Pre-Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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Step 1.1.1

Factor $x$ out of $x^{2}$ .

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

Step 1.1.2

Factor $x$ out of $- 2 x$ .

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

Step 1.1.3

Factor $x$ out of $x \cdot x + x \cdot - 2$ .

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

Step 1.2

Simplify the denominator.

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Step 1.2.1

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

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

Step 1.2.2

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 $b = 2$ .

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

Step 2

To write $\frac{x - 4}{x (x - 2)}$ as a fraction with a common denominator, multiply by $\frac{x + 2}{x + 2}$ .

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

Step 3

To write $\frac{4}{(x + 2) (x - 2)}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

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

Step 4

Write each expression with a common denominator of $x (x - 2) (x + 2)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

Reorder the factors of $x (x - 2) (x + 2)$ .

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

Step 4.4

Reorder the factors of $(x + 2) (x - 2) x$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

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

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Step 6.1.1

Apply the distributive property.

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

Step 6.1.2

Apply the distributive property.

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

Step 6.1.3

Apply the distributive property.

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

Step 6.2

Simplify and combine like terms.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Multiply $x$ by $x$ .

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

Step 6.2.1.2

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

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

Step 6.2.1.3

Multiply $- 4$ by $2$ .

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

Step 6.2.2

Subtract $4 x$ from $2 x$ .

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

Step 6.3

Add $- 2 x$ and $4 x$ .

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

Step 6.4

Factor $x^{2} + 2 x - 8$ using the AC method.

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Step 6.4.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 8$ and whose sum is $2$ .

$- 2, 4$

Step 6.4.2

Write the factored form using these integers.

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

Step 7

Cancel the common factor of $x - 2$ .

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Step 7.1

Cancel the common factor.

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

Step 7.2

Rewrite the expression.

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