To find the difference of the rational expressions [tex]\(\frac{6x}{x-3} - \frac{5}{x}\)[/tex], we will follow a step-by-step approach.
1. Identify the common denominator:
The denominators of the two fractions are [tex]\(x - 3\)[/tex] and [tex]\(x\)[/tex]. To subtract these fractions, we need a common denominator, which is the product of these denominators:
[tex]\[
(x - 3) \cdot x = x(x - 3) = x^2 - 3x
\][/tex]
2. Rewrite each fraction with the common denominator:
[tex]\[
\frac{6x}{x - 3} = \frac{6x \cdot x}{(x - 3) \cdot x} = \frac{6x^2}{x^2 - 3x}
\][/tex]
[tex]\[
\frac{5}{x} = \frac{5 \cdot (x - 3)}{x \cdot (x - 3)} = \frac{5(x - 3)}{x^2 - 3x}
\][/tex]
3. Expand the second term's numerator:
[tex]\[
\frac{5(x - 3)}{x^2 - 3x} = \frac{5x - 15}{x^2 - 3x}
\][/tex]
4. Form the equations with the common denominator:
[tex]\[
\frac{6x^2}{x^2 - 3x} - \frac{5x - 15}{x^2 - 3x}
\][/tex]
5. Combine the numerators under the common denominator:
[tex]\[
\frac{6x^2 - (5x - 15)}{x^2 - 3x}
\][/tex]
6. Simplify the numerator:
[tex]\[
6x^2 - (5x - 15) = 6x^2 - 5x + 15
\][/tex]
7. Write the simplified form of the fraction:
[tex]\[
\frac{6x^2 - 5x + 15}{x^2 - 3x}
\][/tex]
Thus, the difference of the rational expressions [tex]\(\frac{6x}{x-3} - \frac{5}{x}\)[/tex] is:
[tex]\[
\boxed{\frac{6x^2 - 5x + 15}{x^2 - 3x}}
\][/tex]
The correct answer from the given options is:
D. [tex]\(\frac{6x^2 - 5x + 15}{x^2 - 3x}\)[/tex]
