Precalculus Examples

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

Step 1

Simplify the denominator.

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Step 1.1

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

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

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 $b = 2$ .

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

Step 2

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

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

Step 3

Simplify terms.

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Step 3.1

Combine $2$ and $\frac{(x + 2) (x - 2)}{(x + 2) (x - 2)}$ .

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

Step 3.2

Combine the numerators over the common denominator.

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

Step 4

Simplify the numerator.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Multiply $2$ by $2$ .

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

Step 4.3

Expand $(2 x + 4) (x - 2)$ using the FOIL Method.

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Step 4.3.1

Apply the distributive property.

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

Step 4.3.2

Apply the distributive property.

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

Step 4.3.3

Apply the distributive property.

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

Step 4.4

Simplify and combine like terms.

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Step 4.4.1

Simplify each term.

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Step 4.4.1.1

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

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Step 4.4.1.1.1

Move $x$ .

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

Step 4.4.1.1.2

Multiply $x$ by $x$ .

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

Step 4.4.1.2

Multiply $- 2$ by $2$ .

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

Step 4.4.1.3

Multiply $4$ by $- 2$ .

$\frac{1 + 2 x^{2} - 4 x + 4 x - 8}{(x + 2) (x - 2)}$

Step 4.4.2

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

$\frac{1 + 2 x^{2} + 0 - 8}{(x + 2) (x - 2)}$

Step 4.4.3

Add $2 x^{2}$ and $0$ .

$\frac{1 + 2 x^{2} - 8}{(x + 2) (x - 2)}$

Step 4.5

Subtract $8$ from $1$ .

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