Precalculus Examples

$\frac{x}{x - 3} - \frac{x + 1}{x^{2} + 5 x - 24}$

Step 1

Factor $x^{2} + 5 x - 24$ 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 $- 24$ and whose sum is $5$ .

$- 3, 8$

Step 1.2

Write the factored form using these integers.

$\frac{x}{x - 3} - \frac{x + 1}{(x - 3) (x + 8)}$

Step 2

To write $\frac{x}{x - 3}$ as a fraction with a common denominator, multiply by $\frac{x + 8}{x + 8}$ .

$\frac{x}{x - 3} \cdot \frac{x + 8}{x + 8} - \frac{x + 1}{(x - 3) (x + 8)}$

Step 3

Simplify terms.

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Step 3.1

Multiply $\frac{x}{x - 3}$ by $\frac{x + 8}{x + 8}$ .

$\frac{x (x + 8)}{(x - 3) (x + 8)} - \frac{x + 1}{(x - 3) (x + 8)}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{x (x + 8) - (x + 1)}{(x - 3) (x + 8)}$

Step 4

Simplify the numerator.

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Step 4.1

Apply the distributive property.

$\frac{x \cdot x + x \cdot 8 - (x + 1)}{(x - 3) (x + 8)}$

Step 4.2

Multiply $x$ by $x$ .

$\frac{x^{2} + x \cdot 8 - (x + 1)}{(x - 3) (x + 8)}$

Step 4.3

Move $8$ to the left of $x$ .

$\frac{x^{2} + 8 \cdot x - (x + 1)}{(x - 3) (x + 8)}$

Step 4.4

Apply the distributive property.

$\frac{x^{2} + 8 x - x - 1 \cdot 1}{(x - 3) (x + 8)}$

Step 4.5

Multiply $- 1$ by $1$ .

$\frac{x^{2} + 8 x - x - 1}{(x - 3) (x + 8)}$

Step 4.6

Subtract $x$ from $8 x$ .

$\frac{x^{2} + 7 x - 1}{(x - 3) (x + 8)}$