Simplifying 3x(x-12x) + 3x^2 - 2(x-2)^2 A Step-by-Step Guide

Hey guys! Let's dive into simplifying the algebraic expression 3x(x-12x) + 3x^2 - 2(x-2)^2. This might look a bit daunting at first, but trust me, we'll break it down step by step. Understanding how to simplify such expressions is crucial in algebra, and it's super useful for solving more complex problems later on. We will go through each part of the expression, showing you exactly how to simplify it. Let’s get started and make algebra a little less scary, one step at a time! This comprehensive guide will walk you through each stage, ensuring you grasp not just the 'how' but also the 'why' behind every operation. Stick with us, and you'll be simplifying algebraic expressions like a pro in no time!

Initial Expression: Breaking It Down

Let's start by stating our initial expression clearly. We are dealing with: 3x(x - 12x) + 3x^2 - 2(x - 2)^2. The key to tackling any complex expression is to break it down into smaller, more manageable parts. In this case, we have three main terms to consider. The first term is 3x(x - 12x), which involves distribution and combining like terms. The second term is a simple 3x^2, which is already in a simplified form. Lastly, we have -2(x - 2)^2, which requires us to expand the squared binomial before distributing the constant. When approaching any algebraic problem, make sure to deal with each part one by one. This method prevents errors and makes the entire process easier to follow. Think of it like assembling a puzzle – each piece needs individual attention before they can fit together perfectly. By breaking down this expression, we've already taken the first step toward simplification, guys!

Step 1: Simplifying 3x(x - 12x)

The first part of our expression to tackle is 3x(x - 12x). Inside the parentheses, we have (x - 12x). These are like terms, so we can combine them directly. This gives us -11x inside the parentheses. Now our term looks like 3x(-11x). The next step involves multiplying 3x by -11x. When multiplying terms with the same variable, we multiply the coefficients and add the exponents. Here, 3 multiplied by -11 is -33, and x multiplied by x (which is x^1 * x^1) gives us x^2. So, 3x(-11x) simplifies to -33x^2. Breaking down this part step by step helps ensure accuracy and clarity. This is a classic example of using the distributive property combined with simplifying like terms – skills that are fundamental in algebra. By focusing on the details and handling each operation carefully, we've successfully simplified the first term. On to the next!

Step 2: Addressing the Second Term 3x^2

Moving on, let's consider the second term in our original expression: 3x^2. This term is actually already in its simplest form. There are no operations to perform or like terms to combine here. The coefficient 3 is multiplied by the variable x raised to the power of 2, and that's as simple as it gets. So, there's not much to do here except acknowledge it and carry it forward in our simplification process. Sometimes, recognizing that a term is already simplified is just as crucial as knowing how to simplify others. It saves time and prevents unnecessary steps. So, we just note that 3x^2 remains as 3x^2. This might seem like a small step, but it's an important one in maintaining the integrity of our expression as we move towards the final simplified form. Let’s keep going, guys!

Step 3: Expanding and Simplifying -2(x - 2)^2

Now, let's tackle the third and arguably most complex term: -2(x - 2)^2. The first thing we need to do here is expand the squared binomial (x - 2)^2. Remember that squaring a binomial means multiplying it by itself: (x - 2)(x - 2). To multiply this, we use the FOIL method which is First, Outer, Inner, Last. So, we have:

• First: x * x = x^2
• Outer: x * -2 = -2x
• Inner: -2 * x = -2x
• Last: -2 * -2 = 4

Combining these, we get x^2 - 2x - 2x + 4, which simplifies to x^2 - 4x + 4. Now we need to multiply this result by -2. Distributing the -2 across each term in the trinomial gives us:

-2 * x^2 = -2x^2

-2 * -4x = 8x

-2 * 4 = -8

So, -2(x - 2)^2 simplifies to -2x^2 + 8x - 8. This step is a great example of why paying close attention to detail is so important in algebra. From squaring the binomial to distributing the constant, each operation needs to be performed accurately to arrive at the correct simplified form. By breaking it down into these mini-steps, we minimize the chances of making mistakes and keep the process manageable. Great work, guys! We're getting closer to the final answer!

