- Identify the common denominator as \((x-1)(x-3)\).
- Rewrite each fraction to have this common denominator.
- Combine the numerators and simplify the resulting expression to \(\frac{x^2+3}{(x-1)(x-3)}\).
- The correct option is \(\boxed{\frac{x^2+3}{(x-1)(x-3)}}\).
Explanation
- Analyzing the Expression
We are given the rational expression \(\frac{x-3}{x-1} + \frac{6}{x-3}\). Our goal is to simplify this expression by combining the two terms into a single rational fraction. To do this, we need to find a common denominator for the two terms, which is the product of the denominators \((x-1)\) and \((x-3)\). We must also note that the expression is undefined when the denominator is zero, specifically for \(x = 1\) and \(x = 3\).
- Finding a Common Denominator
To add the two fractions, we express both with the common denominator \((x-1)(x-3)\).
For the first term \(\frac{x-3}{x-1}\), we multiply both the numerator and the denominator by \((x-3)\):
\[\frac{x-3}{x-1} = \frac{(x-3)(x-3)}{(x-1)(x-3)} = \frac{x^2 - 6x + 9}{(x-1)(x-3)}\]
For the second term \(\frac{6}{x-3}\), we multiply both the numerator and the denominator by \((x-1)\):
\[\frac{6}{x-3} = \frac{6(x-1)}{(x-1)(x-3)} = \frac{6x - 6}{(x-1)(x-3)}\]
- Combining and Simplifying Fractions
Now that both fractions have the same denominator, we can add their numerators:
\[\frac{x^2 - 6x + 9}{(x-1)(x-3)} + \frac{6x - 6}{(x-1)(x-3)} = \frac{(x^2 - 6x + 9) + (6x - 6)}{(x-1)(x-3)}\]
Next, we simplify the numerator by combining like terms:
\[(x^2) + (-6x + 6x) + (9 - 6) = x^2 + 3\]
Thus, the expression simplifies to \(\frac{x^2+3}{(x-1)(x-3)}\). Comparing this to the given choices, it matches the first option.
Examples
Rational expressions appear frequently in engineering and physics, for example, when calculating the combined resistance of a circuit with parallel components or determining the effective speed of an object moving through a fluid. Simplifying these expressions is a foundational skill that helps in solving complex equations more efficiently. By finding a common denominator, you can consolidate multiple physical factors into a single, manageable formula, making it easier to analyze the behavior of the system.