Problem 13 - Algebra 2 and Trigonometry

Question

In \(3-20\) , perform the indicated additions or subtractions and write the result in simplest form. In each case, list any values of the variables for which the fractions are not defined. $$ a-\frac{3}{2 a} $$

Step-by-Step Solution

Answer

The result is \( \frac{2a^2 - 3}{2a} \), undefined for \( a = 0 \).

Step 1: Identify the Expression

The given expression is \( a - \frac{3}{2a} \). We must perform subtraction and write the result in its simplest form.

Step 2: Common Denominator

To subtract \( \frac{3}{2a} \) from \( a \), we need a common denominator. The first term \( a \) can be rewritten as \( \frac{2a^2}{2a} \) to have the same denominator as the second term.

Step 3: Subtract Fractions

Now that both terms have the same denominator, subtract the numerators: \( \frac{2a^2}{2a} - \frac{3}{2a} = \frac{2a^2 - 3}{2a} \).

Step 4: Simplest Form

The expression is already simplified because \( 2a^2 - 3 \) cannot be factored further, and there are no common factors with the denominator \( 2a \). Thus, the simplest form is \( \frac{2a^2 - 3}{2a} \).

Step 5: Determine Undefined Values

The expression is undefined when the denominator is zero. Solve \( 2a = 0 \) to find that \( a = 0 \) makes the expression undefined.

Key Concepts

Subtraction of Fractions

Subtracting fractions may seem like a tricky task, especially when variables are involved. However, it's much more straightforward once you grasp the underlying process. When subtracting fractions, the key is to focus on the numerators, but only after ensuring both fractions have the same denominator. In the expression \( a - \frac{3}{2a} \), the first step was to rewrite \( a \) as a fraction with a common denominator. This meant converting \( a \) to \( \frac{2a^2}{2a} \) so it shared the base with \( \frac{3}{2a} \). By aligning the denominators, you can then subtract the numerators directly: \( 2a^2 \) from \( 3 \), leading to \( \frac{2a^2 - 3}{2a} \). This process helps streamline the subtraction, making complex algebraic expressions more manageable. Remember, always begin with the denominator when tackling subtraction between fractions.

Common Denominator

Finding a common denominator is a crucial step when dealing with the subtraction or addition of fractions. The primary reason for finding a common denominator is to ensure that we can directly combine the fractions by subtracting or adding their numerators. When the fractions share this common base, it effectively "unifies" them, allowing for straightforward arithmetic between their numerators.

In our expression, \( a - \frac{3}{2a} \), the common denominator was chosen as \( 2a \). Transforming \( a \) to \( \frac{2a^2}{2a} \) was essential because it allowed us to rewrite the entire expression with \( 2a \) as the denominator. This transformation is a powerful way to simplify operations within algebraic fractions, making the math more intuitive and succinct. So, whenever you work with fractions, ensure you find that connecting denominator. It's your ticket to seamless addition or subtraction of even the most daunting algebraic fractions.

Undefined Values

In math, especially when dealing with fractions, it's vital to determine values that make an expression undefined. Typically, an expression is undefined when its denominator equals zero. In algebraic expressions, this can cause the expression to lose its meaning, much like our question's denominator turned void when evaluating \( 2a = 0 \).

For the equation \( \frac{2a^2 - 3}{2a} \), solving for the undefined values means setting \( 2a \) equal to zero and solving for \( a \). The result, \( a = 0 \), tells us this is the undefined point of the expression. Such undefined regions can heavily influence the valid range of solutions or input, often denoted as restrictions or exclusions in algebraic problems.

Always make it a habit to seek out and list these undefined values when dealing with fractions, as they provide crucial insights into the problems' constraints and help ensure the algebraic solutions remain valid and accurate.