Trigonometry Examples

$\frac{x^{2} - 8 x + 15}{x^{2} - 4 x - 5} \cdot \frac{x^{2 - 1}}{x^{2} - 9}$

Step 1

Factor $x^{2} - 8 x + 15$ using the AC method.

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Step 1.1

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 $15$ and whose sum is $- 8$ .

$- 5, - 3$

Step 1.2

Write the factored form using these integers.

$\frac{(x - 5) (x - 3)}{x^{2} - 4 x - 5} \cdot \frac{x^{2 - 1}}{x^{2} - 9}$

Step 2

Factor $x^{2} - 4 x - 5$ 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 $- 5$ and whose sum is $- 4$ .

$- 5, 1$

Step 2.2

Write the factored form using these integers.

$\frac{(x - 5) (x - 3)}{(x - 5) (x + 1)} \cdot \frac{x^{2 - 1}}{x^{2} - 9}$

Step 3

Simplify the numerator.

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Step 3.1

Subtract $1$ from $2$ .

$\frac{(x - 5) (x - 3)}{(x - 5) (x + 1)} \cdot \frac{x^{1}}{x^{2} - 9}$

Step 3.2

Simplify.

$\frac{(x - 5) (x - 3)}{(x - 5) (x + 1)} \cdot \frac{x}{x^{2} - 9}$

Step 4

Simplify the denominator.

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Step 4.1

Rewrite $9$ as $3^{2}$ .

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

Step 4.2

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 $b = 3$ .

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

Step 5

Simplify terms.

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Step 5.1

Cancel the common factor of $x - 3$ .

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Step 5.1.1

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

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

Step 5.1.2

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

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

Step 5.1.3

Cancel the common factor.

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

Step 5.1.4

Rewrite the expression.

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

Step 5.2

Multiply $\frac{x - 5}{(x - 5) (x + 1)}$ by $\frac{x}{x + 3}$ .

$\frac{(x - 5) x}{(x - 5) (x + 1) (x + 3)}$

Step 5.3

Cancel the common factor of $x - 5$ .

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Step 5.3.1

Cancel the common factor.

$\frac{(x - 5) x}{(x - 5) (x + 1) (x + 3)}$

Step 5.3.2

Rewrite the expression.

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