Algebra Examples

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

Step 1

Simplify the denominator.

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Step 1.1

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

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

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 $b = 2$ .

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

Step 2

To write $\frac{x}{x - 1}$ as a fraction with a common denominator, multiply by $\frac{(x + 2) (x - 2)}{(x + 2) (x - 2)}$ .

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

Step 3

To write $\frac{1}{(x + 2) (x - 2)}$ as a fraction with a common denominator, multiply by $\frac{x - 1}{x - 1}$ .

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

Step 4

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

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

Reorder the factors of $(x - 1) ((x + 2) (x - 2))$ .

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

Step 4.4

Reorder the factors of $(x + 2) (x - 2) (x - 1)$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Expand $(x + 2) (x - 2)$ using the FOIL Method.

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Step 6.1.1

Apply the distributive property.

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

Step 6.1.2

Apply the distributive property.

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

Step 6.1.3

Apply the distributive property.

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

Step 6.2

Combine the opposite terms in $x \cdot x + x \cdot - 2 + 2 x + 2 \cdot - 2$ .

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Step 6.2.1

Reorder the factors in the terms $x \cdot - 2$ and $2 x$ .

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

Step 6.2.2

Add $- 2 x$ and $2 x$ .

$\frac{x (x \cdot x + 0 + 2 \cdot - 2) + x - 1}{(x + 2) (x - 1) (x - 2)}$

Step 6.2.3

Add $x \cdot x$ and $0$ .

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

Step 6.3

Simplify each term.

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Step 6.3.1

Multiply $x$ by $x$ .

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

Step 6.3.2

Multiply $2$ by $- 2$ .

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

Step 6.4

Apply the distributive property.

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

Step 6.5

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 6.5.1

Multiply $x$ by $x^{2}$ .

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Step 6.5.1.1

Raise $x$ to the power of $1$ .

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

Step 6.5.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 6.5.2

Add $1$ and $2$ .

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

Step 6.6

Move $- 4$ to the left of $x$ .

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

Step 6.7

Add $- 4 x$ and $x$ .

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