Precalculus Examples

$\frac{x^{2} - x - 20}{x + 4} \cdot \frac{x - 3}{x^{2} - 2 x - 15}$

Step 1

Factor $x^{2} - x - 20$ 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 $- 20$ and whose sum is $- 1$ .

$- 5, 4$

Step 1.2

Write the factored form using these integers.

$\frac{(x - 5) (x + 4)}{x + 4} \cdot \frac{x - 3}{x^{2} - 2 x - 15}$

Step 2

Factor $x^{2} - 2 x - 15$ 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 $- 15$ and whose sum is $- 2$ .

$- 5, 3$

Step 2.2

Write the factored form using these integers.

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

Step 3

Simplify terms.

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Step 3.1

Cancel the common factor of $x - 5$ .

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Step 3.1.1

Cancel the common factor.

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

Step 3.1.2

Rewrite the expression.

$\frac{x + 4}{x + 4} \cdot \frac{x - 3}{x + 3}$

Step 3.2

Multiply $\frac{x + 4}{x + 4}$ by $\frac{x - 3}{x + 3}$ .

$\frac{(x + 4) (x - 3)}{(x + 4) (x + 3)}$

Step 3.3

Cancel the common factor of $x + 4$ .

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Step 3.3.1

Cancel the common factor.

$\frac{(x + 4) (x - 3)}{(x + 4) (x + 3)}$

Step 3.3.2

Rewrite the expression.

$\frac{x - 3}{x + 3}$