Problem 96 - Elementary Algebra Within Reach

Question

Simplify the expression. $$a^{2}+3 a+4-2 a-6$$

Step-by-Step Solution

Answer

The simplified expression is \(a^{2} + a - 2\).

Step 1: Combine Like Terms

Focus on the 'a' terms: \(a^{2}\), \(3a\), and \(-2a\). Adding these together, we get \(a^{2}\) as the 'a' term because \(3a - 2a = a\).

Step 2: Combine Constant Terms

Next, we combine the constants 4 and -6 by subtracting them. This gives us \(-2\) as the result.

Step 3: Write down the Simplified Expression

Now, we combine the 'a' term and constant term again to create the final, simplified expression.

Key Concepts

Combining Like Terms

Understanding how to identify and combine like terms is critical in simplifying algebraic expressions. Like terms refer to variables or constants that have the same variables raised to the same powers. In the given exercise, we look at the terms containing 'a'. We have three terms: a^2, 3a, and -2a. To combine like terms featuring the variable 'a', we add or subtract the coefficients (the numbers in front of the variables) while keeping the variable part intact. Here we have 3a and -2a which can be combined because they both have the variable 'a' to the first power.

Therefore, adding their coefficients gives us 3 - 2 = 1, resulting in a as the simplified term. This process simplifies the expression and makes it more manageable. To master combining like terms, it's helpful to group them visually or mentally and then perform the algebraic operations.

Algebraic Operations

Algebraic operations are the foundation of simplifying expressions in elementary algebra. These operations include addition, subtraction, multiplication, and division, which we use to combine or separate algebraic terms. In the context of our problem, we have to perform both addition and subtraction. After handling the terms with 'a', we focus on the constant terms, which are 4 and -6. Since they do not have any variables attached, they are like terms and can be combined directly.

By performing the subtraction 4 - 6, we get -2. We then write this along with our previously combined term featuring 'a' to achieve the simplified expression a^2 + a - 2. It is important when working with algebraic operations to pay careful attention to the signs (positive or negative) of each term, as they dictate the operation required to combine the terms correctly.

Elementary Algebra

Elementary algebra encompasses the basic principles of algebra, including the use of symbols and letters to represent numbers and quantities in formulas and equations. The goal is to develop the skill to manipulate and resolve these expressions to reach a simplified form. Our exercise illustrates this well: starting from an expanded algebraic expression and applying the principles of like terms and algebraic operations to simplify it.

By systematically grouping like terms and applying the correct operations, we progress from a cluttered expression to a more coherent and simplified one. The final outcome of our exercise a^2 + a - 2 is a manifestation of applying elementary algebra concepts effectively. Learning to recognize patterns in expressions and operating systematically ensures that students can tackle increasingly complex algebraic challenges with confidence.