Algebra Examples

$\frac{x}{x^{2} - 16} - \frac{8}{x^{2} + 5 x + 4}$

Step 1

Simplify each term.

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Step 1.1

Simplify the denominator.

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Step 1.1.1

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

$\frac{x}{x^{2} - 4^{2}} - \frac{8}{x^{2} + 5 x + 4}$

Step 1.1.2

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 $b = 4$ .

$\frac{x}{(x + 4) (x - 4)} - \frac{8}{x^{2} + 5 x + 4}$

Step 1.2

Factor $x^{2} + 5 x + 4$ using the AC method.

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Step 1.2.1

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 $4$ and whose sum is $5$ .

$1, 4$

Step 1.2.2

Write the factored form using these integers.

$\frac{x}{(x + 4) (x - 4)} - \frac{8}{(x + 1) (x + 4)}$

Step 2

To write $\frac{x}{(x + 4) (x - 4)}$ as a fraction with a common denominator, multiply by $\frac{x + 1}{x + 1}$ .

$\frac{x}{(x + 4) (x - 4)} \cdot \frac{x + 1}{x + 1} - \frac{8}{(x + 1) (x + 4)}$

Step 3

To write $- \frac{8}{(x + 1) (x + 4)}$ as a fraction with a common denominator, multiply by $\frac{x - 4}{x - 4}$ .

$\frac{x}{(x + 4) (x - 4)} \cdot \frac{x + 1}{x + 1} - \frac{8}{(x + 1) (x + 4)} \cdot \frac{x - 4}{x - 4}$

Step 4

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

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Step 4.1

Multiply $\frac{x}{(x + 4) (x - 4)}$ by $\frac{x + 1}{x + 1}$ .

$\frac{x (x + 1)}{(x + 4) (x - 4) (x + 1)} - \frac{8}{(x + 1) (x + 4)} \cdot \frac{x - 4}{x - 4}$

Step 4.2

Multiply $\frac{8}{(x + 1) (x + 4)}$ by $\frac{x - 4}{x - 4}$ .

$\frac{x (x + 1)}{(x + 4) (x - 4) (x + 1)} - \frac{8 (x - 4)}{(x + 1) (x + 4) (x - 4)}$

Step 4.3

Reorder the factors of $(x + 4) (x - 4) (x + 1)$ .

$\frac{x (x + 1)}{(x + 1) (x + 4) (x - 4)} - \frac{8 (x - 4)}{(x + 1) (x + 4) (x - 4)}$

Step 5

Combine the numerators over the common denominator.

$\frac{x (x + 1) - 8 (x - 4)}{(x + 1) (x + 4) (x - 4)}$

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

$\frac{x \cdot x + x \cdot 1 - 8 (x - 4)}{(x + 1) (x + 4) (x - 4)}$

Step 6.2

Multiply $x$ by $x$ .

$\frac{x^{2} + x \cdot 1 - 8 (x - 4)}{(x + 1) (x + 4) (x - 4)}$

Step 6.3

Multiply $x$ by $1$ .

$\frac{x^{2} + x - 8 (x - 4)}{(x + 1) (x + 4) (x - 4)}$

Step 6.4

Apply the distributive property.

$\frac{x^{2} + x - 8 x - 8 \cdot - 4}{(x + 1) (x + 4) (x - 4)}$

Step 6.5

Multiply $- 8$ by $- 4$ .

$\frac{x^{2} + x - 8 x + 32}{(x + 1) (x + 4) (x - 4)}$

Step 6.6

Subtract $8 x$ from $x$ .

$\frac{x^{2} - 7 x + 32}{(x + 1) (x + 4) (x - 4)}$