Problem 32 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Perform the indicated operations and simplify. \(2+\frac{1}{a+2}-\frac{2 a}{a-2}\)

Step-by-Step Solution

Answer

The simplified expression for the given problem is: \(-\frac{2a^2 + 3a + 2}{(a+2)(a-2)}\)

Step 1: Identify the common denominator

Identify the common denominator for the fractions. In this case, the denominators are \((a+2)\) and \((a-2)\). The common denominator will be the product of these two, which is \((a+2)(a-2)\).

Step 2: Rewrite the fractions with the common denominator

Now let's rewrite the fractions with the common denominator, \((a+2)(a-2)\): \[\frac{1}{a+2}=\frac{1(a-2)}{(a+2)(a-2)}=\frac{a-2}{(a+2)(a-2)}\] \[\frac{2a}{a-2}=\frac{2a(a+2)}{(a-2)(a+2)}=\frac{2a(a+2)}{(a+2)(a-2)}\]

Step 3: Perform addition and subtraction

Now we perform the addition and subtraction operations of the rewritten fractions: \[2+\frac{a-2}{(a+2)(a-2)}-\frac{2a(a+2)}{(a+2)(a-2)}\] Note that we can simply subtract the numerators, as the denominators are the same: \[\frac{(a-2)-2a(a+2)}{(a+2)(a-2)}\]

Step 4: Simplify the numerator

Now, let's simplify numerator of the fraction: \[(a-2)-2a(a+2)= a - 2 - 2a^2 - 4a = -2a^2 - 3a - 2\]

Step 5: Write the simplified expression

Combine the simplified numerator with the denominator to get the final simplified expression: \[-\frac{2a^2+ 3a + 2}{(a+2)(a-2)}\] As we simplified the expression, this is the final answer.

Key Concepts

Common Denominator

When working with fractions, especially in algebraic expressions, finding a common denominator is crucial. A common denominator is a shared multiple of the denominators of two or more fractions. This step is needed to perform addition or subtraction among these fractions.

In the given problem, the denominators are \(a+2\) and \(a-2\). The simplest common denominator is their product, \( (a+2)(a-2) \). Here’s why:

• Multiplying \(a+2\) by \(a-2\) gives us a single expression that accommodates both fractions.
• This allows for seamless integration when adding or subtracting.

Remember, finding that shared denominator aligns fractions for further operations without changing their value. It's an essential skill for fraction operations in algebra.

Fraction Operations

Operating on fractions (addition, subtraction, etc.) requires particular attention to maintaining a common denominator. This allows changes made to the numerators to be harmonized without altering the overall meaning of the fractions.

Here's how it works with our example:

• Once the common denominator \( (a+2)(a-2) \) is in place, rewrite each fraction accordingly.
• This involves adjusting the numerators so that they align with the new denominator.

In algebraic expressions, this process might include:

• Multiplying the numerator of each fraction by whatever quantity the denominator was multiplied by.
• Ensuring every operation obeys the fundamental rules of algebra, such as distribution.

Once the fractions are rewritten, you can proceed with adding or subtracting them by combining the numerators while keeping the denominator constant.

Expression Simplification

Simplifying an expression is often the goal when handling algebraic problems. It involves reducing an expression to its simplest form while maintaining equivalency. In our continued example, after aligning fractions with a common denominator, we simplified the numerator:

• Using the formula from subtraction, we saw \( (a-2) - 2a(a+2) \) needed simplifying.
• Distribute any multiplication through to clear parentheses: \( -2a^2 - 4a \).
• Combine like terms to collapse the expression further.

This leaves a more concise form: \(-2a^2 - 3a - 2\). Combining these elements into one simplified numerator over the common denominator furnishes us with the simplest form. Simplification not only eases understanding but also makes further calculation straightforward and accurate.