Precalculus Examples

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

Step 1

Simplify each term.

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Step 1.1

Rewrite ${(1 + \frac{1}{x})}^{2}$ as $(1 + \frac{1}{x}) (1 + \frac{1}{x})$ .

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

Step 1.2

Expand $(1 + \frac{1}{x}) (1 + \frac{1}{x})$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

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

Step 1.2.2

Apply the distributive property.

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

Step 1.2.3

Apply the distributive property.

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

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Multiply $1$ by $1$ .

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

Step 1.3.1.2

Multiply $\frac{1}{x}$ by $1$ .

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

Step 1.3.1.3

Multiply $\frac{1}{x}$ by $1$ .

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

Step 1.3.1.4

Multiply $\frac{1}{x} \cdot \frac{1}{x}$ .

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Step 1.3.1.4.1

Multiply $\frac{1}{x}$ by $\frac{1}{x}$ .

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

Step 1.3.1.4.2

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

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

Step 1.3.1.4.3

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

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

Step 1.3.1.4.4

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

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

Step 1.3.1.4.5

Add $1$ and $1$ .

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

Step 1.3.2

Add $\frac{1}{x}$ and $\frac{1}{x}$ .

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

Step 1.4

Combine $2$ and $\frac{1}{x}$ .

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

Step 1.5

Rewrite ${(1 - \frac{1}{x})}^{2}$ as $(1 - \frac{1}{x}) (1 - \frac{1}{x})$ .

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

Step 1.6

Expand $(1 - \frac{1}{x}) (1 - \frac{1}{x})$ using the FOIL Method.

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Step 1.6.1

Apply the distributive property.

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

Step 1.6.2

Apply the distributive property.

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

Step 1.6.3

Apply the distributive property.

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

Step 1.7

Simplify and combine like terms.

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Step 1.7.1

Simplify each term.

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Step 1.7.1.1

Multiply $1$ by $1$ .

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

Step 1.7.1.2

Multiply $- \frac{1}{x}$ by $1$ .

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

Step 1.7.1.3

Multiply $- 1$ by $1$ .

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

Step 1.7.1.4

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

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Step 1.7.1.4.1

Multiply $- 1$ by $- 1$ .

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

Step 1.7.1.4.2

Multiply $\frac{1}{x}$ by $1$ .

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

Step 1.7.1.4.3

Multiply $\frac{1}{x}$ by $\frac{1}{x}$ .

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

Step 1.7.1.4.4

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

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

Step 1.7.1.4.5

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

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

Step 1.7.1.4.6

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

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

Step 1.7.1.4.7

Add $1$ and $1$ .

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

Step 1.7.2

Subtract $\frac{1}{x}$ from $- \frac{1}{x}$ .

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

Step 1.8

Simplify each term.

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Step 1.8.1

Combine $- 2$ and $\frac{1}{x}$ .

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

Step 1.8.2

Move the negative in front of the fraction.

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

Step 1.9

Apply the distributive property.

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

Step 1.10

Simplify.

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Step 1.10.1

Multiply $- 1$ by $1$ .

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

Step 1.10.2

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

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Step 1.10.2.1

Multiply $- 1$ by $- 1$ .

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

Step 1.10.2.2

Multiply $\frac{2}{x}$ by $1$ .

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

Step 2

Simplify terms.

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Step 2.1

Combine the opposite terms in $1 + \frac{2}{x} + \frac{1}{x^{2}} - 1 + \frac{2}{x} - \frac{1}{x^{2}}$ .

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Step 2.1.1

Subtract $1$ from $1$ .

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

Step 2.1.2

Add $\frac{2}{x} + \frac{1}{x^{2}}$ and $0$ .

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

Step 2.1.3

Subtract $\frac{1}{x^{2}}$ from $\frac{1}{x^{2}}$ .

$\frac{2}{x} + \frac{2}{x} + 0$

Step 2.1.4

Add $\frac{2}{x} + \frac{2}{x}$ and $0$ .

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

Step 2.2

Combine the numerators over the common denominator.

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

Step 2.3

Add $2$ and $2$ .

$\frac{4}{x}$