Step 4: Combining Like Terms

Okay, we've simplified each part of our expression individually. Now it's time to bring everything together and combine like terms. Our expression currently looks like this:

-33x^2 + 3x^2 - 2x^2 + 8x - 8

We have three terms involving x^2: -33x^2, 3x^2, and -2x^2. Let's combine these first. -33x^2 + 3x^2 gives us -30x^2. Then, -30x^2 - 2x^2 results in -32x^2. So, the x^2 terms combined simplify to -32x^2.

Next, we have the term 8x. There are no other x terms to combine it with, so it stays as it is.

Finally, we have the constant term -8. Again, there are no other constant terms, so it remains -8.

Putting it all together, our simplified expression is -32x^2 + 8x - 8. This step is where all our previous work comes together, and it's super satisfying to see the expression finally simplified. Combining like terms is a fundamental skill in algebra, and it's essential for solving equations and simplifying more complex expressions. We've done a fantastic job getting here, guys! But we have one more step to ensure our result is in the best form.

Step 5: The Final Simplified Expression

After combining like terms, we've arrived at the expression -32x^2 + 8x - 8. This is the simplified form of our original expression, guys! We've taken a complex algebraic expression and, through careful step-by-step simplification, reduced it to its most basic form. This final expression is much easier to work with in further algebraic manipulations, such as solving equations or graphing functions. Simplifying expressions is a key skill in algebra, and mastering it opens the door to tackling more advanced topics. We've covered expanding, distributing, combining like terms, and paying attention to detail – all crucial elements in this process. So, congratulations on making it to the end! You've successfully simplified the expression 3x(x - 12x) + 3x^2 - 2(x - 2)^2. Keep practicing, and you'll become even more confident in your algebra skills!

True Statements About the Process and Simplified Product

Now, let's circle back to the original question and identify the true statements about the process we followed and the simplified product we obtained. Remember, the original question asked us to select three options that accurately describe the simplification process and the final result. Here’s a recap of our journey:

1. First, we broke down the complex expression into smaller, manageable parts.
2. Next, we simplified 3x(x - 12x) by combining like terms inside the parentheses and then distributing.
3. Then, we acknowledged that 3x^2 was already in its simplest form.
4. After that, we tackled -2(x - 2)^2 by first expanding the squared binomial (x - 2)^2 using the FOIL method and then distributing the -2.
5. Finally, we combined all like terms to arrive at the simplified expression -32x^2 + 8x - 8.

Based on these steps, we can confidently identify several true statements. Let's analyze some potential statements to see which ones fit:

• Statement 1: The term -2(x-2)^2 is simplified by first squaring the expression x-2. This statement is TRUE. We indeed started by expanding (x - 2)^2 before multiplying by -2.
• Statement 2: The simplified product is [insert product here]. To determine the truth of this statement, you'd need to compare the provided product with our result, -32x^2 + 8x - 8. If they match, the statement is true; otherwise, it's false.
• Statement 3: [Another statement about the process or product]. Analyze this statement based on the steps we took and the final result we obtained. Does it accurately describe a step in our process or a characteristic of the final simplified expression?

Remember, the key here is to carefully review each statement in the context of the steps we performed and the final expression we derived. Select the three statements that are undeniably true based on our work. You got this, guys!

Final Thoughts

Simplifying algebraic expressions can seem like a puzzle, but with the right approach and a bit of practice, it becomes a whole lot easier. We've walked through each step of simplifying 3x(x - 12x) + 3x^2 - 2(x - 2)^2, from breaking down the expression to combining like terms and arriving at the final simplified form: -32x^2 + 8x - 8. Remember, the key is to take things one step at a time, pay close attention to detail, and double-check your work along the way. Algebra is a foundational skill in mathematics, and mastering these simplification techniques will serve you well in more advanced math courses and real-world problem-solving. So, keep practicing, keep exploring, and don't be afraid to tackle those algebraic challenges head-on. You're doing great, guys, and every expression you simplify brings you one step closer to math mastery! Keep up the awesome work!