Expanding & Simplifying: A Math Problem Explained

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Expanding & Simplifying: A Math Problem Explained

Hey guys! Let's dive into this math problem together. It looks a bit intimidating at first, but we'll break it down step by step so it's super clear. We're going to tackle expanding and simplifying the expression: -2x(9x-5)-(8x-3)(5x+6). This is a classic algebra problem that combines the distributive property and combining like terms. So, grab your pencils, and let's get started!

Understanding the Problem

Before we jump into the calculations, let's make sure we understand what the problem is asking. We have an algebraic expression with parentheses, multiplication, and subtraction. Our goal is to expand the expression by getting rid of the parentheses and then simplify it by combining any like terms. Think of it like untangling a knot – we're taking something complex and making it simpler and easier to work with.

Expanding means multiplying the terms inside the parentheses by the terms outside. This uses the distributive property, which is a fundamental concept in algebra. Simplifying involves combining terms that have the same variable and exponent. For example, 3x^2 and 5x^2 are like terms, but 3x^2 and 5x are not. Combining like terms makes the expression cleaner and easier to understand.

The expression -2x(9x-5)-(8x-3)(5x+6) has two main parts. The first part is -2x(9x-5), and the second part is -(8x-3)(5x+6). We'll need to expand each part separately and then combine them. The negative sign in front of the second part is crucial – we need to remember to distribute it carefully to avoid mistakes. Math can be fun, especially when you understand the basics, so let’s continue our journey to solving the problem.

Step-by-Step Solution

Okay, let's break down the solution step-by-step. We'll start with the first part of the expression, -2x(9x-5), and then move on to the second part, -(8x-3)(5x+6). Remember, the key is to take it one step at a time and be super careful with the signs.

Part 1: Expanding -2x(9x-5)

To expand this, we'll use the distributive property. This means we multiply -2x by each term inside the parentheses:

  • -2x * 9x = -18x^2
  • -2x * -5 = +10x

So, the first part expands to -18x^2 + 10x. See? Not so scary when we break it down!

Part 2: Expanding -(8x-3)(5x+6)

This part is a bit trickier because we have two sets of parentheses. We need to multiply each term in the first set of parentheses by each term in the second set. It’s like a mini multiplication table. Let's do it carefully:

  • 8x * 5x = 40x^2
  • 8x * 6 = 48x
  • -3 * 5x = -15x
  • -3 * 6 = -18

So, (8x-3)(5x+6) expands to 40x^2 + 48x - 15x - 18. But wait, we're not done yet! We need to remember the negative sign in front of the parentheses. This means we need to multiply every term in this expansion by -1:

  • -(40x^2) = -40x^2
  • -(48x) = -48x
  • -(-15x) = +15x
  • -(-18) = +18

So, the second part expands and distributes the negative sign to become -40x^2 - 48x + 15x + 18.

Combining the Expanded Parts

Now that we've expanded both parts, we can combine them. We have:

(-18x^2 + 10x) + (-40x^2 - 48x + 15x + 18)

Let's rewrite this without the extra parentheses to make it clearer:

-18x^2 + 10x - 40x^2 - 48x + 15x + 18

Simplifying the Expression

The final step is to simplify the expression by combining like terms. Remember, like terms have the same variable and exponent. Let’s group them together:

Combining x^2 terms

We have -18x^2 and -40x^2. Adding these together, we get:

  • -18x^2 - 40x^2 = -58x^2

Combining x terms

We have 10x, -48x, and 15x. Adding these together, we get:

  • 10x - 48x + 15x = -23x

Constant term

We only have one constant term, which is 18. So, it stays as it is.

Putting it all Together

Now we combine the simplified terms:

  • -58x^2 - 23x + 18

So, the simplified expression is -58x^2 - 23x + 18.

Final Answer

After expanding and simplifying the expression -2x(9x-5)-(8x-3)(5x+6), we arrive at the final answer:

-58x^2 - 23x + 18

And that's it! We've successfully tackled this algebraic problem. Remember, the key to success in math is to break down complex problems into smaller, manageable steps. Let’s recap our steps in a friendly way:

Recapping the Steps

To make sure we've got it all down, let's quickly recap the steps we took to solve this problem. It's always a good idea to review, especially in math, to make sure everything is crystal clear.

