Problem 74 - Elementary and Intermediate Algebra

Question

Perform the operations. $$ (a+12)(a-12) $$

Step-by-Step Solution

Answer

\((a+12)(a-12) = a^2 - 144\).

Step 1: Recognize the Pattern

The expression \((a+12)(a-12)\) is an example of the difference of squares. This is a pattern where the product of a sum and a difference, \((x+y)(x-y)\), results in the difference of two squares, \(x^2-y^2\). Identify \(x = a\) and \(y = 12\).

Step 2: Apply the Difference of Squares Formula

Substitute \(x = a\) and \(y = 12\) into the difference of squares formula, \(x^2 - y^2\). Calculate \(x^2\) as \(a^2\) and \(y^2\) as \(12^2\), which equals 144. The expression becomes \(a^2 - 144\).

Step 3: Write the Simplified Expression

Having used the difference of squares pattern, we simplify the original expression \((a+12)(a-12)\) to \(a^2 - 144\).

Key Concepts

Understanding Algebraic Expressions

Algebraic expressions are a fundamental part of algebra, involving combinations of variables, numbers, and operation symbols. These expressions represent a value or set of values and often feature one or more variables, such as "a" in our example. The expression

• \((a+12)(a-12)\) involves two binomials, which are polynomials with two terms each.

Each binomial consists of a variable "a" and a constant (in this case, 12).The power of algebraic expressions lies in their ability to be manipulated and rewritten using algebraic rules. This is crucial for solving equations and simplifying expressions. When faced with an expression like

• \((a+12)(a-12)\), the goal is to simplify it to find an equivalent, often more straightforward, form.

Understanding these building blocks provides a strong foundation for tackling more complicated problems in algebra.

Simplifying Expressions with Difference of Squares

Simplifying expressions involves rewriting them to reveal underlying patterns or to make them easier to work with. A common pattern is the difference of squares, which simplifies expressions like

• \((a+12)(a-12)\).

This pattern emerges when you multiply a sum and a difference, resulting in

• \(x^2-y^2\).

Here, \((a+12)(a-12) = a^2 - 12^2\), and since \(12^2=144\), it becomes \(a^2-144\).This simplification makes the expression simpler and more elegant. Through this process, complex expressions can often transform into straightforward polynomials or other algebraic forms that are easier to interpret and solve. The key step is recognizing that such patterns exist and applying algebraic rules, like those for the difference of squares, systematically.

Pattern Recognition in Algebra

Recognizing patterns in algebra is essential for efficiently solving problems and simplifying expressions. Patterns in algebra, such as the difference of squares, allow students to transform complex tasks into simpler steps. For instance, identifying that

• \((x+y)(x-y)\)

results in

• \(x^2-y^2\)

can save time and reduce errors. Looking at our initial expression,

• \((a+12)(a-12)\)

noticing the presence of a sum \((a+12)\) and a difference \((a-12)\), hints at this familiar pattern.Practice with pattern recognition improves a student's ability to quickly solve similar problems and strengthens their understanding of how algebraic expressions work. By finding and using reliable patterns, students can tackle algebraic challenges with more confidence and accuracy.