Basic Math Examples

$\frac{9 r}{r^{2} - 25} + \frac{r}{r - 5}$

Step 1

Simplify the denominator.

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Step 1.1

Rewrite $25$ as $5^{2}$ .

$\frac{9 r}{r^{2} - 5^{2}} + \frac{r}{r - 5}$

Step 1.2

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 $b = 5$ .

$\frac{9 r}{(r + 5) (r - 5)} + \frac{r}{r - 5}$

Step 2

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

$\frac{9 r}{(r + 5) (r - 5)} + \frac{r}{r - 5} \cdot \frac{r + 5}{r + 5}$

Step 3

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

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Step 3.1

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

$\frac{9 r}{(r + 5) (r - 5)} + \frac{r (r + 5)}{(r - 5) (r + 5)}$

Step 3.2

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

$\frac{9 r}{(r + 5) (r - 5)} + \frac{r (r + 5)}{(r + 5) (r - 5)}$

Step 4

Combine the numerators over the common denominator.

$\frac{9 r + r (r + 5)}{(r + 5) (r - 5)}$

Step 5

Simplify the numerator.

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Step 5.1

Factor $r$ out of $9 r + r (r + 5)$ .

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Step 5.1.1

Factor $r$ out of $9 r$ .

$\frac{r \cdot 9 + r (r + 5)}{(r + 5) (r - 5)}$

Step 5.1.2

Factor $r$ out of $r \cdot 9 + r (r + 5)$ .

$\frac{r (9 + r + 5)}{(r + 5) (r - 5)}$

Step 5.2

Add $9$ and $5$ .

$\frac{r (r + 14)}{(r + 5) (r - 5)}$