  1. Understanding the Problem: We started by identifying what we needed to do – expand and simplify the expression. We recognized the distributive property and the importance of combining like terms.
  2. Expanding the First Part: We expanded -2x(9x-5) by multiplying -2x by each term inside the parentheses, resulting in -18x^2 + 10x. We focused on applying the distributive property correctly.
  3. Expanding the Second Part: This was a bit trickier. We expanded (8x-3)(5x+6) and then distributed the negative sign. This gave us -40x^2 - 48x + 15x + 18. We paid extra attention to the signs to avoid errors.
  4. Combining the Expanded Parts: We combined the expanded forms of both parts of the expression, making sure to keep track of all the terms.
  5. Simplifying the Expression: We combined like terms (x^2 terms, x terms, and constants) to get the final simplified form: -58x^2 - 23x + 18. This step involved adding and subtracting coefficients of like terms.
  6. Final Answer: We clearly stated the final simplified expression.

By following these steps, you can tackle similar problems with confidence. Remember, practice makes perfect, so don't be afraid to try more examples. Now, let's talk about some common mistakes to watch out for.

Common Mistakes to Avoid

When tackling problems like this, it's easy to make little mistakes that can throw off the whole answer. Let's chat about some common pitfalls to watch out for. Knowing these can save you a lot of headaches (and points on a test!).

Sign Errors

One of the most frequent mistakes is messing up the signs, especially when distributing a negative number. For example, in the second part of our problem, -(8x-3)(5x+6), it’s super important to distribute that negative sign to every term after you've expanded the parentheses. Forgetting to do so can flip the signs and change your answer completely. So, always double-check your signs! It's a small thing that makes a big difference.

Incorrect Distribution

Another common mistake is not distributing correctly. Remember, when you have something like -2x(9x-5), you need to multiply -2x by both 9x and -5. Sometimes, people forget to multiply by one of the terms, which leads to an incorrect expansion. Take your time and make sure you're hitting every term inside the parentheses.

Combining Unlike Terms

It's tempting to combine terms that aren't actually like terms, especially when you're in a hurry. But remember, you can only combine terms that have the same variable and the same exponent. So, you can combine 3x^2 and 5x^2, but you can't combine 3x^2 and 5x. Mixing these up will give you the wrong simplified expression. Take that extra moment to check.

Forgetting to Simplify

Sometimes, you might expand the expression correctly but forget to simplify it by combining like terms. This means you've done most of the work but haven't quite finished the job. Always look for like terms after you've expanded and make sure to combine them. It's like baking a cake and forgetting the frosting – it's still good, but not quite complete!

Skipping Steps

It might seem faster to skip steps, but in algebra, that can often lead to mistakes. Writing out each step, especially when you're learning, helps you keep track of what you're doing and reduces the chance of errors. Think of it like following a recipe – each step is important, and skipping one can mess up the final result.

By being aware of these common mistakes, you can avoid them and improve your accuracy. Math is all about precision, so a little extra care can go a long way. Now, let’s see how these concepts apply to other similar problems.

Practice Problems

Alright, guys, let's put our knowledge to the test with a few practice problems. Remember, the key to mastering algebra is practice, practice, practice! We'll work through these together, reinforcing the steps we've learned and building our confidence. So, grab your pencils, and let's dive in!

Practice Problem 1

Expand and simplify: 3x(2x + 4) - (x - 1)(x + 2)

Let’s tackle this one step-by-step:

  1. Expand the first part: 3x(2x + 4) = 6x^2 + 12x
  2. Expand the second part: (x - 1)(x + 2) = x^2 + 2x - x - 2 = x^2 + x - 2
  3. Distribute the negative sign: -(x^2 + x - 2) = -x^2 - x + 2
  4. Combine the expanded parts: 6x^2 + 12x - x^2 - x + 2
  5. Simplify by combining like terms: (6x^2 - x^2) + (12x - x) + 2 = 5x^2 + 11x + 2

So, the simplified expression is 5x^2 + 11x + 2.

Practice Problem 2

Expand and simplify: -2(4x - 3) + (2x + 1)(3x - 2)

Let's break this down:

  1. Expand the first part: -2(4x - 3) = -8x + 6
  2. Expand the second part: (2x + 1)(3x - 2) = 6x^2 - 4x + 3x - 2 = 6x^2 - x - 2
  3. Combine the expanded parts: -8x + 6 + 6x^2 - x - 2
  4. Simplify by combining like terms: 6x^2 + (-8x - x) + (6 - 2) = 6x^2 - 9x + 4

So, the simplified expression is 6x^2 - 9x + 4.

