Trigonometry Examples

$\frac{\frac{36 - x^{2}}{{(x + 6)}^{2}}}{\frac{x - 3}{x^{2} + 3 x - 18}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\frac{36 - x^{2}}{{(x + 6)}^{2}} \cdot \frac{x^{2} + 3 x - 18}{x - 3}$

Step 2

Simplify the numerator.

Tap for more steps...

Step 2.1

Rewrite $36$ as $6^{2}$ .

$\frac{6^{2} - x^{2}}{{(x + 6)}^{2}} \cdot \frac{x^{2} + 3 x - 18}{x - 3}$

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 $a = 6$ and $b = x$ .

$\frac{(6 + x) (6 - x)}{{(x + 6)}^{2}} \cdot \frac{x^{2} + 3 x - 18}{x - 3}$

Step 3

Cancel the common factor of $6 + x$ and ${(x + 6)}^{2}$ .

Tap for more steps...

Step 3.1

Reorder terms.

$\frac{(x + 6) (6 - x)}{{(x + 6)}^{2}} \cdot \frac{x^{2} + 3 x - 18}{x - 3}$

Step 3.2

Cancel the common factors.

Tap for more steps...

Step 3.2.1

Factor $x + 6$ out of ${(x + 6)}^{2}$ .

$\frac{(x + 6) (6 - x)}{(x + 6) (x + 6)} \cdot \frac{x^{2} + 3 x - 18}{x - 3}$

Step 3.2.2

Cancel the common factor.

$\frac{(x + 6) (6 - x)}{(x + 6) (x + 6)} \cdot \frac{x^{2} + 3 x - 18}{x - 3}$

Step 3.2.3

Rewrite the expression.

$\frac{6 - x}{x + 6} \cdot \frac{x^{2} + 3 x - 18}{x - 3}$

Step 4

Factor $x^{2} + 3 x - 18$ using the AC method.

Tap for more steps...

Step 4.1

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 $3$ .

$- 3, 6$

Step 4.2

Write the factored form using these integers.

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

Step 5

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 5.1

Cancel the common factor of $x + 6$ .

Tap for more steps...

Step 5.1.1

Factor $x + 6$ out of $(x - 3) (x + 6)$ .

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

Step 5.1.2

Cancel the common factor.

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

Step 5.1.3

Rewrite the expression.

$(6 - x) \frac{x - 3}{x - 3}$

Step 5.2

Cancel the common factor of $x - 3$ .

Tap for more steps...

Step 5.2.1

Cancel the common factor.

$(6 - x) \frac{x - 3}{x - 3}$

Step 5.2.2

Rewrite the expression.

$(6 - x) \cdot 1$

Step 5.3

Multiply $6 - x$ by $1$ .

$6 - x$