Hey guys! Ever stumbled upon an algebraic expression that looks like a jumbled mess? Don't worry, we've all been there. Today, we're going to break down a common type of problem: simplifying the expression (3p + 4q) multiplied by (6p + 6q). This might seem intimidating at first, but with a little bit of algebraic know-how, we can solve it together. So, grab your pencils and paper, and let's dive in!

Understanding the Basics: The Distributive Property

Before we jump into the main problem, let's quickly refresh our memory on a fundamental concept: the distributive property. This property is the key to unlocking expressions like the one we're tackling today. In simple terms, the distributive property states that a(b + c) = ab + ac. What does this mean? It means that if you have a number (or variable) multiplied by a sum inside parentheses, you need to multiply that number (or variable) by each term inside the parentheses.

Think of it like this: you're hosting a party, and you need to give each of your guests a goodie bag. The 'a' is you, and the '(b + c)' represents your guests. The distributive property says that you need to make sure each guest ('b' and 'c') gets their own individual goodie bag ('ab' and 'ac').

For example, let's say we have 2(x + 3). Using the distributive property, we multiply 2 by both 'x' and '3'. This gives us 2 * x + 2 * 3, which simplifies to 2x + 6. See? Not so scary after all!

This property is crucial when dealing with expressions like (3p + 4q) * (6p + 6q) because it allows us to break down the problem into smaller, more manageable pieces. Mastering the distributive property is like having a superpower in algebra – it allows you to conquer all sorts of seemingly complex expressions with ease. So, make sure you have a solid understanding of this concept before moving on!

Applying the Distributive Property: The FOIL Method

Now that we've got the distributive property under our belts, let's move on to a handy technique called FOIL. FOIL is an acronym that stands for First, Outer, Inner, Last. It's a mnemonic device that helps us remember how to apply the distributive property when multiplying two binomials (expressions with two terms), like our (3p + 4q) and (6p + 6q).

Here's how FOIL works:

• First: Multiply the first terms of each binomial. In our case, that's 3p * 6p.
• Outer: Multiply the outer terms of the expression. That's 3p * 6q.
• Inner: Multiply the inner terms of the expression. That's 4q * 6p.
• Last: Multiply the last terms of each binomial. That's 4q * 6q.

Let's apply this to our problem:

(3p + 4q) * (6p + 6q)

• First: 3p * 6p = 18p²
• Outer: 3p * 6q = 18pq
• Inner: 4q * 6p = 24pq
• Last: 4q * 6q = 24q²

So, after applying the FOIL method, we get: 18p² + 18pq + 24pq + 24q². Notice that we've broken down the original problem into four simpler terms. Now, all that's left to do is simplify! The FOIL method is just a structured way of applying the distributive property, ensuring that we multiply each term in the first binomial by each term in the second binomial. It's a powerful tool for simplifying these types of expressions.

Combining Like Terms: Simplifying the Expression

Okay, we're almost there! We've applied the distributive property (using the FOIL method) and now we have: 18p² + 18pq + 24pq + 24q². The final step is to combine like terms. Like terms are terms that have the same variable(s) raised to the same power. In our expression, the like terms are 18pq and 24pq. They both have the variables 'p' and 'q' raised to the power of 1.

To combine like terms, we simply add their coefficients (the numbers in front of the variables). So, 18pq + 24pq = (18 + 24)pq = 42pq.

Now, let's rewrite our expression with the combined like terms: 18p² + 42pq + 24q². And that's it! We've successfully simplified the expression (3p + 4q) * (6p + 6q).

Therefore, the final answer is 18p² + 42pq + 24q².

Combining like terms is an essential step in simplifying algebraic expressions. It allows us to reduce the expression to its most concise form, making it easier to understand and work with. Always remember to look for like terms after applying the distributive property. Mastering this skill will greatly improve your ability to solve algebraic problems.

Let's Recap: A Step-by-Step Guide

To make sure we're all on the same page, let's quickly recap the steps we took to simplify the expression (3p + 4q) * (6p + 6q):

1. Understand the Distributive Property: Make sure you know how to multiply a term by a sum inside parentheses.
2. Apply the FOIL Method: Use FOIL (First, Outer, Inner, Last) to multiply the two binomials.
3. Combine Like Terms: Add the coefficients of terms with the same variable(s) raised to the same power.

By following these steps, you can simplify a wide variety of algebraic expressions. Remember, practice makes perfect! The more you work with these concepts, the more comfortable you'll become with them.

Common Mistakes to Avoid

While simplifying algebraic expressions might seem straightforward, there are a few common mistakes that students often make. Let's take a look at some of them so you can avoid them:

• Forgetting to Distribute: One of the biggest mistakes is forgetting to multiply every term inside the parentheses by the term outside. Remember, the distributive property applies to all terms inside the parentheses.
• Incorrectly Combining Like Terms: Make sure you only combine terms that have the same variable(s) raised to the same power. For example, you can combine 2x and 3x, but you can't combine 2x and 3x².
• Sign Errors: Pay close attention to the signs (positive and negative) when multiplying and combining terms. A simple sign error can throw off your entire answer.
• Rushing Through the Problem: Take your time and double-check your work. It's better to be accurate than fast. Rushing can lead to careless mistakes.

By being aware of these common mistakes, you can increase your chances of getting the correct answer and avoid unnecessary frustration. Always double-check your work and pay attention to detail!

Practice Problems

Now that you've learned the steps and know what mistakes to avoid, it's time to put your knowledge to the test! Here are a few practice problems for you to try:

1. (2x + 3y) * (4x + y)
2. (5a - 2b) * (a + 3b)
3. (x - 4) * (x + 4)

Work through these problems step-by-step, using the techniques we've discussed. Check your answers with a friend or teacher to make sure you're on the right track. The more you practice, the more confident you'll become in your ability to simplify algebraic expressions.

Conclusion

So there you have it! Simplifying the expression (3p + 4q) * (6p + 6q) is a breeze once you understand the distributive property, the FOIL method, and how to combine like terms. Remember to take your time, double-check your work, and practice, practice, practice!

Algebra can be a challenging subject, but with the right tools and techniques, you can conquer any problem that comes your way. Keep practicing, stay curious, and don't be afraid to ask for help when you need it. You've got this! Now go out there and simplify some expressions, guys!