To simplify the given expression [tex]\(\frac{1}{2 x^2-4 x}-\frac{2}{x}\)[/tex], let's follow these steps:
### Step 1: Factor the Denominator of the First Term
The first term is [tex]\(\frac{1}{2 x^2-4 x}\)[/tex]. We can factor the denominator as follows:
[tex]\[
2 x^2 - 4 x = 2x(x-2)
\][/tex]
So, the expression becomes:
[tex]\[
\frac{1}{2x(x-2)}
\][/tex]
### Step 2: Rewriting the Original Expression
The original expression is now:
[tex]\[
\frac{1}{2x(x-2)} - \frac{2}{x}
\][/tex]
### Step 3: Finding a Common Denominator
The common denominator for the two terms is [tex]\(2x(x-2)\)[/tex]. So, we rewrite each fraction with this common denominator.
For the term [tex]\(\frac{2}{x}\)[/tex], we need to multiply the numerator and the denominator by [tex]\(2(x-2)\)[/tex]:
[tex]\[
\frac{2}{x} = \frac{2 \cdot 2(x-2)}{x \cdot 2(x-2)} = \frac{4(x-2)}{2x(x-2)}
\][/tex]
### Step 4: Combining the Fractions
Combine the fractions over the common denominator [tex]\(2x(x-2)\)[/tex]:
[tex]\[
\frac{1}{2x(x-2)} - \frac{4(x-2)}{2x(x-2)} = \frac{1 - 4(x-2)}{2x(x-2)}
\][/tex]
### Step 5: Simplify the Numerator
Now simplify the numerator:
[tex]\[
1 - 4(x-2) = 1 - 4x + 8 = 9 - 4x
\][/tex]
### Step 6: Writing the Final Expression
Combining these results, we get:
[tex]\[
\frac{1 - 4(x-2)}{2x(x-2)} = \frac{9 - 4x}{2x(x-2)}
\][/tex]
So the simplified expression is:
[tex]\[
\frac{9 - 4x}{2x(x-2)}
\][/tex]
### Conclusion
Therefore, the answer is:
[tex]\[
\boxed{\frac{9 - 4x}{2x(x-2)}}
\][/tex]
Now, let's match this with one of the given options. We see that option D is [tex]\(\frac{9 - 4x}{2 x(x-2)}\)[/tex], which matches our simplified expression. Hence, the answer is D.
