Precalculus Examples

$\frac{\frac{x - 3}{3 x - 6}}{\frac{x^{2} - x - 6}{x^{2} - x - 2}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

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

Step 2

Factor $3$ out of $3 x - 6$ .

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Step 2.1

Factor $3$ out of $3 x$ .

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

Step 2.2

Factor $3$ out of $- 6$ .

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

Step 2.3

Factor $3$ out of $3 x + 3 \cdot - 2$ .

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

Step 3

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

$- 2, 1$

Step 3.2

Write the factored form using these integers.

$\frac{x - 3}{3 (x - 2)} \cdot \frac{(x - 2) (x + 1)}{x^{2} - x - 6}$

Step 4

Factor $x^{2} - x - 6$ 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 $- 6$ and whose sum is $- 1$ .

$- 3, 2$

Step 4.2

Write the factored form using these integers.

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

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

Cancel the common factor.

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

Step 5.1.2

Rewrite the expression.

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

Step 5.2

Cancel the common factor of $x - 2$ .

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Step 5.2.1

Factor $x - 2$ out of $3 (x - 2)$ .

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

Step 5.2.2

Cancel the common factor.

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

Step 5.2.3

Rewrite the expression.

$\frac{1}{3} \cdot \frac{x + 1}{x + 2}$

Step 5.3

Multiply $\frac{1}{3}$ by $\frac{x + 1}{x + 2}$ .

$\frac{x + 1}{3 (x + 2)}$