Trigonometry Examples

$\frac{3 x}{x - 2} - \frac{6}{x^{2} - 4}$

Step 1

Simplify the denominator.

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Step 1.1

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

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

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 $b = 2$ .

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

Step 2

To write $\frac{3 x}{x - 2}$ as a fraction with a common denominator, multiply by $\frac{x + 2}{x + 2}$ .

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

Step 3

Write each expression with a common denominator of $(x - 2) (x + 2)$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

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

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

Step 3.2

Reorder the factors of $(x - 2) (x + 2)$ .

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

Factor $3$ out of $3 x (x + 2) - 6$ .

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Step 5.1.1

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

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

Step 5.1.2

Factor $3$ out of $- 6$ .

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

Step 5.1.3

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

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

Step 5.2

Apply the distributive property.

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

Step 5.3

Multiply $x$ by $x$ .

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

Step 5.4

Move $2$ to the left of $x$ .

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