To simplify the expression [tex]\(\frac{x-3}{x-1}+\frac{6}{x-3}\)[/tex], let's follow a detailed, step-by-step process.
1. Find a common denominator:
The denominators of the fractions are [tex]\(x-1\)[/tex] and [tex]\(x-3\)[/tex]. The common denominator is [tex]\((x-1)(x-3)\)[/tex].
2. Rewrite each fraction with the common denominator:
[tex]\[
\frac{x-3}{x-1} = \frac{(x-3)(x-3)}{(x-1)(x-3)}
\][/tex]
[tex]\[
\frac{6}{x-3} = \frac{6(x-1)}{(x-1)(x-3)}
\][/tex]
3. Combine the fractions into a single fraction:
[tex]\[
\frac{(x-3)(x-3) + 6(x-1)}{(x-1)(x-3)}
\][/tex]
4. Expand and simplify the numerator:
First, expand [tex]\((x-3)(x-3)\)[/tex]:
[tex]\[
(x-3)(x-3) = x^2 - 6x + 9
\][/tex]
Then, expand [tex]\(6(x-1)\)[/tex]:
[tex]\[
6(x-1) = 6x - 6
\][/tex]
Combine these results in the numerator:
[tex]\[
x^2 - 6x + 9 + 6x - 6
\][/tex]
Simplify the numerator by combining like terms:
[tex]\[
x^2 - 6x + 6x + 9 - 6 = x^2 + 3
\][/tex]
5. Write the simplified form of the fraction:
[tex]\[
\frac{x^2 + 3}{(x-1)(x-3)}
\][/tex]
Thus, the simplest form of the expression is:
[tex]\[
\boxed{\frac{x^2 + 3}{(x-1)(x-3)}}
\][/tex]
Therefore, the correct answer is:
A. [tex]\(\frac{x^2+3}{(x-1)(x-3)}\)[/tex]
