Simplifying Expressions: A Step-by-Step Guide

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Hey guys! Ever feel like math problems are a bit like a tangled ball of yarn? Well, simplifying expressions is all about untangling those knots and making things nice and clear. Today, we're going to break down how to simplify the expression (-3x^2 + x + 5) - (4x^2 - 2x). Don't worry, it's not as scary as it looks. We'll go through it step by step, making sure you understand every move. Ready? Let's dive in and make math a little less intimidating. This guide will walk you through the process, making sure you grasp the concepts and feel confident in tackling similar problems. Let's make math a bit more fun, shall we?

Understanding the Basics of Simplifying Expressions

Before we jump into the problem, let's quickly recap what simplifying expressions means. Simplifying an expression means rewriting it in a more concise form. It's like taking a long sentence and shortening it without losing the original meaning. In math, we do this by combining like terms. Like terms are terms that have the same variables raised to the same powers. For example, 3x^2 and -5x^2 are like terms because they both have x^2. On the other hand, x^2 and x are not like terms because they have different powers of x. When we simplify, we can only add or subtract like terms. This is the golden rule, folks! Now, why is this important? Because simplifying makes it easier to see what an expression represents and to perform further calculations. It's like organizing your closet before finding your favorite shirt; it just makes life easier. In our case, the goal is to combine all similar terms so that the expression becomes as succinct as possible. Let’s get into the main example now. In our case, simplifying is really about combining like terms to get to the simplest form possible.

Now, let's look at the expression (-3x^2 + x + 5) - (4x^2 - 2x). The first thing we need to do is to remove the parentheses. This means we'll apply the subtraction to each term inside the second set of parentheses. Remember that subtracting a term is the same as adding its negative. So, we'll change the sign of each term in (4x^2 - 2x). It becomes -4x^2 + 2x. So, now the expression looks like this: -3x^2 + x + 5 - 4x^2 + 2x. Notice that the signs of the terms inside the second set of parentheses have changed. This is the crucial step where many people might make a mistake, so pay close attention! Once we've done this, the next step is to group like terms together. This makes it easier to combine them. We'll group the x^2 terms, the x terms, and the constant terms separately. You can think of it like sorting different types of fruits: apples with apples, oranges with oranges, and bananas with bananas. This keeps everything organized and prevents any mix-ups. This process helps us keep our work organized and easier to follow.

Step-by-Step Simplification

Alright, let’s get down to business. We’ll follow the steps to simplify the expression (-3x^2 + x + 5) - (4x^2 - 2x) and give you a great explanation!

  • Step 1: Distribute the Negative Sign

    As we mentioned before, the first step is to get rid of those parentheses. Because we're subtracting the entire second expression, we change the sign of each term inside the parentheses. So, -(4x^2 - 2x) becomes -4x^2 + 2x. Our expression now looks like this: -3x^2 + x + 5 - 4x^2 + 2x. This step is all about making sure we account for the subtraction correctly. It is a fundamental process, and it sets the stage for combining like terms. Guys, pay close attention to this part, since it is a common place to make mistakes.

  • Step 2: Group Like Terms

    Next up, let's group those like terms together. This means putting all the x^2 terms together, all the x terms together, and any constants together. We'll rearrange the expression to make it easier to combine them: (-3x^2 - 4x^2) + (x + 2x) + 5. This step is all about organization. Grouping the like terms helps us see exactly which terms we can combine. It makes the next step much simpler and reduces the chance of making a mistake. It is important to group like terms because they are the only ones you can combine.

  • Step 3: Combine Like Terms

    Now that we've grouped our like terms, it’s time to combine them. Remember, we only add or subtract the coefficients (the numbers in front of the variables), and the variables and their exponents stay the same. Let's combine the x^2 terms: -3x^2 - 4x^2 = -7x^2. Then, combine the x terms: x + 2x = 3x. Finally, we have the constant term, which stays as +5. So, putting it all together, we have -7x^2 + 3x + 5. This step is where the magic happens! We're simplifying the expression by performing the actual addition and subtraction. When you add or subtract the coefficients, you are making the expression more concise and easier to understand. The result we get is the simplest form of the initial expression. It’s what we've been aiming for!

Final Answer and Explanation

So, after simplifying (-3x^2 + x + 5) - (4x^2 - 2x), we get -7x^2 + 3x + 5. This matches option D. Congratulations, you've successfully simplified the expression! That wasn't so bad, right?

Let's recap what we did:

  1. We distributed the negative sign to remove the parentheses.
  2. We grouped like terms together.
  3. We combined the like terms by adding or subtracting their coefficients.

This method can be applied to simplify a wide range of algebraic expressions. With practice, you'll become more confident in simplifying expressions, and they'll become a piece of cake. This process not only solves the problem but also provides a systematic method for solving similar problems.

Remember, practice is key. Try solving similar problems on your own to solidify your understanding. The more you practice, the more comfortable you'll become with these types of problems. Now go forth and conquer those expressions!

Answer: D. −7x2+3x+5-7x^2 + 3x + 5

Tips for Success

Here are some quick tips to help you ace these types of problems:

  • Pay attention to the signs: The biggest mistake people make is often related to the signs. Always double-check when distributing the negative sign. Making this step clear will prevent many mistakes!
  • Write out each step: Don't try to skip steps, especially when you're just starting out. Writing out each step clearly helps you avoid errors and keeps your work organized. This will significantly boost your chances of getting the right answers.
  • Double-check your work: Once you've finished, go back and review your work. Make sure you haven't missed any terms or made any calculation errors. Proofreading is as important as the solving steps.
  • Practice, practice, practice: The more you practice, the easier and faster it will become. Work through as many examples as possible. When you do, you start to spot patterns and gain confidence.

By following these tips, you'll be well on your way to mastering the art of simplifying expressions. Remember, guys, math can be fun! With a bit of patience and practice, you can get it.