Expanding & Simplifying: (3x + 4)(3x - 4) Explained!

by Alex Johnson 53 views

Let's dive into expanding and simplifying the algebraic expression (3x + 4)(3x - 4). This type of problem frequently appears in mathematics, particularly in algebra, and mastering it is crucial for solving more complex equations and problems. In this guide, we will break down the steps involved, explain the underlying principles, and provide examples to ensure a solid understanding. So, grab your pencil and paper, and let's get started!

Understanding the Basics: The Distributive Property and FOIL Method

Before we tackle the specific expression, let's quickly review some foundational concepts. The primary tool we'll use is the distributive property, which states that a(b + c) = ab + ac. This property allows us to multiply a single term by a group of terms within parentheses. For example, 2(x + 3) becomes 2x + 6.

Another helpful technique is the FOIL method, an acronym that stands for First, Outer, Inner, Last. It's a mnemonic device that helps us remember how to multiply two binomials (expressions with two terms). When multiplying (a + b)(c + d), FOIL tells us to:

  • First: Multiply the first terms in each binomial (a * c).
  • Outer: Multiply the outer terms (a * d).
  • Inner: Multiply the inner terms (b * c).
  • Last: Multiply the last terms (b * d).

After applying FOIL, we combine like terms to simplify the expression. While FOIL is useful, it's essentially an application of the distributive property. Let's see how these concepts apply to our problem.

Expanding (3x + 4)(3x - 4): A Step-by-Step Approach

Now, let's apply our knowledge to expand and simplify (3x + 4)(3x - 4). We can use either the distributive property or the FOIL method. Let's use FOIL for this example:

  1. First: Multiply the first terms: (3x) * (3x) = 9x²
  2. Outer: Multiply the outer terms: (3x) * (-4) = -12x
  3. Inner: Multiply the inner terms: (4) * (3x) = 12x
  4. Last: Multiply the last terms: (4) * (-4) = -16

So, after applying FOIL, we have: 9x² - 12x + 12x - 16. The next step is to combine like terms.

Simplifying the Expression: Combining Like Terms

In the expanded expression 9x² - 12x + 12x - 16, we have two terms that are like terms: -12x and +12x. These terms have the same variable (x) raised to the same power (1). When we combine them, we get:

-12x + 12x = 0

Therefore, the simplified expression becomes:

9x² - 16

Notice that the middle terms canceled out. This is a special case we'll discuss in the next section.

Recognizing the Difference of Squares Pattern

The expression (3x + 4)(3x - 4) is an example of a special pattern called the difference of squares. This pattern occurs when we multiply two binomials that are identical except for the sign between the terms. In general, the pattern is:

(a + b)(a - b) = a² - b²

In our case, a = 3x and b = 4. So, applying the difference of squares pattern directly, we get:

(3x + 4)(3x - 4) = (3x)² - (4)² = 9x² - 16

Recognizing this pattern can save you time because you can skip the FOIL method and go straight to the simplified form. It's a valuable tool to have in your mathematical arsenal.

Examples and Practice Problems

To solidify your understanding, let's look at a few more examples:

Example 1: Expand and simplify (2y + 5)(2y - 5)

Using the difference of squares pattern:

(2y + 5)(2y - 5) = (2y)² - (5)² = 4y² - 25

Example 2: Expand and simplify (x - 3)(x + 3)

Using the difference of squares pattern:

(x - 3)(x + 3) = (x)² - (3)² = x² - 9

Example 3: Expand and simplify (4z + 1)(4z - 1)

Using the difference of squares pattern:

(4z + 1)(4z - 1) = (4z)² - (1)² = 16z² - 1

Now, let's try some practice problems:

  1. (a + 7)(a - 7)
  2. (5b - 2)(5b + 2)
  3. (2c + 9)(2c - 9)

Try solving these problems using the difference of squares pattern. The answers are provided at the end of this guide.

Common Mistakes to Avoid

When expanding and simplifying expressions, it's easy to make mistakes. Here are some common pitfalls to avoid:

  • Forgetting the Sign: Pay close attention to the signs (positive and negative) when multiplying terms. A misplaced sign can lead to an incorrect answer.
  • Incorrectly Applying the Distributive Property: Ensure you multiply each term inside the parentheses by the term outside.
  • Combining Unlike Terms: You can only combine terms that have the same variable raised to the same power (like terms). For example, you cannot combine x² and x.
  • Misunderstanding the Difference of Squares: The difference of squares pattern only applies when the binomials are identical except for the sign between the terms. Be sure to check that this condition is met before applying the pattern.

By being mindful of these common mistakes, you can improve your accuracy and avoid errors.

Real-World Applications of Expanding and Simplifying

You might be wondering,