What is the difference of the rational expressions below?

[tex]\[
\frac{9}{x^2} - \frac{2x + 1}{8x}
\][/tex]

A. [tex]\(\frac{-2x^2 + x + 72}{8x^2}\)[/tex]

B. [tex]\(\frac{-2x + 10}{8x^2}\)[/tex]

C. [tex]\(\frac{-2x + 8}{8x^2}\)[/tex]

D. [tex]\(\frac{-2x^2 - x + 72}{8x^2}\)[/tex]



Answer :

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]