Pre-Algebra Examples

$\frac{- 3}{x^{2} - 2 x - 8} + \frac{x}{x^{2} - 16}$

Step 1

Simplify each term.

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Step 1.1

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

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Step 1.1.1

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 $- 2$ .

$- 4, 2$

Step 1.1.2

Write the factored form using these integers.

$\frac{- 3}{(x - 4) (x + 2)} + \frac{x}{x^{2} - 16}$

Step 1.2

Move the negative in front of the fraction.

$- \frac{3}{(x - 4) (x + 2)} + \frac{x}{x^{2} - 16}$

Step 1.3

Simplify the denominator.

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Step 1.3.1

Rewrite $16$ as $4^{2}$ .

$- \frac{3}{(x - 4) (x + 2)} + \frac{x}{x^{2} - 4^{2}}$

Step 1.3.2

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 $b = 4$ .

$- \frac{3}{(x - 4) (x + 2)} + \frac{x}{(x + 4) (x - 4)}$

Step 2

To write $- \frac{3}{(x - 4) (x + 2)}$ as a fraction with a common denominator, multiply by $\frac{x + 4}{x + 4}$ .

$- \frac{3}{(x - 4) (x + 2)} \cdot \frac{x + 4}{x + 4} + \frac{x}{(x + 4) (x - 4)}$

Step 3

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

$- \frac{3}{(x - 4) (x + 2)} \cdot \frac{x + 4}{x + 4} + \frac{x}{(x + 4) (x - 4)} \cdot \frac{x + 2}{x + 2}$

Step 4

Write each expression with a common denominator of $(x - 4) (x + 2) (x + 4)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{3}{(x - 4) (x + 2)}$ by $\frac{x + 4}{x + 4}$ .

$- \frac{3 (x + 4)}{(x - 4) (x + 2) (x + 4)} + \frac{x}{(x + 4) (x - 4)} \cdot \frac{x + 2}{x + 2}$

Step 4.2

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

$- \frac{3 (x + 4)}{(x - 4) (x + 2) (x + 4)} + \frac{x (x + 2)}{(x + 4) (x - 4) (x + 2)}$

Step 4.3

Reorder the factors of $(x - 4) (x + 2) (x + 4)$ .

$- \frac{3 (x + 4)}{(x + 2) (x + 4) (x - 4)} + \frac{x (x + 2)}{(x + 4) (x - 4) (x + 2)}$

Step 4.4

Reorder the factors of $(x + 4) (x - 4) (x + 2)$ .

$- \frac{3 (x + 4)}{(x + 2) (x + 4) (x - 4)} + \frac{x (x + 2)}{(x + 2) (x + 4) (x - 4)}$

Step 5

Combine the numerators over the common denominator.

$\frac{- 3 (x + 4) + x (x + 2)}{(x + 2) (x + 4) (x - 4)}$

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

$\frac{- 3 x - 3 \cdot 4 + x (x + 2)}{(x + 2) (x + 4) (x - 4)}$

Step 6.2

Multiply $- 3$ by $4$ .

$\frac{- 3 x - 12 + x (x + 2)}{(x + 2) (x + 4) (x - 4)}$

Step 6.3

Apply the distributive property.

$\frac{- 3 x - 12 + x \cdot x + x \cdot 2}{(x + 2) (x + 4) (x - 4)}$

Step 6.4

Multiply $x$ by $x$ .

$\frac{- 3 x - 12 + x^{2} + x \cdot 2}{(x + 2) (x + 4) (x - 4)}$

Step 6.5

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

$\frac{- 3 x - 12 + x^{2} + 2 \cdot x}{(x + 2) (x + 4) (x - 4)}$

Step 6.6

Add $- 3 x$ and $2 x$ .

$\frac{- x - 12 + x^{2}}{(x + 2) (x + 4) (x - 4)}$

Step 6.7

Reorder terms.

$\frac{x^{2} - x - 12}{(x + 2) (x + 4) (x - 4)}$

Step 6.8

Factor $x^{2} - x - 12$ using the AC method.

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Step 6.8.1

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 $- 12$ and whose sum is $- 1$ .

$- 4, 3$

Step 6.8.2

Write the factored form using these integers.

$\frac{(x - 4) (x + 3)}{(x + 2) (x + 4) (x - 4)}$

Step 7

Cancel the common factor of $x - 4$ .

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Step 7.1

Cancel the common factor.

$\frac{(x - 4) (x + 3)}{(x + 2) (x + 4) (x - 4)}$

Step 7.2

Rewrite the expression.

$\frac{x + 3}{(x + 2) (x + 4)}$