Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

Factor $3$ out of $3 x - 6$ .

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Step 1.1.1

Factor $3$ out of $3 x$ .

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

Step 1.1.2

Factor $3$ out of $- 6$ .

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

Step 1.1.3

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

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

Step 1.2

Simplify the denominator.

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Step 1.2.1

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

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

Step 1.2.2

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 $b = 2$ .

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

Step 1.3

Cancel the common factor of $x - 2$ .

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Step 1.3.1

Cancel the common factor.

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

Step 1.3.2

Rewrite the expression.

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

Step 2

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

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

Step 3

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

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

Step 4

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 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

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

Step 6.2

Multiply $3$ by $- 2$ .

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

Step 6.3

Apply the distributive property.

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

Step 6.4

Multiply $- 2$ by $2$ .

$\frac{3 x - 6 - 2 x - 4}{(x + 2) (x - 2)}$

Step 6.5

Subtract $2 x$ from $3 x$ .

$\frac{x - 6 - 4}{(x + 2) (x - 2)}$

Step 6.6

Subtract $4$ from $- 6$ .

$\frac{x - 10}{(x + 2) (x - 2)}$