Simplify r/(r^2-9)-1/(r^2+8r+15)

$\frac{r}{r^{2} - 9} - \frac{1}{r^{2} + 8 r + 15}$

Step 1

Simplify each term.

Simplify the denominator.

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

$\frac{r}{r^{2} - 3^{2}} - \frac{1}{r^{2} + 8 r + 15}$

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

$\frac{r}{(r + 3) (r - 3)} - \frac{1}{r^{2} + 8 r + 15}$

Factor $r^{2} + 8 r + 15$ 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 $8$ .

$3, 5$

Write the factored form using these integers.

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

Step 2

To write $\frac{r}{(r + 3) (r - 3)}$ as a fraction with a common denominator, multiply by $\frac{r + 5}{r + 5}$ .

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

Step 3

To write $- \frac{1}{(r + 3) (r + 5)}$ as a fraction with a common denominator, multiply by $\frac{r - 3}{r - 3}$ .

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

Step 4

Write each expression with a common denominator of $(r + 3) (r - 3) (r + 5)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{r}{(r + 3) (r - 3)}$ by $\frac{r + 5}{r + 5}$ .

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

Multiply $\frac{1}{(r + 3) (r + 5)}$ by $\frac{r - 3}{r - 3}$ .

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

Reorder the factors of $(r + 3) (r - 3) (r + 5)$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

Apply the distributive property.

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

Multiply $r$ by $r$ .

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

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

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

Apply the distributive property.

$\frac{r^{2} + 5 r - r - - 3}{(r + 3) (r + 5) (r - 3)}$

Multiply $- 1$ by $- 3$ .

$\frac{r^{2} + 5 r - r + 3}{(r + 3) (r + 5) (r - 3)}$

Subtract $r$ from $5 r$ .

$\frac{r^{2} + 4 r + 3}{(r + 3) (r + 5) (r - 3)}$

Factor $r^{2} + 4 r + 3$ 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 $3$ and whose sum is $4$ .

$1, 3$

Write the factored form using these integers.

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

Step 7

Cancel the common factor of $r + 3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify r/(r^2-9)-1/(r^2+8r+15)

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