Algebra Examples

$\frac{x}{x^{2} + 6 x - 16} - \frac{5}{x + 8}$

Step 1

Factor $x^{2} + 6 x - 16$ using the AC method.

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Step 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 $- 16$ and whose sum is $6$ .

$- 2, 8$

Step 1.2

Write the factored form using these integers.

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

Step 2

To write $- \frac{5}{x + 8}$ as a fraction with a common denominator, multiply by $\frac{x - 2}{x - 2}$ .

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

Step 3

Write each expression with a common denominator of $(x - 2) (x + 8)$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{5}{x + 8}$ by $\frac{x - 2}{x - 2}$ .

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

Step 3.2

Reorder the factors of $(x - 2) (x + 8)$ .

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Multiply $- 5$ by $- 2$ .

$\frac{x - 5 x + 10}{(x + 8) (x - 2)}$

Step 5.3

Subtract $5 x$ from $x$ .

$\frac{- 4 x + 10}{(x + 8) (x - 2)}$

Step 5.4

Factor $2$ out of $- 4 x + 10$ .

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Step 5.4.1

Factor $2$ out of $- 4 x$ .

$\frac{2 (- 2 x) + 10}{(x + 8) (x - 2)}$

Step 5.4.2

Factor $2$ out of $10$ .

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

Step 5.4.3

Factor $2$ out of $2 (- 2 x) + 2 (5)$ .

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

Step 6

Simplify with factoring out.

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Step 6.1

Factor $- 1$ out of $- 2 x$ .

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

Step 6.2

Rewrite $5$ as $- 1 (- 5)$ .

$\frac{2 (- (2 x) - 1 (- 5))}{(x + 8) (x - 2)}$

Step 6.3

Factor $- 1$ out of $- (2 x) - 1 (- 5)$ .

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

Step 6.4

Simplify the expression.

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Step 6.4.1

Rewrite $- (2 x - 5)$ as $- 1 (2 x - 5)$ .

$\frac{2 (- 1 (2 x - 5))}{(x + 8) (x - 2)}$

Step 6.4.2

Move the negative in front of the fraction.

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