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

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

Step 1

Simplify the denominator.

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

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

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 $b = 5$ .

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

Step 2

To write $\frac{x^{2}}{x - 5}$ as a fraction with a common denominator, multiply by $\frac{x + 5}{x + 5}$ .

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

Step 3

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

Multiply $\frac{x^{2}}{x - 5}$ by $\frac{x + 5}{x + 5}$ .

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

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

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

Factor $x$ out of $x + x^{2} (x + 5)$ .

Raise $x$ to the power of $1$ .

$\frac{x^{1} + x^{2} (x + 5)}{(x + 5) (x - 5)}$

Factor $x$ out of $x^{1}$ .

$\frac{x \cdot 1 + x^{2} (x + 5)}{(x + 5) (x - 5)}$

Factor $x$ out of $x^{2} (x + 5)$ .

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

Factor $x$ out of $x \cdot 1 + x (x (x + 5))$ .

$\frac{x (1 + x (x + 5))}{(x + 5) (x - 5)}$

Apply the distributive property.

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

Multiply $x$ by $x$ .

$\frac{x (1 + x^{2} + x \cdot 5)}{(x + 5) (x - 5)}$

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

$\frac{x (1 + x^{2} + 5 \cdot x)}{(x + 5) (x - 5)}$

Reorder terms.

$\frac{x (x^{2} + 5 x + 1)}{(x + 5) (x - 5)}$