Simplify 5/(x^2-3x-28)+7/(2x-14)

$\frac{5}{x^{2} - 3 x - 28} + \frac{7}{2 x - 14}$

Step 1

Simplify each term.

Factor $x^{2} - 3 x - 28$ using the AC method.

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 $- 28$ and whose sum is $- 3$ .

$- 7, 4$

Write the factored form using these integers.

$\frac{5}{(x - 7) (x + 4)} + \frac{7}{2 x - 14}$

Factor $2$ out of $2 x - 14$ .

Factor $2$ out of $2 x$ .

$\frac{5}{(x - 7) (x + 4)} + \frac{7}{2 (x) - 14}$

Factor $2$ out of $- 14$ .

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

Factor $2$ out of $2 x + 2 \cdot - 7$ .

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

Step 2

To write $\frac{5}{(x - 7) (x + 4)}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 3

To write $\frac{7}{2 (x - 7)}$ as a fraction with a common denominator, multiply by $\frac{x + 4}{x + 4}$ .

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

Step 4

Write each expression with a common denominator of $(x - 7) (x + 4) \cdot 2$ , by multiplying each by an appropriate factor of $1$ .

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

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

Multiply $\frac{7}{2 (x - 7)}$ by $\frac{x + 4}{x + 4}$ .

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

Reorder the factors of $(x - 7) (x + 4) \cdot 2$ .

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

Reorder the factors of $2 (x - 7) (x + 4)$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

Multiply $5$ by $2$ .

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

Apply the distributive property.

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

Multiply $7$ by $4$ .

$\frac{10 + 7 x + 28}{2 (x + 4) (x - 7)}$

Add $10$ and $28$ .

$\frac{7 x + 38}{2 (x + 4) (x - 7)}$

Simplify 5/(x^2-3x-28)+7/(2x-14)

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