Precalculus Examples

$\frac{x + 2}{x - 1} - \frac{2}{x + 6} - \frac{14}{x^{2} + 5 x - 6}$

Step 1

Factor $x^{2} + 5 x - 6$ 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 $- 6$ and whose sum is $5$ .

$- 1, 6$

Step 1.2

Write the factored form using these integers.

$\frac{x + 2}{x - 1} - \frac{2}{x + 6} - \frac{14}{(x - 1) (x + 6)}$

Step 2

To write $\frac{x + 2}{x - 1}$ as a fraction with a common denominator, multiply by $\frac{x + 6}{x + 6}$ .

$\frac{x + 2}{x - 1} \cdot \frac{x + 6}{x + 6} - \frac{2}{x + 6} - \frac{14}{(x - 1) (x + 6)}$

Step 3

To write $- \frac{2}{x + 6}$ as a fraction with a common denominator, multiply by $\frac{x - 1}{x - 1}$ .

$\frac{x + 2}{x - 1} \cdot \frac{x + 6}{x + 6} - \frac{2}{x + 6} \cdot \frac{x - 1}{x - 1} - \frac{14}{(x - 1) (x + 6)}$

Step 4

Write each expression with a common denominator of $(x - 1) (x + 6)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{x + 2}{x - 1}$ by $\frac{x + 6}{x + 6}$ .

$\frac{(x + 2) (x + 6)}{(x - 1) (x + 6)} - \frac{2}{x + 6} \cdot \frac{x - 1}{x - 1} - \frac{14}{(x - 1) (x + 6)}$

Step 4.2

Multiply $\frac{2}{x + 6}$ by $\frac{x - 1}{x - 1}$ .

$\frac{(x + 2) (x + 6)}{(x - 1) (x + 6)} - \frac{2 (x - 1)}{(x + 6) (x - 1)} - \frac{14}{(x - 1) (x + 6)}$

Step 4.3

Reorder the factors of $(x - 1) (x + 6)$ .

$\frac{(x + 2) (x + 6)}{(x + 6) (x - 1)} - \frac{2 (x - 1)}{(x + 6) (x - 1)} - \frac{14}{(x - 1) (x + 6)}$

Step 5

Simplify terms.

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Step 5.1

Combine the numerators over the common denominator.

$\frac{(x + 2) (x + 6) - 2 (x - 1)}{(x + 6) (x - 1)} - \frac{14}{(x - 1) (x + 6)}$

Step 5.2

Combine the numerators over the common denominator.

$\frac{(x + 2) (x + 6) - 2 (x - 1) - 14}{(x + 6) (x - 1)}$

Step 6

Simplify each term.

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Step 6.1

Expand $(x + 2) (x + 6)$ using the FOIL Method.

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Step 6.1.1

Apply the distributive property.

$\frac{x (x + 6) + 2 (x + 6) - 2 (x - 1) - 14}{(x + 6) (x - 1)}$

Step 6.1.2

Apply the distributive property.

$\frac{x \cdot x + x \cdot 6 + 2 (x + 6) - 2 (x - 1) - 14}{(x + 6) (x - 1)}$

Step 6.1.3

Apply the distributive property.

$\frac{x \cdot x + x \cdot 6 + 2 x + 2 \cdot 6 - 2 (x - 1) - 14}{(x + 6) (x - 1)}$

Step 6.2

Simplify and combine like terms.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Multiply $x$ by $x$ .

$\frac{x^{2} + x \cdot 6 + 2 x + 2 \cdot 6 - 2 (x - 1) - 14}{(x + 6) (x - 1)}$

Step 6.2.1.2

Move $6$ to the left of $x$ .

$\frac{x^{2} + 6 \cdot x + 2 x + 2 \cdot 6 - 2 (x - 1) - 14}{(x + 6) (x - 1)}$

Step 6.2.1.3

Multiply $2$ by $6$ .

$\frac{x^{2} + 6 x + 2 x + 12 - 2 (x - 1) - 14}{(x + 6) (x - 1)}$

Step 6.2.2

Add $6 x$ and $2 x$ .

$\frac{x^{2} + 8 x + 12 - 2 (x - 1) - 14}{(x + 6) (x - 1)}$

Step 6.3

Apply the distributive property.

$\frac{x^{2} + 8 x + 12 - 2 x - 2 \cdot - 1 - 14}{(x + 6) (x - 1)}$

Step 6.4

Multiply $- 2$ by $- 1$ .

$\frac{x^{2} + 8 x + 12 - 2 x + 2 - 14}{(x + 6) (x - 1)}$

Step 7

Simplify terms.

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Step 7.1

Subtract $2 x$ from $8 x$ .

$\frac{x^{2} + 6 x + 12 + 2 - 14}{(x + 6) (x - 1)}$

Step 7.2

Add $12$ and $2$ .

$\frac{x^{2} + 6 x + 14 - 14}{(x + 6) (x - 1)}$

Step 7.3

Combine the opposite terms in $x^{2} + 6 x + 14 - 14$ .

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Step 7.3.1

Subtract $14$ from $14$ .

$\frac{x^{2} + 6 x + 0}{(x + 6) (x - 1)}$

Step 7.3.2

Add $x^{2} + 6 x$ and $0$ .

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

Step 7.4

Factor $x$ out of $x^{2} + 6 x$ .

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Step 7.4.1

Factor $x$ out of $x^{2}$ .

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

Step 7.4.2

Factor $x$ out of $6 x$ .

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

Step 7.4.3

Factor $x$ out of $x \cdot x + x \cdot 6$ .

$\frac{x (x + 6)}{(x + 6) (x - 1)}$

Step 7.5

Cancel the common factor of $x + 6$ .

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Step 7.5.1

Cancel the common factor.

$\frac{x (x + 6)}{(x + 6) (x - 1)}$

Step 7.5.2

Rewrite the expression.

$\frac{x}{x - 1}$