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Magazine Chapter 0: Problem 81
Simplify. $$ \frac{x+3}{x-5}+\frac{5}{x-3} $$
Short Answer
Expert verified
\(\frac{x^2 + 5x - 34}{(x-5)(x-3)}\)
Step by step solution
01
Identify the common denominator
To combine the two fractions, we need a common denominator. The denominators we have are \(x-5\) and \(x-3\). The least common denominator (LCD) is the product of these two expressions, \((x-5)(x-3)\).
02
Rewrite fractions with the common denominator
Adjust each fraction to have the common denominator \((x-5)(x-3)\). For the first fraction, \(\frac{x+3}{x-5}\), multiply both the numerator and denominator by \(x-3\). It becomes \(\frac{(x+3)(x-3)}{(x-5)(x-3)}\). For the second fraction, \(\frac{5}{x-3}\), multiply both the numerator and denominator by \(x-5\). It becomes \(\frac{5(x-5)}{(x-3)(x-5)}\).
03
Expand the numerators
Expand the numerators of both fractions: \( (x+3)(x-3) = x^2 - 9 \) and \( 5(x-5) = 5x - 25 \). This gives us \(\frac{x^2 - 9}{(x-5)(x-3)}\) for the first fraction and \(\frac{5x - 25}{(x-3)(x-5)}\) for the second fraction.
04
Add the fractions
Since both fractions now have a common denominator, we can add them by combining their numerators over the common denominator: \(\frac{x^2 - 9 + 5x - 25}{(x-5)(x-3)}\).
05
Simplify the combined numerator
Combine like terms in the numerator: \(x^2 + 5x - 9 - 25 = x^2 + 5x - 34\). So the expression is \(\frac{x^2 + 5x - 34}{(x-5)(x-3)}\).
06
Write the final simplified expression
The expression cannot be factored further so the final simplified expression is \(\frac{x^2 + 5x - 34}{(x-5)(x-3)}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Common Denominator
In algebraic fractions, a common denominator is pivotal for combining fractions effectively. It is especially important because fractions can only be added or subtracted when their denominators match. Here's how you find it:
- Identify the unique denominators in the problem. Here, they are \(x-5\) and \(x-3\).
- Calculate the least common denominator (LCD), which is often the product of the denominators when dealing with polynomial expressions. For our example, the LCD is \((x-5)(x-3)\).
Simplifying Expressions
Simplifying expressions often follows the process of expanding and combining terms to make the equation as concise and clear as possible. Here's a breakdown:
- After establishing a common denominator, rewrite each expression such that their denominators match. Multiply the numerator and the denominator by the necessary terms to achieve this.
- For instance, the expressions in the example are rewritten with the common denominator: \(\frac{(x+3)(x-3)}{(x-5)(x-3)}\) and \(\frac{5(x-5)}{(x-3)(x-5)}\).
- Expand the numerators: simplify \((x+3)(x-3)\) to get \(x^2 - 9\) and \(5(x-5)\) to get \(5x - 25\).
Combining Fractions
Combining fractions involves carrying out addition or subtraction after rewriting them to have a common denominator. Here's how it works:
- With both fractions now sharing the common denominator \((x-5)(x-3)\), adding them becomes straightforward.
- Add their numerators together: \(x^2 - 9\) from the first fraction and \(5x - 25\) from the second. This results in the new numerator: \(x^2 + 5x - 34\).
- The combined fraction then becomes \(\frac{x^2 + 5x - 34}{(x-5)(x-3)}\).
