Problem 70 - Elementary and Intermediate Algebra

Question

Perform the operations. $$ (t+2)(t-2) $$

Step-by-Step Solution

Answer

The result is \(t^2 - 4\).

Step 1: Understand the Expression

We have the expression \((t+2)(t-2)\). This is a multiplication of two binomials, and can be solved efficiently using the difference of squares formula.

Step 2: Use the Difference of Squares Formula

The expression \((t+2)(t-2)\) matches the pattern of \((a+b)(a-b)\), which is a special case in algebra known as the difference of squares. The formula for the difference of squares is: \((a+b)(a-b) = a^2 - b^2\).

Step 3: Identify 'a' and 'b'

In our expression, \(a\) is \(t\) and \(b\) is \(2\). Therefore, the terms match the pattern for applying the difference of squares formula.

Step 4: Apply the Formula

Using the values of \(a\) and \(b\) in the formula, we have: \(a^2 - b^2\) which becomes \(t^2 - 2^2\).

Step 5: Simplify the Expression

Calculate \(2^2\):- \(2^2 = 4\).Thus, the expression simplifies to \(t^2 - 4\).

Key Concepts

Algebraic Expressions

In algebra, an expression is a combination of numbers, variables, and operators (like plus and minus signs). When dealing with algebraic expressions, you are working with symbols, mainly letters, that stand in for numbers. These are called variables.
For example, in the expression \( (t+2)(t-2) \), the variable is \( t \), and we have two additional constant terms, 2.Algebraic expressions can be either simple or complex, and are a fundamental part of algebra. They can be:

• Monomial: A single term, like \( 5x \) or \( -3 \).
• Binomial: Two terms, like \( x + y \) or \( a - b \).
• Polynomial: Multiple terms, like \( x^2 + 3x - 5 \).

Understanding these expressions and how they behave when operations are performed on them is key to solving algebraic problems efficiently.

Binomial Multiplication

Binomial multiplication involves multiplying two binomials together. In our specific problem, we had the expression \((t+2)(t-2)\). This is an example of binomial multiplication. There are a few methods to tackle binomial multiplication, but one of the most efficient ways in this particular case is by using the formula for the difference of squares.

The Difference of Squares Formula

The difference of squares is a concept in algebra where you multiply two binomials that have the same terms but opposite signs. The general formula is: \[(a+b)(a-b) = a^2 - b^2\]In our case:

• \( a = t \)
• \( b = 2 \)

This formula provides a shortcut that simplifies the multiplication because it reduces the process to subtraction of squares rather than having to multiply each term individually. It shows the beauty of algebraic manipulation and how formulas can simplify complex operations.

Simplifying Expressions

Once we apply the difference of squares formula, the next step is to simplify the given expression. Simplification makes an expression easier to read and use in calculations. In our example, after the multiplication, we apply the formula:\[t^2 - 2^2\]This simplifies further by evaluating the square of 2, which is simply 4. Hence, the expression becomes:\[t^2 - 4\]

Why Simplify?

Simplification is essential because:

• It reduces the complexity of expressions, making them easier to understand and solve.
• Simplified expressions are easier to work with in further calculations or when substituting values.
• They reveal the core mathematical relationships and patterns.

Thus, simplifying expressions is a crucial skill in algebra, making it easier to analyze and solve problems effectively.