Problem 68 - Algebra for College Students

Question

Perform the indicated operations. When possible write down only the answer. $$\frac{\frac{b^{2}-4 a}{2}}{a}$$

Step-by-Step Solution

Answer

\(\frac{b^{2}}{2a} - 2\)

Step 1: Simplify the Numerator

The numerator is \(\frac{b^{2}-4a}{2}\). It consists of a single fraction. There are no operations within the numerator to perform, so we move to the next step.

Step 2: Set up the Division

We need to divide the fraction \(\frac{b^{2}-4a}{2}\) by \(a\). The expression becomes \[\frac{\frac{b^{2}-4a}{2}}{a} = \frac{b^{2}-4a}{2a}\].

Step 3: Simplify the Expression

Now, we simplify the fraction by dividing both the numerator and the denominator by their common factor, 2. The expression simplifies to \[\frac{b^{2}-4a}{2a} = \frac{b^{2}}{2a} - \frac{2 \cdot 2a}{2a} = \frac{b^{2}}{2a} - 2\].

Key Concepts

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form. This makes them easier to work with.
In the given exercise, the expression \(\frac{b^2 - 4a}{2a}\) needs simplification. Simplifying expressions usually involves:

• Identifying any common factors
• Reducing fractions
• Combining like terms

In our case, we first look at the numerator and see that \(\frac{b^2 - 4a}{2}\) is a single fraction, so no further simplification inside the numerator is needed.
Moving on, we express the whole fraction as \(\frac{b^2 - 4a}{2} \div a\) or easier \(\frac{b^2 - 4a}{2a}\). Now see if there is a common factor in the numerator and the denominator. Here, there isn’t one except through individual terms, leading to the simplified form \(\frac{b^2}{2a} - 2\). This step-by-step process helps in breaking down the expression into manageable parts.

Fractions

Understanding fractions is crucial when dealing with rational expressions. A fraction consists of a numerator (top part) and a denominator (bottom part). In the expression \(\frac{b^2 - 4a}{2a}\), \((b^2 - 4a)\) is the numerator and \((2a)\) is the denominator.
Remember that dividing a numerator by a denominator essentially means splitting the numerator into equal parts as indicated by the denominator. In our example, we are dividing \(\frac{b^2 - 4a}{2}\) by \((a)\), turning it into \(\frac{b^2 - 4a}{2a}\).
Fractions can be simplified by finding common factors between the numerator and the denominator. In this problem, you can split \(\frac{b^2 - 4a}{2a}\) into two separate fractions for simplification: \(\frac{b^2}{2a} - \frac{4a}{2a}\), reducing to \(\frac{b^2}{2a} - 2\). This intermediate step helps visualize the separation and simplification of terms.

Algebraic Operations

Algebraic operations involve the basic math operations: addition, subtraction, multiplication, and division, but with algebraic terms. In this particular exercise, the initial operation required is division. We start with the fraction \(\frac{\frac{b^2 - 4a}{2}}{a}\). To solve it, we first recognize that dividing by a variable, in this case, \((a)\), means we distribute it across all terms in the numerator.
By converting the nested fraction to a single fraction, \(\frac{b^2 - 4a}{2a}\), we then simplify it step-by-step. We see that dividing both the numerator and denominator by the same factor (like factoring \((2)\) out and canceling terms) will streamline the process. Finally, simplifying individually leads to \(\frac{b^2}{2a} - 2\).
Consistent practice with these individual steps will help in mastering similar operations in more complex algebraic expressions. Using these basic principles ensures a clear path through seemingly complicated problems.