Algebra Examples

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

Step 1

Simplify the denominator.

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Step 1.1

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

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

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 $b = 2$ .

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

Step 2

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

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

Step 3

Combine the numerators over the common denominator.

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

Step 4

Simplify the numerator.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Multiply $3$ by $2$ .

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

Step 4.3

Add $3 x$ and $x$ .

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

Step 4.4

Factor $2$ out of $4 x + 6$ .

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Step 4.4.1

Factor $2$ out of $4 x$ .

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

Step 4.4.2

Factor $2$ out of $6$ .

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

Step 4.4.3

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

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

Step 5

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

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

Step 6

Simplify terms.

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Step 6.1

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

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

Step 6.2

Combine the numerators over the common denominator.

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

Step 7

Simplify the numerator.

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Step 7.1

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

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Step 7.1.1

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

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

Step 7.1.2

Factor $2$ out of $- 2$ .

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

Step 7.1.3

Factor $2$ out of $2 ((2 x + 3) (x - 2)) + 2 (- 1)$ .

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

Step 7.2

Expand $(2 x + 3) (x - 2)$ using the FOIL Method.

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Step 7.2.1

Apply the distributive property.

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

Step 7.2.2

Apply the distributive property.

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

Step 7.2.3

Apply the distributive property.

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

Step 7.3

Simplify and combine like terms.

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Step 7.3.1

Simplify each term.

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Step 7.3.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 7.3.1.1.1

Move $x$ .

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

Step 7.3.1.1.2

Multiply $x$ by $x$ .

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

Step 7.3.1.2

Multiply $- 2$ by $2$ .

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

Step 7.3.1.3

Multiply $3$ by $- 2$ .

$\frac{2 (2 x^{2} - 4 x + 3 x - 6 - 1)}{(x + 2) (x - 2)}$

Step 7.3.2

Add $- 4 x$ and $3 x$ .

$\frac{2 (2 x^{2} - x - 6 - 1)}{(x + 2) (x - 2)}$

Step 7.4

Subtract $1$ from $- 6$ .

$\frac{2 (2 x^{2} - x - 7)}{(x + 2) (x - 2)}$