 4.1.1

Factor $9 x$ out of $9 x^{3}$ .

$\frac{9}{(x - 9) (x + 9)} \cdot \frac{9 x (x^{2}) - 729 x}{4 x}$

Step 4.1.2

Factor $9 x$ out of $- 729 x$ .

$\frac{9}{(x - 9) (x + 9)} \cdot \frac{9 x (x^{2}) + 9 x (- 81)}{4 x}$

Step 4.1.3

Factor $9 x$ out of $9 x (x^{2}) + 9 x (- 81)$ .

$\frac{9}{(x - 9) (x + 9)} \cdot \frac{9 x (x^{2} - 81)}{4 x}$

Step 4.2

Rewrite $81$ as $9^{2}$ .

$\frac{9}{(x - 9) (x + 9)} \cdot \frac{9 x (x^{2} - 9^{2})}{4 x}$

Step 4.3

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 $b = 9$ .

$\frac{9}{(x - 9) (x + 9)} \cdot \frac{9 x (x + 9) (x - 9)}{4 x}$

Step 5

Simplify terms.

Tap for more steps...

Step 5.1

Cancel the common factor of $(x + 9) (x - 9)$ .

Tap for more steps...

Step 5.1.1

Factor $(x + 9) (x - 9)$ out of $(x - 9) (x + 9)$ .

$\frac{9}{(x + 9) (x - 9) (1)} \cdot \frac{9 x (x + 9) (x - 9)}{4 x}$

Step 5.1.2

Factor $(x + 9) (x - 9)$ out of $9 x (x + 9) (x - 9)$ .

$\frac{9}{(x + 9) (x - 9) (1)} \cdot \frac{(x + 9) (x - 9) (9 x)}{4 x}$

Step 5.1.3

Cancel the common factor.

$\frac{9}{(x + 9) (x - 9) \cdot 1} \cdot \frac{(x + 9) (x - 9) (9 x)}{4 x}$

Step 5.1.4

Rewrite the expression.

$9 \frac{9 x}{4 x}$

Step 5.2

Combine $9$ and $\frac{9 x}{4 x}$ .

$\frac{9 (9 x)}{4 x}$

Step 5.3

Multiply $9$ by $9$ .

$\frac{81 x}{4 x}$

Step 5.4

Cancel the common factor of $x$ .

Tap for more steps...

Step 5.4.1

Cancel the common factor.

$\frac{81 x}{4 x}$

Step 5.4.2

Rewrite the expression.

$\frac{81}{4}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{81}{4}$

Decimal Form:

$20.25$

Mixed Number Form:

$20 \frac{1}{4}$