Practice Problem 3

Expand and simplify: (5x - 2)(x + 3) - 4x(2x - 1)

Let’s work through it:

  1. Expand the first part: (5x - 2)(x + 3) = 5x^2 + 15x - 2x - 6 = 5x^2 + 13x - 6
  2. Expand the second part: -4x(2x - 1) = -8x^2 + 4x
  3. Combine the expanded parts: 5x^2 + 13x - 6 - 8x^2 + 4x
  4. Simplify by combining like terms: (5x^2 - 8x^2) + (13x + 4x) - 6 = -3x^2 + 17x - 6

So, the simplified expression is -3x^2 + 17x - 6.

By working through these practice problems, you're not just getting the answers; you're building a deeper understanding of the process. Each problem helps reinforce the steps and concepts we've discussed. The next section gives you tips on how to continue practicing and improving your skills.

Tips for Continued Practice

So, we've tackled some tough problems together, and you're getting the hang of expanding and simplifying algebraic expressions. But math is a skill that gets better with practice, like playing a musical instrument or learning a new language. Here are some tips to keep improving and feeling confident in your abilities.

Work on a Variety of Problems

It's tempting to stick with the types of problems you feel comfortable with, but challenging yourself with different types of questions is really important. Try problems with more parentheses, different signs, or even fractions. The more variety you see, the better you'll become at adapting your skills to any situation. Think of it like a workout for your brain – you need to exercise all your mental muscles!

Use Online Resources and Workbooks

There are tons of amazing resources online that can help you practice. Websites like Khan Academy, IXL, and others offer endless practice problems with step-by-step solutions. These are great for getting immediate feedback and seeing where you might be going wrong. Workbooks are also fantastic because they often have a structured approach, gradually increasing the difficulty of the problems. It's like having a personal tutor at your fingertips!

Practice Regularly

Consistency is key. Even short, regular practice sessions are more effective than long, sporadic ones. Try setting aside 15-30 minutes each day to work on math problems. It's like brushing your teeth – a little bit each day keeps the problems away! Regular practice helps reinforce the concepts in your brain, making them stick better.

Review Mistakes

Everyone makes mistakes, and that's totally okay! The important thing is to learn from them. When you get a problem wrong, take the time to understand why. Go back through your steps and see where you went wrong. Maybe it was a sign error, or maybe you didn't distribute correctly. Identifying your mistakes and understanding how to avoid them is one of the best ways to improve. Mistakes are just stepping stones to success!

Collaborate with Others

Studying with friends or classmates can be super helpful. You can explain concepts to each other, work through problems together, and catch each other's mistakes. Teaching someone else is one of the best ways to solidify your own understanding. Plus, it can make studying more fun! Just make sure you're actually working together and not just copying answers.

Seek Help When Needed

Don't be afraid to ask for help if you're struggling. Talk to your teacher, a tutor, or a friend who's good at math. There's no shame in asking for help – it's a sign that you're serious about learning. Sometimes, a different explanation or a fresh perspective is all you need to break through a difficult concept.

By following these tips, you'll be well on your way to mastering algebra and feeling confident in your math skills. Remember, practice makes progress, and every problem you solve is a step forward.

Conclusion

Alright, guys, we've reached the end of our journey through expanding and simplifying algebraic expressions! We started with a challenging problem, broke it down step-by-step, and learned a ton along the way. Remember, math might seem intimidating at first, but with a little patience and a lot of practice, you can conquer any problem. We've covered everything from understanding the distributive property to combining like terms, and we've even talked about common mistakes to avoid.

The key takeaway is that breaking down complex problems into smaller, manageable steps makes them much easier to handle. We expanded expressions piece by piece, paying close attention to signs and making sure to distribute correctly. Then, we simplified by combining like terms, which cleaned up the expressions and made them easier to work with. It’s like organizing a messy room – tackle one area at a time, and before you know it, everything is tidy and in its place.

We also emphasized the importance of practice. Just like any skill, math improves with consistent effort. By working through a variety of problems, using online resources and workbooks, and reviewing your mistakes, you'll build your confidence and become a math whiz in no time. And don't forget to collaborate with others – teaching and learning from your peers is a fantastic way to solidify your understanding.

So, the next time you see an algebraic expression that looks daunting, remember the steps we've learned. Take a deep breath, break it down, and tackle it one step at a time. You've got this! Keep practicing, keep learning, and most importantly, keep believing in yourself. Math can be fun, rewarding, and totally within your reach. Now, go out there and conquer those equations!