Simplify 7/(x+6)-x/(x^2-x-42)

$\frac{7}{x + 6} - \frac{x}{x^{2} - x - 42}$

Factor $x^{2} - x - 42$ 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 $- 42$ and whose sum is $- 1$ .

$- 7, 6$

Write the factored form using these integers.

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

To write $\frac{7}{x + 6}$ as a fraction with a common denominator, multiply by $\frac{x - 7}{x - 7}$ .

$\frac{7}{x + 6} \cdot \frac{x - 7}{x - 7} - \frac{x}{(x - 7) (x + 6)}$

Write each expression with a common denominator of $(x + 6) (x - 7)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{7}{x + 6}$ by $\frac{x - 7}{x - 7}$ .

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

Reorder the factors of $(x - 7) (x + 6)$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Apply the distributive property.

$\frac{7 x + 7 \cdot - 7 - x}{(x + 6) (x - 7)}$

Multiply $7$ by $- 7$ .

$\frac{7 x - 49 - x}{(x + 6) (x - 7)}$

Subtract $x$ from $7 x$ .

$\frac{6 x - 49}{(x + 6) (x - 7)}$

Simplify 7/(x+6)-x/(x^2-x-42)

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