Simplify ((x^2+9x+18)/(7x+42))÷((x^2-9)/x)

$\frac{x^{2} + 9 x + 18}{7 x + 42} \div \frac{x^{2} - 9}{x}$

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{2} + 9 x + 18}{7 x + 42} \cdot \frac{x}{x^{2} - 9}$

Factor $x^{2} + 9 x + 18$ 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 $18$ and whose sum is $9$ .

$3, 6$

Write the factored form using these integers.

$\frac{(x + 3) (x + 6)}{7 x + 42} \cdot \frac{x}{x^{2} - 9}$

Simplify terms.

Factor $7$ out of $7 x + 42$ .

Factor $7$ out of $7 x$ .

$\frac{(x + 3) (x + 6)}{7 (x) + 42} \cdot \frac{x}{x^{2} - 9}$

Factor $7$ out of $42$ .

$\frac{(x + 3) (x + 6)}{7 x + 7 \cdot 6} \cdot \frac{x}{x^{2} - 9}$

Factor $7$ out of $7 x + 7 \cdot 6$ .

$\frac{(x + 3) (x + 6)}{7 (x + 6)} \cdot \frac{x}{x^{2} - 9}$

Cancel the common factor of $x + 6$ .

Cancel the common factor.

$\frac{(x + 3) (x + 6)}{7 (x + 6)} \cdot \frac{x}{x^{2} - 9}$

Rewrite the expression.

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

Simplify the denominator.

Rewrite $9$ as $3^{2}$ .

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

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 3$ .

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

Simplify terms.

Cancel the common factor of $x + 3$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{1}{7} \cdot \frac{x}{x - 3}$

Multiply $\frac{1}{7}$ by $\frac{x}{x - 3}$ .

$\frac{x}{7 (x - 3)}$

Simplify ((x^2+9x+18)/(7x+42))÷((x^2-9)/x)

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