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Chapter 6: Problem 45

Perform the indicated operations. Simplify when possible $$ \frac{a+2}{a-4}+\frac{a-2}{a+3} $$

Short Answer

Expert verified
The simplified form is \( \frac{2a^2 - a + 14}{(a-4)(a+3)} \).

Step by step solution

01

- Find common denominator

To add the fractions \(\frac{a+2}{a-4} + \frac{a-2}{a+3} \), first find a common denominator. The common denominator is \( (a-4)(a+3) \).
02

- Rewrite each fraction with the common denominator

Rewrite each fraction so they have the common denominator \( (a-4)(a+3) \). This gives us: \(\frac{(a+2)(a+3)}{(a-4)(a+3)} + \frac{(a-2)(a-4)}{(a-4)(a+3)} \).
03

- Expand the numerators

Expand each numerator: \( (a+2)(a+3) \) and \( (a-2)(a-4) \). This results in: \( \frac{a^2 + 5a + 6}{(a-4)(a+3)} + \frac{a^2 - 6a + 8}{(a-4)(a+3)} \).
04

- Combine the fractions

Since the fractions now have a common denominator, combine them: \(\frac{a^2 + 5a + 6 + a^2 - 6a + 8}{(a-4)(a+3)} \).
05

- Simplify the numerator

Combine like terms in the numerator: \( a^2 + a^2 + 5a - 6a + 6 + 8 = 2a^2 - a + 14 \). So, the expression is \(\frac{2a^2 - a + 14}{(a-4)(a+3)} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Common Denominator
When working with algebraic fractions, the first key step is finding a common denominator. This ensures that the fractions are comparable and can be combined.

In our exercise, we started with the fractions \(\frac{a+2}{a-4}\) and \(\frac{a-2}{a+3}\). To add these fractions, we need the same denominator for both.

The common denominator is simply the product of the distinct denominators: \(a-4\) and \(a+3\). This gives us \((a-4)(a+3)\). Once we have this common denominator, we rewrite each fraction with it. This enables us to combine the fractions in the next step.
Simplifying Fractions
Simplifying fractions is essential for making expressions more manageable.

In our current problem, we rewrite each fraction so they share the common denominator \((a-4)(a+3)\).

After rewriting, we need to expand the numerators:
  • For \(a+2\) and \(a+3\), we get \( (a+2)(a+3) \rightarrow a^2 + 5a + 6 \).
  • For \(a-2\) and \(a-4\), we get \( (a-2)(a-4) \rightarrow a^2 - 6a + 8 \).
Now, both fractions have the common denominator, and we combine them as a single fraction: \(\frac{a^2 + 5a + 6 + a^2 - 6a + 8}{(a-4)(a+3)}\). Simplifying is vital at each step to keep the expression concise.
Combining Like Terms
After obtaining a common denominator and rewriting the fractions, combining like terms is the next step to simplify our expression further.

With the combined fraction \(\frac{a^2 + a^2 + 5a - 6a + 6 + 8}{(a-4)(a+3)}\), we notice terms in the numerator that can be merged.

Summing up similar terms results in:
  • \(a^2 + a^2 \rightarrow 2a^2\)
  • \(5a - 6a \rightarrow -a\)
  • \(6 + 8 \rightarrow 14\)
Now, our expression simplifies to \(\frac{2a^2 - a + 14}{(a-4)(a+3)}\).

Understanding combining like terms allows students to reduce expressions efficiently and solve algebraic problems effectively.

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