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

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

Step 1

Simplify the numerator.

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

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

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 $b = 5$ .

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

Step 2

Simplify each term.

Factor $x$ out of $x^{2} - 5 x$ .

Factor $x$ out of $x^{2}$ .

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

Factor $x$ out of $- 5 x$ .

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

Factor $x$ out of $x \cdot x + x \cdot - 5$ .

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

Simplify the denominator.

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

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

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 $b = 5$ .

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

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

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

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

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

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

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

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

Step 6

Combine the numerators over the common denominator.

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

Step 7

Simplify the numerator.

Raise $x + 5$ to the power of $1$ .

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

Raise $x + 5$ to the power of $1$ .

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

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Add $1$ and $1$ .

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

Rewrite $x \cdot x$ as $x^{2}$ .

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

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x + 5$ and $b = x$ .

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

Simplify.

Add $x$ and $x$ .

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

Subtract $x$ from $x$ .

$\frac{(2 x + 5) (0 + 5)}{x (x + 5) (x - 5)} \cdot \frac{(x + 5) (x - 5)}{5}$

Add $0$ and $5$ .

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

Step 8

Reduce the expression by cancelling the common factors.

Cancel the common factor of $5$ .

Factor $5$ out of $(2 x + 5) \cdot 5$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $(x + 5) (x - 5)$ .

Factor $(x + 5) (x - 5)$ out of $x (x + 5) (x - 5)$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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