Precalculus Examples

$\frac{\frac{x^{2} - 9 x - 22}{x^{2} + 5 x - 24}}{\frac{x + 2}{x - 3}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\frac{x^{2} - 9 x - 22}{x^{2} + 5 x - 24} \cdot \frac{x - 3}{x + 2}$

Step 2

Factor $x^{2} - 9 x - 22$ 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 $- 22$ and whose sum is $- 9$ .

$- 11, 2$

Step 2.2

Write the factored form using these integers.

$\frac{(x - 11) (x + 2)}{x^{2} + 5 x - 24} \cdot \frac{x - 3}{x + 2}$

Step 3

Factor $x^{2} + 5 x - 24$ using the AC method.

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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 $- 24$ and whose sum is $5$ .

$- 3, 8$

Step 3.2

Write the factored form using these integers.

$\frac{(x - 11) (x + 2)}{(x - 3) (x + 8)} \cdot \frac{x - 3}{x + 2}$

Step 4

Cancel the common factor of $x + 2$ .

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Step 4.1

Factor $x + 2$ out of $(x - 11) (x + 2)$ .

$\frac{(x + 2) (x - 11)}{(x - 3) (x + 8)} \cdot \frac{x - 3}{x + 2}$

Step 4.2

Cancel the common factor.

$\frac{(x + 2) (x - 11)}{(x - 3) (x + 8)} \cdot \frac{x - 3}{x + 2}$

Step 4.3

Rewrite the expression.

$\frac{x - 11}{(x - 3) (x + 8)} (x - 3)$

Step 5

Cancel the common factor of $x - 3$ .

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Step 5.1

Cancel the common factor.

$\frac{x - 11}{(x - 3) (x + 8)} (x - 3)$

Step 5.2

Rewrite the expression.

$\frac{x - 11}{x + 8}$