Algebra Examples

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

Step 1

To divide by a fraction, multiply by its reciprocal.

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

Step 2

Simplify terms.

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Step 2.1

Cancel the common factor of $x - 2$ .

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Step 2.1.1

Factor $x - 2$ out of ${(x - 2)}^{2}$ .

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

Step 2.1.2

Cancel the common factor.

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

Step 2.1.3

Rewrite the expression.

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

Step 2.2

Multiply $\frac{x - 2}{x + 2}$ by $x^{2} - 4$ .

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

Step 3

Simplify the numerator.

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Step 3.1

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

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

Step 3.2

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 $b = 2$ .

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

Step 3.3

Combine exponents.

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Step 3.3.1

Raise $x - 2$ to the power of $1$ .

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

Step 3.3.2

Raise $x - 2$ to the power of $1$ .

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

Step 3.3.3

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

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

Step 3.3.4

Add $1$ and $1$ .

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

Step 4

Reduce the expression by cancelling the common factors.

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Step 4.1

Cancel the common factor of $x + 2$ .

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Step 4.1.1

Cancel the common factor.

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

Step 4.1.2

Divide ${(x - 2)}^{2}$ by $1$ .

${(x - 2)}^{2}$

Step 4.2

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

$(x - 2) (x - 2)$

Step 5

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

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Step 5.1

Apply the distributive property.

$x (x - 2) - 2 (x - 2)$

Step 5.2

Apply the distributive property.

$x \cdot x + x \cdot - 2 - 2 (x - 2)$

Step 5.3

Apply the distributive property.

$x \cdot x + x \cdot - 2 - 2 x - 2 \cdot - 2$

Step 6

Simplify and combine like terms.

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Step 6.1

Simplify each term.

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Step 6.1.1

Multiply $x$ by $x$ .

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

Step 6.1.2

Move $- 2$ to the left of $x$ .

$x^{2} - 2 \cdot x - 2 x - 2 \cdot - 2$

Step 6.1.3

Multiply $- 2$ by $- 2$ .

$x^{2} - 2 x - 2 x + 4$

Step 6.2

Subtract $2 x$ from $- 2 x$ .

$x^{2} - 4 x + 4$