Pre-Algebra Examples

$\frac{x - 7}{x^{2} + 2 x - 8} + \frac{5}{x + 4}$

Step 1

Factor $x^{2} + 2 x - 8$ 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 $- 8$ and whose sum is $2$ .

$- 2, 4$

Step 1.2

Write the factored form using these integers.

$\frac{x - 7}{(x - 2) (x + 4)} + \frac{5}{x + 4}$

Step 2

To write $\frac{5}{x + 4}$ as a fraction with a common denominator, multiply by $\frac{x - 2}{x - 2}$ .

$\frac{x - 7}{(x - 2) (x + 4)} + \frac{5}{x + 4} \cdot \frac{x - 2}{x - 2}$

Step 3

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

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Step 3.1

Multiply $\frac{5}{x + 4}$ by $\frac{x - 2}{x - 2}$ .

$\frac{x - 7}{(x - 2) (x + 4)} + \frac{5 (x - 2)}{(x + 4) (x - 2)}$

Step 3.2

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

$\frac{x - 7}{(x + 4) (x - 2)} + \frac{5 (x - 2)}{(x + 4) (x - 2)}$

Step 4

Combine the numerators over the common denominator.

$\frac{x - 7 + 5 (x - 2)}{(x + 4) (x - 2)}$

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

$\frac{x - 7 + 5 x + 5 \cdot - 2}{(x + 4) (x - 2)}$

Step 5.2

Multiply $5$ by $- 2$ .

$\frac{x - 7 + 5 x - 10}{(x + 4) (x - 2)}$

Step 5.3

Add $x$ and $5 x$ .

$\frac{6 x - 7 - 10}{(x + 4) (x - 2)}$

Step 5.4

Subtract $10$ from $- 7$ .

$\frac{6 x - 17}{(x + 4) (x - 2)}$