Simplifying Algebraic Expressions How To Simplify (2x-9)(x+6)

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Hey guys! Let's dive into simplifying algebraic expressions. In this article, we're going to break down how to simplify the expression (2x - 9)(x + 6). This kind of problem is super common in algebra, and mastering it will definitely boost your math skills. We'll go through the step-by-step process, so you'll be able to tackle similar problems with confidence. So, grab your pencils, and let's get started!

Understanding the Basics of Expression Simplification

Before we jump into the specific problem, let's quickly recap the fundamental principles of simplifying expressions. At its heart, simplifying an expression means rewriting it in a more manageable or understandable form. In the context of algebraic expressions, this often involves expanding products, combining like terms, and applying the distributive property. Think of it like decluttering your room – you're taking a messy situation and organizing it into something neat and tidy. When we talk about simplifying expressions like (2x - 9)(x + 6), we're essentially looking to get rid of the parentheses and combine any terms that can be combined. This not only makes the expression easier to work with but also helps in solving equations or understanding the behavior of functions. Remember, the goal is always to make the expression as simple and clear as possible without changing its value. This skill is crucial not just for algebra but also for higher-level math like calculus and beyond. The main techniques we'll be using are distribution and combining like terms, so make sure you're comfortable with these concepts before moving forward.

Step-by-Step Expansion Using the Distributive Property

Now, let's get into the nitty-gritty of simplifying (2x - 9)(x + 6). The first thing we need to do is expand the expression. This means we're going to multiply each term in the first set of parentheses by each term in the second set. The most common method for this is often called the FOIL method, which stands for First, Outer, Inner, Last. It’s a handy way to make sure we don't miss any terms. Here’s how it works:

  1. First: Multiply the first terms in each parenthesis: 2x * x = 2x²
  2. Outer: Multiply the outer terms: 2x * 6 = 12x
  3. Inner: Multiply the inner terms: -9 * x = -9x
  4. Last: Multiply the last terms: -9 * 6 = -54

So, after applying the FOIL method, we get: 2x² + 12x - 9x - 54. It looks a bit more complex now, but don't worry! We're just one step away from simplifying it completely. The key here is to take it one step at a time and make sure each term is multiplied correctly. A small mistake in this stage can throw off the entire solution, so double-check your work as you go. Remember, practice makes perfect, and the more you expand these expressions, the easier it will become. Next, we’ll combine the like terms to reach our final simplified expression.

Combining Like Terms for Final Simplification

Okay, we've expanded the expression to 2x² + 12x - 9x - 54. Now, the next step is to combine like terms. What does that mean? Well, like terms are terms that have the same variable raised to the same power. In our expression, we have two terms with x to the power of 1: 12x and -9x. These are our like terms, and we can combine them by simply adding or subtracting their coefficients (the numbers in front of the x).

So, let’s do it: 12x - 9x = 3x. Now we can rewrite our expression as 2x² + 3x - 54. Notice that the 2x² term and the -54 term don't have any like terms, so they stay as they are. This is our final simplified expression! We've taken the original expression (2x - 9)(x + 6) and transformed it into a much cleaner form: 2x² + 3x - 54. This process of combining like terms is essential because it reduces the expression to its simplest form, making it easier to work with in further calculations or problem-solving. Always remember to look for those like terms – they're the key to simplification. And with that, we've simplified the expression! Let's recap our steps and then check our answer against the given options.

Checking the Solution Against the Given Options

Alright, we've gone through the entire simplification process, and we've arrived at the expression 2x² + 3x - 54. Now, the crucial step is to check our solution against the options provided in the question. This is super important because it helps us ensure that we haven't made any mistakes along the way. Let's take a look at the options:

  • A. 2x² + 3x - 54
  • B. 2x² - 54
  • C. 2x² - 3x - 54
  • D. 2x² + 21x - 54

Comparing our solution, 2x² + 3x - 54, with the options, we can clearly see that it matches option A. This is fantastic news! It means we've followed all the steps correctly and our simplification is accurate. Checking against the options is not just a formality; it’s a vital part of the problem-solving process. It gives us confidence in our answer and helps us catch any errors we might have overlooked. So, always make it a habit to double-check your solution. Now that we've confirmed our answer, let's highlight the correct option and wrap up this problem.

Final Answer and Conclusion

So, after meticulously simplifying the expression (2x - 9)(x + 6) and checking our solution against the given options, we've confidently arrived at the correct answer. The simplified form of the expression is 2x² + 3x - 54, which corresponds to option A.

Therefore, the final answer is:

  • *A. 2x² + 3x - 54

Great job, guys! We've successfully tackled this algebraic problem step by step. We started by understanding the basic principles of simplifying expressions, then we expanded the expression using the distributive property (or FOIL method), combined like terms, and finally, checked our solution against the options provided. This process not only helps us solve the problem at hand but also reinforces our understanding of algebraic manipulation. Remember, practice is key in mathematics. The more you solve these kinds of problems, the more comfortable and confident you'll become. Keep up the fantastic work, and you'll be mastering algebraic expressions in no time!

Additional Practice Problems

To really nail this skill, let's look at some extra practice problems. Working through these will help solidify your understanding and boost your confidence. Remember, math is all about practice!

  1. Simplify: (3x + 2)(x - 4)
  2. Simplify: (x - 5)(2x + 1)
  3. Simplify: (4x - 3)(x + 7)
  4. Simplify: (2x + 6)(3x - 2)

Try working these out on your own, following the steps we discussed earlier. Expand the expressions, combine like terms, and double-check your answers. If you get stuck, don't worry! Go back through the steps we covered in this article, and you'll get there. Practice makes perfect, so keep at it!

Common Mistakes to Avoid

When simplifying expressions, it's easy to make small errors that can lead to the wrong answer. Here are some common mistakes to watch out for:

  • Forgetting to distribute: Make sure you multiply each term in the first set of parentheses by each term in the second set.
  • Incorrectly multiplying signs: Pay close attention to positive and negative signs when multiplying. A wrong sign can change the entire answer.
  • Not combining like terms: Remember to combine terms with the same variable and exponent.
  • Making arithmetic errors: Double-check your calculations to avoid simple math mistakes.

By being aware of these common pitfalls, you can significantly reduce your chances of making errors and improve your accuracy. Always take your time, show your work, and double-check each step. With practice and attention to detail, you'll be simplifying expressions like a pro!

Conclusion

Simplifying algebraic expressions like (2x - 9)(x + 6) might seem challenging at first, but with a step-by-step approach and plenty of practice, it becomes much easier. We've walked through the entire process, from expanding the expression using the distributive property to combining like terms and checking our solution. Remember, the key is to break down the problem into manageable steps and pay close attention to detail.

By mastering this skill, you'll be well-equipped to tackle more complex algebraic problems in the future. Keep practicing, stay focused, and don't be afraid to ask for help when you need it. You've got this! And with that, we wrap up this article. Keep practicing and keep learning!