Simplify 10/(x^2-10x)-x/(10x-100)

$\frac{10}{x^{2} - 10 x} - \frac{x}{10 x - 100}$

Step 1

Simplify each term.

Factor $x$ out of $x^{2} - 10 x$ .

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

$\frac{10}{x \cdot x - 10 x} - \frac{x}{10 x - 100}$

Factor $x$ out of $- 10 x$ .

$\frac{10}{x \cdot x + x \cdot - 10} - \frac{x}{10 x - 100}$

Factor $x$ out of $x \cdot x + x \cdot - 10$ .

$\frac{10}{x (x - 10)} - \frac{x}{10 x - 100}$

Factor $10$ out of $10 x - 100$ .

Factor $10$ out of $10 x$ .

$\frac{10}{x (x - 10)} - \frac{x}{10 (x) - 100}$

Factor $10$ out of $- 100$ .

$\frac{10}{x (x - 10)} - \frac{x}{10 x + 10 \cdot - 10}$

Factor $10$ out of $10 x + 10 \cdot - 10$ .

$\frac{10}{x (x - 10)} - \frac{x}{10 (x - 10)}$

Step 2

To write $\frac{10}{x (x - 10)}$ as a fraction with a common denominator, multiply by $\frac{10}{10}$ .

$\frac{10}{x (x - 10)} \cdot \frac{10}{10} - \frac{x}{10 (x - 10)}$

Step 3

To write $- \frac{x}{10 (x - 10)}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{10}{x (x - 10)} \cdot \frac{10}{10} - \frac{x}{10 (x - 10)} \cdot \frac{x}{x}$

Step 4

Write each expression with a common denominator of $x (x - 10) \cdot 10$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{10}{x (x - 10)}$ by $\frac{10}{10}$ .

$\frac{10 \cdot 10}{x (x - 10) \cdot 10} - \frac{x}{10 (x - 10)} \cdot \frac{x}{x}$

Multiply $\frac{x}{10 (x - 10)}$ by $\frac{x}{x}$ .

$\frac{10 \cdot 10}{x (x - 10) \cdot 10} - \frac{x \cdot x}{10 (x - 10) x}$

Reorder the factors of $x (x - 10) \cdot 10$ .

$\frac{10 \cdot 10}{10 x (x - 10)} - \frac{x \cdot x}{10 (x - 10) x}$

Reorder the factors of $10 (x - 10) x$ .

$\frac{10 \cdot 10}{10 x (x - 10)} - \frac{x \cdot x}{10 x (x - 10)}$

Step 5

Combine the numerators over the common denominator.

$\frac{10 \cdot 10 - x \cdot x}{10 x (x - 10)}$

Step 6

Simplify the numerator.

Multiply $10$ by $10$ .

$\frac{100 - x \cdot x}{10 x (x - 10)}$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$\frac{100 - (x \cdot x)}{10 x (x - 10)}$

Multiply $x$ by $x$ .

$\frac{100 - x^{2}}{10 x (x - 10)}$

Rewrite $100 - x^{2}$ in a factored form.

Rewrite $100$ as $10^{2}$ .

$\frac{10^{2} - x^{2}}{10 x (x - 10)}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 10$ and $b = x$ .

$\frac{(10 + x) (10 - x)}{10 x (x - 10)}$

Step 7

Reduce the expression by cancelling the common factors.

Cancel the common factor of $10 - x$ and $x - 10$ .

Rewrite $10$ as $- 1 (- 10)$ .

$\frac{(10 + x) (- 1 (- 10) - x)}{10 x (x - 10)}$

Factor $- 1$ out of $- x$ .

$\frac{(10 + x) (- 1 (- 10) - (x))}{10 x (x - 10)}$

Factor $- 1$ out of $- 1 (- 10) - (x)$ .

$\frac{(10 + x) (- 1 (- 10 + x))}{10 x (x - 10)}$

Reorder terms.

$\frac{(10 + x) (- 1 (x - 10))}{10 x (x - 10)}$

Cancel the common factor.

$\frac{(10 + x) (- 1 (x - 10))}{10 x (x - 10)}$

Rewrite the expression.

$\frac{(10 + x) \cdot (- 1)}{10 x}$

Simplify the expression.

Move $- 1$ to the left of $10 + x$ .

$\frac{- 1 \cdot (10 + x)}{10 x}$

Move the negative in front of the fraction.

$- \frac{10 + x}{10 x}$

Simplify 10/(x^2-10x)-x/(10x-100)

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