Precalculus Examples

$\frac{3 x + 4}{x^{2} - 16} - \frac{2}{x - 4}$

Step 1

Simplify the denominator.

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Step 1.1

Rewrite $16$ as $4^{2}$ .

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

Step 1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 4$ .

$\frac{3 x + 4}{(x + 4) (x - 4)} - \frac{2}{x - 4}$

Step 2

To write $- \frac{2}{x - 4}$ as a fraction with a common denominator, multiply by $\frac{x + 4}{x + 4}$ .

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

Step 3

Write each expression with a common denominator of $(x + 4) (x - 4)$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{2}{x - 4}$ by $\frac{x + 4}{x + 4}$ .

$\frac{3 x + 4}{(x + 4) (x - 4)} - \frac{2 (x + 4)}{(x - 4) (x + 4)}$

Step 3.2

Reorder the factors of $(x - 4) (x + 4)$ .

$\frac{3 x + 4}{(x + 4) (x - 4)} - \frac{2 (x + 4)}{(x + 4) (x - 4)}$

Step 4

Combine the numerators over the common denominator.

$\frac{3 x + 4 - 2 (x + 4)}{(x + 4) (x - 4)}$

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Multiply $- 2$ by $4$ .

$\frac{3 x + 4 - 2 x - 8}{(x + 4) (x - 4)}$

Step 5.3

Subtract $2 x$ from $3 x$ .

$\frac{x + 4 - 8}{(x + 4) (x - 4)}$

Step 5.4

Subtract $8$ from $4$ .

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

Step 6

Cancel the common factor of $x - 4$ .

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Step 6.1

Cancel the common factor.

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

Step 6.2

Rewrite the expression.

$\frac{1}{x + 4}$