Error In Expanding (x-2)(3x+4): Find The Mistake!

by Alex Johnson 50 views

Have you ever been working on an algebra problem and felt like you were on the right track, only to end up with the wrong answer? It's a frustrating experience, but a common one! Math, especially algebra, requires careful attention to detail. Even a small slip-up can throw off the entire solution. Let's dive into a problem where we need to identify the exact step where a mistake was made. We'll break down each step, making sure we understand the logic and calculations involved. By the end of this, you'll not only know the answer but also have a better grasp of how to avoid similar errors in the future. So, let’s put on our detective hats and find the mistake!

The Problem Unveiled

Let's consider the problem of expanding the expression (x-2)(3x+4). This is a classic algebra problem that involves the distributive property, a fundamental concept in mathematics. The goal is to multiply these two binomials (expressions with two terms) together correctly. Seems straightforward, right? But as we'll see, there are plenty of opportunities for small errors to creep in. We're given a step-by-step solution, but somewhere along the way, a mistake happened. Our job is to pinpoint exactly where. It's like finding a needle in a haystack, but with our understanding of algebra, we can do it! Each step has a purpose, and each calculation needs to be accurate. Let’s look at the proposed steps and break them down one by one to identify the culprit.

Step-by-Step Breakdown: Finding the Flaw

Here's the breakdown of the steps we're given, followed by a detailed analysis to help you understand the correct approach and spot the error:

  1. (x-2)(3x) + (x-2)(4)

    • What's happening here? This step uses the distributive property, which is the cornerstone of expanding expressions like this. The distributive property states that a(b+c) = ab + ac. In our case, we're distributing the (x-2) term across the (3x+4) term. This means we multiply (x-2) by 3x and then multiply (x-2) by 4, adding the results together. Think of it like sharing – everything inside the first set of parentheses needs to be multiplied by each term inside the second set.
    • Is it correct? Yes! This step correctly applies the distributive property. We've taken the original expression and rewritten it in a way that's mathematically equivalent but sets us up for the next stage of simplification. No errors detected here!

  2. (x)(3x) + (-2)(3x) + (x)(4) + (-2)(4)

    • The next level of distribution: This step takes the result from Step 1 and applies the distributive property again, but this time within each of the two terms. We're essentially expanding each product. We multiply x by 3x, -2 by 3x, x by 4, and -2 by 4. It’s like we're peeling away the layers of the expression, making it more detailed.
    • Is it correct? Absolutely! Each multiplication is performed correctly, maintaining the integrity of the expression. We're breaking it down piece by piece, and so far, so good. The signs are correct, and the terms are properly multiplied.

  3. 3x² + 6x + 4x - 8

    • Simplifying the terms: This is where things get interesting! In this step, we're multiplying the individual terms from Step 2. Let's take a close look. (x)(3x) becomes 3x², (-2)(3x) should be -6x, (x)(4) becomes 4x, and (-2)(4) becomes -8. This step involves basic multiplication and combining like terms, which are fundamental skills in algebra.
    • Spot the mistake! Did you see it? The error lies in the second term. (-2)(3x) is incorrectly written as +6x. It should be -6x. This seemingly small error is enough to throw off the entire rest of the calculation. This is why paying attention to detail is so crucial in math.

  4. 3x² + 10x + 8

    • Combining like terms (incorrectly): This step attempts to combine the like terms from Step 3. Like terms are those that have the same variable raised to the same power. In this case, 6x and 4x are like terms. However, because of the error in Step 3, the combination is also incorrect. If Step 3 had been correct (3x² - 6x + 4x - 8), this step would have involved combining -6x and 4x.
    • Incorrect due to previous error: This step is incorrect because it's based on the faulty result from Step 3. The incorrect +6x term leads to an incorrect final expression. Had Step 3 been correct, this step would have been different and, hopefully, accurate.

The Verdict: Step 3 is the Culprit

After careful examination, the mistake occurs in Step 3. The term (-2)(3x) was incorrectly calculated as +6x instead of -6x. This single error cascades through the rest of the solution, leading to an incorrect final answer. This highlights the importance of carefully checking each step in a mathematical problem. Even a small sign error can have significant consequences.

Correcting the Mistake: A Fresh Start

Let's correct the mistake and solve the problem properly. This will solidify our understanding and show how a single correction can lead to the right answer.

  1. (x-2)(3x) + (x-2)(4) (Correct - Distributive Property)
  2. (x)(3x) + (-2)(3x) + (x)(4) + (-2)(4) (Correct - Expanding Each Term)
  3. 3x² - 6x + 4x - 8 (Corrected - Notice the -6x)
  4. 3x² - 2x - 8 (Correct - Combining Like Terms)

By correcting the error in Step 3, we arrive at the correct expanded form of the expression, which is 3x² - 2x - 8. This demonstrates the ripple effect of errors in math and the importance of accuracy at each step.

Why These Mistakes Happen (and How to Avoid Them)

Sign errors are a very common type of mistake in algebra. They often happen when we're working quickly or trying to juggle multiple steps in our head. Here are a few tips to help you avoid these kinds of errors:

  • Write every step: Don't try to skip steps, especially when dealing with negative signs. Writing each step out explicitly helps you keep track of the details.
  • Double-check your signs: Before moving on to the next step, take a moment to double-check that you've handled the signs correctly. It's a small effort that can save you a lot of trouble.
  • Use the distributive property carefully: When distributing, make sure you multiply each term inside the parentheses by the term outside. Pay close attention to the signs.
  • Stay organized: Keep your work neat and organized. This makes it easier to spot errors and follow your own logic.
  • Practice makes perfect: The more you practice, the more comfortable you'll become with these concepts and the less likely you'll be to make mistakes.

Mastering the Distributive Property: Your Key to Algebraic Success

The distributive property is a cornerstone of algebra. It's used in many different types of problems, from expanding expressions to solving equations. The better you understand and can apply this property, the more successful you'll be in algebra. Make sure you practice using the distributive property with different types of expressions, including those with negative signs and multiple terms. Try working through a variety of problems, and don't be afraid to make mistakes – they're a natural part of the learning process. The key is to learn from your mistakes and keep practicing.

Wrapping Up: Spotting Errors and Building Confidence

Finding the mistake in this problem was like detective work! We carefully analyzed each step, looking for the one small error that threw off the entire solution. By identifying the mistake in Step 3, we not only corrected the problem but also reinforced our understanding of the distributive property and the importance of accuracy in algebra. Remember, everyone makes mistakes sometimes. The key is to learn from them and develop strategies to avoid them in the future. Math can be challenging, but with practice and a careful approach, you can build your confidence and succeed. Keep practicing, keep asking questions, and don't give up!

For further learning and practice on similar algebraic concepts, explore resources like Khan Academy's Algebra Section.