Algebra Examples

$\frac{x - 1}{x^{2} - 4} + \frac{2}{x - 2}$

Step 1

Simplify the denominator.

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Step 1.1

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

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

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 $b = 2$ .

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

Step 2

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

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

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

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

Step 3.2

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

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Multiply $2$ by $2$ .

$\frac{x - 1 + 2 x + 4}{(x + 2) (x - 2)}$

Step 5.3

Add $x$ and $2 x$ .

$\frac{3 x - 1 + 4}{(x + 2) (x - 2)}$

Step 5.4

Add $- 1$ and $4$ .

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

Step 5.5

Factor $3$ out of $3 x + 3$ .

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Step 5.5.1

Factor $3$ out of $3 x$ .

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

Step 5.5.2

Factor $3$ out of $3$ .

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

Step 5.5.3

Factor $3$ out of $3 (x) + 3 (1)$ .

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