Trigonometry Examples

$\frac{x^{2} + 13 x + 42}{x^{2} + x - 42} \cdot \frac{x^{2} - 49}{x^{2} - x - 42}$

Step 1

Factor $x^{2} + 13 x + 42$ 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 $42$ and whose sum is $13$ .

$6, 7$

Step 1.2

Write the factored form using these integers.

$\frac{(x + 6) (x + 7)}{x^{2} + x - 42} \cdot \frac{x^{2} - 49}{x^{2} - x - 42}$

Step 2

Factor $x^{2} + x - 42$ 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 $- 42$ and whose sum is $1$ .

$- 6, 7$

Step 2.2

Write the factored form using these integers.

$\frac{(x + 6) (x + 7)}{(x - 6) (x + 7)} \cdot \frac{x^{2} - 49}{x^{2} - x - 42}$

Step 3

Simplify the numerator.

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Step 3.1

Rewrite $49$ as $7^{2}$ .

$\frac{(x + 6) (x + 7)}{(x - 6) (x + 7)} \cdot \frac{x^{2} - 7^{2}}{x^{2} - x - 42}$

Step 3.2

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 $b = 7$ .

$\frac{(x + 6) (x + 7)}{(x - 6) (x + 7)} \cdot \frac{(x + 7) (x - 7)}{x^{2} - x - 42}$

Step 4

Factor $x^{2} - x - 42$ using the AC method.

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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 $- 42$ and whose sum is $- 1$ .

$- 7, 6$

Step 4.2

Write the factored form using these integers.

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

Step 5

Simplify terms.

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Step 5.1

Cancel the common factor of $x + 6$ .

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Step 5.1.1

Factor $x + 6$ out of $(x - 7) (x + 6)$ .

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

Step 5.1.2

Cancel the common factor.

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

Step 5.1.3

Rewrite the expression.

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

Step 5.2

Cancel the common factor of $x + 7$ .

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Step 5.2.1

Factor $x + 7$ out of $(x - 6) (x + 7)$ .

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

Step 5.2.2

Cancel the common factor.

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

Step 5.2.3

Rewrite the expression.

$\frac{x + 7}{x - 6} \cdot \frac{x - 7}{x - 7}$

Step 5.3

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

$\frac{(x + 7) (x - 7)}{(x - 6) (x - 7)}$

Step 5.4

Cancel the common factor of $x - 7$ .

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Step 5.4.1

Cancel the common factor.

$\frac{(x + 7) (x - 7)}{(x - 6) (x - 7)}$

Step 5.4.2

Rewrite the expression.

$\frac{x + 7}{x - 6}$