Problem 31 - Elementary Algebra

Question

Evaluate. \(a_{2}-b_{2},\) where \(a=2\) and \(b=3\)

Step-by-Step Solution

Answer

The result is -5.

Step 1: Understand the Expression

The given expression is related to the squares of the numbers: \(a_{2}\) means \(a^2\), and \(b_{2}\) means \(b^2\). We need to find \(a^2 - b^2\).

Step 2: Substitute Known Values

Insert \(a = 2\) and \(b = 3\) into the expression, resulting in \(2^2 - 3^2\).

Step 3: Calculate the Squares

Compute the squares: \(2^2 = 4\) and \(3^2 = 9\).

Step 4: Perform the Subtraction

Subtract the square of \(b\) from the square of \(a\): \(4 - 9\).

Step 5: Solve the Expression

Perform the calculation: \(4 - 9 = -5\).

Key Concepts

Understanding Squares of Numbers

Squares of numbers are quite simple to understand once you break it down. When you square a number, you multiply that number by itself. For example, the square of 2 is calculated as follows:
\[2^2 = 2 \times 2 = 4\]
Similarly, the square of 3 is:
\[3^2 = 3 \times 3 = 9\]
In algebra, when we say \(a^2\), it means the square of \(a\), and when we say \(b^2\), it means the square of \(b\).

• The squared term is always non-negative.
• The result of squaring a positive or negative number will always be positive.

This concept is useful in a wide range of mathematical problems and algebraic expressions.

Delving into the Substitution Method

The substitution method is an important technique used in algebra to simplify expressions and solve equations. It involves replacing a variable with a known value.
This method requires you to carefully substitute given values into an expression or equation without altering its fundamental structure.
Here's how it works in practice:

• Start with the given expression: \(a^2 - b^2\).
• Identify the values for \(a\) and \(b\) that you need to substitute, in this case, \(a = 2\) and \(b = 3\).
• Substitute these values into the expression to get \(2^2 - 3^2\).

After substitution, you then proceed to solve the expression using arithmetic operations. This method is crucial for solving algebraic problems efficiently.

Simplifying Expressions Step by Step

Simplifying expressions involves breaking down a complex expression into simpler parts to make calculations easier. In this case, we started with the expression \(2^2 - 3^2\).
Here's the step-by-step process:

• Calculate each component's square individually. You find \(2^2 = 4\) and \(3^2 = 9\).
• Next, perform the operation indicated between the squares of these numbers. Here, it's subtraction: \(4 - 9\).
• The result of the subtraction is \(-5\).

By simplifying the expression, each operation is handled one by one, ensuring clarity at every stage. This method is a key part of algebraic problem-solving, helping break down problems into manageable steps.