Simplify ((9-s^2)/(3rs))÷((s^2+2s-15)/(5rs))

$\frac{9 - s^{2}}{3 r s} \div \frac{s^{2} + 2 s - 15}{5 r s}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{9 - s^{2}}{3 r s} \cdot \frac{5 r s}{s^{2} + 2 s - 15}$

Step 2

Simplify terms.

Cancel the common factor of $r s$ .

Factor $r s$ out of $3 r s$ .

$\frac{9 - s^{2}}{r s (3)} \cdot \frac{5 r s}{s^{2} + 2 s - 15}$

Factor $r s$ out of $5 r s$ .

$\frac{9 - s^{2}}{r s (3)} \cdot \frac{r s (5)}{s^{2} + 2 s - 15}$

Cancel the common factor.

$\frac{9 - s^{2}}{r s \cdot 3} \cdot \frac{r s \cdot 5}{s^{2} + 2 s - 15}$

Rewrite the expression.

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

Multiply $\frac{9 - s^{2}}{3}$ by $\frac{5}{s^{2} + 2 s - 15}$ .

$\frac{(9 - s^{2}) \cdot 5}{3 (s^{2} + 2 s - 15)}$

Step 3

Simplify the numerator.

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

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

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

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

Step 4

Factor $s^{2} + 2 s - 15$ 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 $2$ .

$- 3, 5$

Write the factored form using these integers.

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

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

Step 5

Cancel the common factor of $3 - s$ and $s - 3$ .

Rewrite $3$ as $- 1 (- 3)$ .

$\frac{(3 + s) (- 1 (- 3) - s) \cdot 5}{3 (s - 3) (s + 5)}$

Factor $- 1$ out of $- s$ .

$\frac{(3 + s) (- 1 (- 3) - (s)) \cdot 5}{3 (s - 3) (s + 5)}$

Factor $- 1$ out of $- 1 (- 3) - (s)$ .

$\frac{(3 + s) (- 1 (- 3 + s)) \cdot 5}{3 (s - 3) (s + 5)}$

Reorder terms.

$\frac{(3 + s) (- 1 (s - 3)) \cdot 5}{3 (s - 3) (s + 5)}$

Cancel the common factor.

$\frac{(3 + s) (- 1 (s - 3)) \cdot 5}{3 (s - 3) (s + 5)}$

Rewrite the expression.

$\frac{((3 + s) \cdot (- 1)) \cdot 5}{3 (s + 5)}$

Step 6

Simplify the expression.

Multiply $5$ by $- 1$ .

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

Move $- 5$ to the left of $3 + s$ .

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

Move the negative in front of the fraction.

$- \frac{5 (3 + s)}{3 (s + 5)}$

Simplify ((9-s^2)/(3rs))÷((s^2+2s-15)/(5rs))

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