Simplify ((x-3)/(x^2-5x-14))÷((x^2-x-6)/(x-7))

$\frac{x - 3}{x^{2} - 5 x - 14} \div \frac{x^{2} - x - 6}{x - 7}$

To divide by a fraction, multiply by its reciprocal.

$\frac{x - 3}{x^{2} - 5 x - 14} \cdot \frac{x - 7}{x^{2} - x - 6}$

Factor $x^{2} - 5 x - 14$ 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 $- 5$ .

$- 7, 2$

Write the factored form using these integers.

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

Simplify terms.

Cancel the common factor of $x - 7$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{x - 3}{x + 2} \cdot \frac{1}{x^{2} - x - 6}$

Multiply $\frac{x - 3}{x + 2}$ by $\frac{1}{x^{2} - x - 6}$ .

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

Simplify the denominator.

Factor $x^{2} - x - 6$ 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 $- 6$ and whose sum is $- 1$ .

$- 3, 2$

Write the factored form using these integers.

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

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

Combine exponents.

Raise $x + 2$ to the power of $1$ .

$\frac{x - 3}{{(x + 2)}^{1} (x + 2) (x - 3)}$

Raise $x + 2$ to the power of $1$ .

$\frac{x - 3}{{(x + 2)}^{1} {(x + 2)}^{1} (x - 3)}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{x - 3}{{(x + 2)}^{1 + 1} (x - 3)}$

Add $1$ and $1$ .

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

Cancel the common factor of $x - 3$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{1}{{(x + 2)}^{2}}$

Simplify ((x-3)/(x^2-5x-14))÷((x^2-x-6)/(x-7))

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