Simplify 13/(x-5)+x/(x^2-25)

$\frac{13}{x - 5} + \frac{x}{x^{2} - 25}$

Simplify the denominator.

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

$\frac{13}{x - 5} + \frac{x}{x^{2} - 5^{2}}$

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 $b = 5$ .

$\frac{13}{x - 5} + \frac{x}{(x + 5) (x - 5)}$

To write $\frac{13}{x - 5}$ as a fraction with a common denominator, multiply by $\frac{x + 5}{x + 5}$ .

$\frac{13}{x - 5} \cdot \frac{x + 5}{x + 5} + \frac{x}{(x + 5) (x - 5)}$

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

Multiply $\frac{13}{x - 5}$ by $\frac{x + 5}{x + 5}$ .

$\frac{13 (x + 5)}{(x - 5) (x + 5)} + \frac{x}{(x + 5) (x - 5)}$

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

$\frac{13 (x + 5)}{(x + 5) (x - 5)} + \frac{x}{(x + 5) (x - 5)}$

Combine the numerators over the common denominator.

$\frac{13 (x + 5) + x}{(x + 5) (x - 5)}$

Simplify the numerator.

Apply the distributive property.

$\frac{13 x + 13 \cdot 5 + x}{(x + 5) (x - 5)}$

Multiply $13$ by $5$ .

$\frac{13 x + 65 + x}{(x + 5) (x - 5)}$

Add $13 x$ and $x$ .

$\frac{14 x + 65}{(x + 5) (x - 5)}$