To find the difference of the rational expressions [tex]\(\frac{9}{x^2} - \frac{2x + 1}{8x}\)[/tex], we need to follow these steps:
1. Find a common denominator:
The denominators in the given expressions are [tex]\(x^2\)[/tex] and [tex]\(8x\)[/tex]. The least common denominator (LCD) of these expressions is [tex]\(8x^2\)[/tex].
2. Rewrite each expression with the common denominator:
The first expression:
[tex]\[
\frac{9}{x^2} = \frac{9 \cdot 8}{x^2 \cdot 8} = \frac{72}{8x^2}
\][/tex]
The second expression:
[tex]\[
\frac{2x + 1}{8x} = \frac{2x + 1 \cdot x}{8x \cdot x} = \frac{2x^2 + x}{8x^2}
\][/tex]
3. Subtract the second expression from the first:
Now perform the subtraction:
[tex]\[
\frac{72}{8x^2} - \frac{2x^2 + x}{8x^2}
\][/tex]
Since the denominators are the same, we can combine the numerators:
[tex]\[
\frac{72 - (2x^2 + x)}{8x^2} = \frac{72 - 2x^2 - x}{8x^2}
\][/tex]
4. Simplify the numerator:
Rewrite the expression clearly:
[tex]\[
\frac{-2x^2 - x + 72}{8x^2}
\][/tex]
Therefore, the simplified form of the given subtraction is:
[tex]\[
\frac{-2x^2 - x + 72}{8x^2}
\][/tex]
Hence, the correct answer is:
D. [tex]\(\frac{-2x^2 - x + 72}{8x^2}\)[/tex]
