Algebra Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $x$ by $x$ .

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

Step 3.1.2

Rewrite using the commutative property of multiplication.

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

Step 3.1.3

Cancel the common factor of $x$ .

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Step 3.1.3.1

Factor $x$ out of $- x$ .

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

Step 3.1.3.2

Cancel the common factor.

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

Step 3.1.3.3

Rewrite the expression.

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

Step 3.1.4

Cancel the common factor of $x$ .

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Step 3.1.4.1

Move the leading negative in $- \frac{1}{x}$ into the numerator.

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

Step 3.1.4.2

Cancel the common factor.

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

Step 3.1.4.3

Rewrite the expression.

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

Step 3.1.5

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

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Step 3.1.5.1

Multiply $- 1$ by $- 1$ .

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

Step 3.1.5.2

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

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

Step 3.1.5.3

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

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

Step 3.1.5.4

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

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

Step 3.1.5.5

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

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

Step 3.1.5.6

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

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

Step 3.1.5.7

Add $1$ and $1$ .

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

Step 3.2

Subtract $1$ from $- 1$ .

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