Simplify 3/(x+7)-(x-8)/(x^2+9x+14)

$\frac{3}{x + 7} - \frac{x - 8}{x^{2} + 9 x + 14}$

Factor $x^{2} + 9 x + 14$ 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 $14$ and whose sum is $9$ .

$2, 7$

Write the factored form using these integers.

$\frac{3}{x + 7} - \frac{x - 8}{(x + 2) (x + 7)}$

To write $\frac{3}{x + 7}$ as a fraction with a common denominator, multiply by $\frac{x + 2}{x + 2}$ .

$\frac{3}{x + 7} \cdot \frac{x + 2}{x + 2} - \frac{x - 8}{(x + 2) (x + 7)}$

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

Multiply $\frac{3}{x + 7}$ by $\frac{x + 2}{x + 2}$ .

$\frac{3 (x + 2)}{(x + 7) (x + 2)} - \frac{x - 8}{(x + 2) (x + 7)}$

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

$\frac{3 (x + 2)}{(x + 2) (x + 7)} - \frac{x - 8}{(x + 2) (x + 7)}$

Combine the numerators over the common denominator.

$\frac{3 (x + 2) - (x - 8)}{(x + 2) (x + 7)}$

Simplify the numerator.

Apply the distributive property.

$\frac{3 x + 3 \cdot 2 - (x - 8)}{(x + 2) (x + 7)}$

Multiply $3$ by $2$ .

$\frac{3 x + 6 - (x - 8)}{(x + 2) (x + 7)}$

Apply the distributive property.

$\frac{3 x + 6 - x - - 8}{(x + 2) (x + 7)}$

Multiply $- 1$ by $- 8$ .

$\frac{3 x + 6 - x + 8}{(x + 2) (x + 7)}$

Subtract $x$ from $3 x$ .

$\frac{2 x + 6 + 8}{(x + 2) (x + 7)}$

Add $6$ and $8$ .

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

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

Factor $2$ out of $2 x$ .

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

Factor $2$ out of $14$ .

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

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

$\frac{2 (x + 7)}{(x + 2) (x + 7)}$

Cancel the common factor of $x + 7$ .

Cancel the common factor.

$\frac{2 (x + 7)}{(x + 2) (x + 7)}$

Rewrite the expression.

$\frac{2}{x + 2}$

Simplify 3/(x+7)-(x-8)/(x^2+9x+14)

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