Simplify 5/(x^2-2x)-4/(x^2-x)

$\frac{5}{x^{2} - 2 x} - \frac{4}{x^{2} - x}$

Step 1

Simplify each term.

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

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

$\frac{5}{x \cdot x - 2 x} - \frac{4}{x^{2} - x}$

Factor $x$ out of $- 2 x$ .

$\frac{5}{x \cdot x + x \cdot - 2} - \frac{4}{x^{2} - x}$

Factor $x$ out of $x \cdot x + x \cdot - 2$ .

$\frac{5}{x (x - 2)} - \frac{4}{x^{2} - x}$

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

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

$\frac{5}{x (x - 2)} - \frac{4}{x \cdot x - x}$

Factor $x$ out of $- x$ .

$\frac{5}{x (x - 2)} - \frac{4}{x \cdot x + x \cdot - 1}$

Factor $x$ out of $x \cdot x + x \cdot - 1$ .

$\frac{5}{x (x - 2)} - \frac{4}{x (x - 1)}$

Step 2

To write $\frac{5}{x (x - 2)}$ as a fraction with a common denominator, multiply by $\frac{x - 1}{x - 1}$ .

$\frac{5}{x (x - 2)} \cdot \frac{x - 1}{x - 1} - \frac{4}{x (x - 1)}$

Step 3

To write $- \frac{4}{x (x - 1)}$ as a fraction with a common denominator, multiply by $\frac{x - 2}{x - 2}$ .

$\frac{5}{x (x - 2)} \cdot \frac{x - 1}{x - 1} - \frac{4}{x (x - 1)} \cdot \frac{x - 2}{x - 2}$

Step 4

Write each expression with a common denominator of $x (x - 2) (x - 1)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{5}{x (x - 2)}$ by $\frac{x - 1}{x - 1}$ .

$\frac{5 (x - 1)}{x (x - 2) (x - 1)} - \frac{4}{x (x - 1)} \cdot \frac{x - 2}{x - 2}$

Multiply $\frac{4}{x (x - 1)}$ by $\frac{x - 2}{x - 2}$ .

$\frac{5 (x - 1)}{x (x - 2) (x - 1)} - \frac{4 (x - 2)}{x (x - 1) (x - 2)}$

Reorder the factors of $x (x - 2) (x - 1)$ .

$\frac{5 (x - 1)}{x (x - 1) (x - 2)} - \frac{4 (x - 2)}{x (x - 1) (x - 2)}$

Step 5

Combine the numerators over the common denominator.

$\frac{5 (x - 1) - 4 (x - 2)}{x (x - 1) (x - 2)}$

Step 6

Simplify the numerator.

Apply the distributive property.

$\frac{5 x + 5 \cdot - 1 - 4 (x - 2)}{x (x - 1) (x - 2)}$

Multiply $5$ by $- 1$ .

$\frac{5 x - 5 - 4 (x - 2)}{x (x - 1) (x - 2)}$

Apply the distributive property.

$\frac{5 x - 5 - 4 x - 4 \cdot - 2}{x (x - 1) (x - 2)}$

Multiply $- 4$ by $- 2$ .

$\frac{5 x - 5 - 4 x + 8}{x (x - 1) (x - 2)}$

Subtract $4 x$ from $5 x$ .

$\frac{x - 5 + 8}{x (x - 1) (x - 2)}$

Add $- 5$ and $8$ .

$\frac{x + 3}{x (x - 1) (x - 2)}$

Simplify 5/(x^2-2x)-4/(x^2-x)

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