Question

select the correct answer. what is the simplest form of this expression? \\(\boldsymbol{\frac{x - 3}{x - 1} + \frac{6}{x - 3}}\\) a. \\(\frac{x + 3}{(x - 1)(x - 3)}\\) b. \\(\frac{x^2 + 12x + 15}{(x - 1)(x - 3)}\\) c. \\(\frac{x^2 - 6x + 3}{(x - 1)(x - 3)}\\) d. \\(\frac{x^2 + 3}{(x - 1)(x - 3)}\\)

Explanation:

Step1: Find common denominator

The common denominator is $(x-1)(x-3)$. Rewrite each fraction:
$\frac{(x-3)(x-3)}{(x-1)(x-3)} + \frac{6(x-1)}{(x-1)(x-3)}$

Step2: Expand numerators

Calculate each expanded numerator:
$(x-3)^2 = x^2 - 6x + 9$, $6(x-1) = 6x - 6$

Step3: Combine numerators

Add the expanded numerators over the common denominator:
$\frac{(x^2 - 6x + 9) + (6x - 6)}{(x-1)(x-3)}$

Step4: Simplify numerator

Combine like terms in the numerator:
$x^2 - 6x + 9 + 6x - 6 = x^2 + 3$

Answer:

D. $\frac{x^2 + 3}{(x-1)(x-3)}$