Precalculus Examples

$\frac{x}{x^{2} - x - 30} - \frac{1}{x + 5}$

Step 1

Factor $x^{2} - x - 30$ 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 $- 30$ and whose sum is $- 1$ .

$- 6, 5$

Step 1.2

Write the factored form using these integers.

$\frac{x}{(x - 6) (x + 5)} - \frac{1}{x + 5}$

Step 2

To write $- \frac{1}{x + 5}$ as a fraction with a common denominator, multiply by $\frac{x - 6}{x - 6}$ .

$\frac{x}{(x - 6) (x + 5)} - \frac{1}{x + 5} \cdot \frac{x - 6}{x - 6}$

Step 3

Write each expression with a common denominator of $(x - 6) (x + 5)$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{1}{x + 5}$ by $\frac{x - 6}{x - 6}$ .

$\frac{x}{(x - 6) (x + 5)} - \frac{x - 6}{(x + 5) (x - 6)}$

Step 3.2

Reorder the factors of $(x - 6) (x + 5)$ .

$\frac{x}{(x + 5) (x - 6)} - \frac{x - 6}{(x + 5) (x - 6)}$

Step 4

Combine the numerators over the common denominator.

$\frac{x - (x - 6)}{(x + 5) (x - 6)}$

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

$\frac{x - x - - 6}{(x + 5) (x - 6)}$

Step 5.2

Multiply $- 1$ by $- 6$ .

$\frac{x - x + 6}{(x + 5) (x - 6)}$

Step 5.3

Subtract $x$ from $x$ .

$\frac{0 + 6}{(x + 5) (x - 6)}$

Step 5.4

Add $0$ and $6$ .

$\frac{6}{(x + 5) (x - 6)}$