Simplify 1/(2x^2-4x)-2/x

$\frac{1}{2 x^{2} - 4 x} - \frac{2}{x}$

Step 1

Factor $2 x$ out of $2 x^{2} - 4 x$ .

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

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

Factor $2 x$ out of $- 4 x$ .

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

Factor $2 x$ out of $2 x (x) + 2 x (- 2)$ .

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

Step 2

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

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

Step 3

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

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

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

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

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

Multiply $- 2$ by $2$ .

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

Apply the distributive property.

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

Multiply $- 4$ by $- 2$ .

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

Add $1$ and $8$ .

$\frac{- 4 x + 9}{2 x (x - 2)}$

Step 6

Simplify with factoring out.

Factor $- 1$ out of $- 4 x$ .

$\frac{- (4 x) + 9}{2 x (x - 2)}$

Rewrite $9$ as $- 1 (- 9)$ .

$\frac{- (4 x) - 1 (- 9)}{2 x (x - 2)}$

Factor $- 1$ out of $- (4 x) - 1 (- 9)$ .

$\frac{- (4 x - 9)}{2 x (x - 2)}$

Simplify the expression.

Rewrite $- (4 x - 9)$ as $- 1 (4 x - 9)$ .

$\frac{- 1 (4 x - 9)}{2 x (x - 2)}$

Move the negative in front of the fraction.

$- \frac{4 x - 9}{2 x (x - 2)}$