Simplify ((5x)/(x+3))÷((x^2-8x+15)/(x^2-9))

$\frac{5 x}{x + 3} \div \frac{x^{2} - 8 x + 15}{x^{2} - 9}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{5 x}{x + 3} \cdot \frac{x^{2} - 9}{x^{2} - 8 x + 15}$

Step 2

Simplify the numerator.

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

$\frac{5 x}{x + 3} \cdot \frac{x^{2} - 3^{2}}{x^{2} - 8 x + 15}$

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 $b = 3$ .

$\frac{5 x}{x + 3} \cdot \frac{(x + 3) (x - 3)}{x^{2} - 8 x + 15}$

Step 3

Factor $x^{2} - 8 x + 15$ 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 $15$ and whose sum is $- 8$ .

$- 5, - 3$

Write the factored form using these integers.

$\frac{5 x}{x + 3} \cdot \frac{(x + 3) (x - 3)}{(x - 5) (x - 3)}$

Step 4

Simplify terms.

Cancel the common factor of $x + 3$ .

Cancel the common factor.

$\frac{5 x}{x + 3} \cdot \frac{(x + 3) (x - 3)}{(x - 5) (x - 3)}$

Rewrite the expression.

$5 x \frac{x - 3}{(x - 5) (x - 3)}$

Combine $5$ and $\frac{x - 3}{(x - 5) (x - 3)}$ .

$x \frac{5 (x - 3)}{(x - 5) (x - 3)}$

Combine $x$ and $\frac{5 (x - 3)}{(x - 5) (x - 3)}$ .

$\frac{x (5 (x - 3))}{(x - 5) (x - 3)}$

Cancel the common factor of $x - 3$ .

Cancel the common factor.

$\frac{x (5 (x - 3))}{(x - 5) (x - 3)}$

Rewrite the expression.

$\frac{x \cdot (5)}{x - 5}$

Move $5$ to the left of $x$ .

$\frac{5 x}{x - 5}$

Simplify ((5x)/(x+3))÷((x^2-8x+15)/(x^2-9))

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