Problem 13 - Algebra 2 and Trigonometry

Question

Perform the indicated operations and write the result in simplest form. \((a+3)(a-3)\)

Step-by-Step Solution

Answer

The simplified form is \(a^2 - 9\).

Step 1: Understanding the Problem

We need to perform the operations indicated by the expression \((a+3)(a-3)\) and simplify the result. This expression is the product of two binomials.

Step 2: Recognizing the Pattern

The expression \((a+3)(a-3)\) is an example of the difference of squares formula, which states that \((x+y)(x-y) = x^2 - y^2\). Here, \(x = a\) and \(y = 3\).

Step 3: Applying the Difference of Squares Formula

Using the difference of squares formula, we write \((a+3)(a-3)\) as \(a^2 - 3^2\).

Step 4: Simplifying the Expression

Now compute the squares: \(a^2 - 3^2 = a^2 - 9\). This is the simplest form of the expression after performing the operations.

Key Concepts

Binomials

Binomials are mathematical expressions that consist of two terms connected by a plus or minus sign. They are a fundamental building block in algebra and are expressed in the form \(x + y\) or \(x - y\). Binomials appear frequently in algebraic expressions and equations, and they can represent a wide range of mathematical scenarios.

When dealing with problems involving binomials, such as the expression \((a+3)(a-3)\), it is essential to recognize the structure and the operations involved. Identifying the expression as the product of two binomials helps us apply known formulas and simplifies the process of solving the problem. Understanding binomials is crucial, as they serve as the basis for more complex algebraic operations, including multiplication, factoring, and applying various algebraic identities like the difference of squares formula.

Recognizing patterns in binomials can lead to more efficient problem-solving techniques. For example, the expression \((a+3)(a-3)\) is a straightforward application of the difference of squares, which can be solved efficiently by applying the formula directly.

Simplification

Simplification involves reducing an algebraic expression to its simplest form. The goal is to make the expression as basic as possible, while preserving its original value. This simplifies further operations and makes it easier to understand and analyze.

In our example with \((a+3)(a-3)\), simplification is achieved by applying the difference of squares formula. This formula is specifically designed to handle expressions of the form \((x+y)(x-y)\). By recognizing this pattern, we can simplify \((a+3)(a-3)\) to \(a^2 - 9\).

The simplification process allows us to eliminate potential complexities presented by multiplication of binomials. Instead of expanding and combining like terms, recognizing the structure allows for a more direct path to the simplest form. Simplification not only makes expressions easier to work with, but it also highlights the mathematical relationships and properties within those expressions.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations (such as addition, subtraction, multiplication, and division). They form the backbone of algebra and are used to generalize mathematical problems and solutions.

In the problem at hand, the expression \((a+3)(a-3)\) is an algebraic expression that requires simplification to its most basic form, \(a^2 - 9\). This expression is composed of variable \(a\) and number \(3\), manipulated through the binomial operations.

Understanding algebraic expressions includes recognizing their components and how they can be manipulated. Working with expressions such as the difference of squares allows learners to see how particular forms lend themselves to specific formulas, greatly enhancing their ability to solve algebraic equations efficiently.

Algebraic expressions often require patience and careful manipulation. A clear understanding of foundational concepts like binomials and simplification techniques aids in breaking down these expressions into understandable and manageable forms.