Simplifying Algebraic Expressions A Step By Step Guide To 2x(x-6)-7x^2-(13x-3)
Hey there, math enthusiasts! Ever stumbled upon an algebraic expression that looks like a tangled mess? Well, today, we're diving deep into simplifying one such expression. Our mission, should we choose to accept it, is to find the simplest form of the expression 2x(x-6)-7x^2-(13x-3). We'll break it down step by step, ensuring that by the end of this guide, you'll be a pro at simplifying similar expressions. So, grab your pencils, and let's get started!
Understanding the Expression
Before we jump into the nitty-gritty, let's take a good look at the expression we're dealing with: 2x(x-6)-7x^2-(13x-3). At first glance, it might seem a bit intimidating, but don't worry, we'll tackle it piece by piece. The expression involves terms with the variable 'x,' constants, and various mathematical operations such as multiplication, subtraction, and exponentiation. Our goal is to combine like terms and simplify the expression into its most basic form. This involves distributing, combining like terms, and paying close attention to signs and operations. Remember, simplifying algebraic expressions is like solving a puzzle – each step brings us closer to the final answer. And just like any puzzle, it's all about having the right strategy and a keen eye for detail. So, let’s get our strategy in place and start simplifying this expression together!
Initial Breakdown and the Order of Operations
When we first encounter an expression like 2x(x-6)-7x^2-(13x-3), it's crucial to understand the order of operations. You might have heard of the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). This is our golden rule for simplifying expressions. First up, we need to tackle the parentheses. In our expression, we have 2x(x-6) and (13x-3). The first set of parentheses requires us to distribute the 2x across the terms inside. This means we multiply 2x by both x and -6. The second set of parentheses, (13x-3), is preceded by a minus sign, which means we need to distribute the negative sign across these terms. Understanding this initial breakdown is key because it sets the stage for the rest of our simplification process. Missteps here can lead to incorrect answers, so let's take our time and make sure we've got this first step down pat. With a clear understanding of the order of operations and the initial breakdown, we’re well-prepared to move on to the next stage of simplifying our expression.
Step-by-Step Simplification
Now comes the exciting part where we roll up our sleeves and get into the actual simplification process. Remember our expression? It's 2x(x-6)-7x^2-(13x-3). We've already broken it down and understood the order of operations. Now, let’s put that knowledge to work.
Distributing the Terms
Our first task is to distribute the terms. As we discussed, this means multiplying the term outside the parentheses by each term inside. Let's start with 2x(x-6). We multiply 2x by x, which gives us 2x^2. Then, we multiply 2x by -6, resulting in -12x. So, 2x(x-6) simplifies to 2x^2 - 12x. Next, we tackle the second part of the expression, -(13x-3). Here, we're essentially distributing a -1 across the terms inside the parentheses. This means we multiply -1 by 13x, which gives us -13x, and then we multiply -1 by -3, which gives us +3. So, -(13x-3) simplifies to -13x + 3. Now, our expression looks like this: 2x^2 - 12x - 7x^2 - 13x + 3. We’ve successfully distributed all the terms, and the expression is starting to look a little less tangled already! Remember, the key to distribution is careful multiplication and keeping track of those pesky negative signs. With this step completed, we’re ready to move on to the next phase: combining like terms.
Combining Like Terms
With the distribution done, we've got 2x^2 - 12x - 7x^2 - 13x + 3. Now, it's time to gather our like terms. Like terms are those that have the same variable raised to the same power. In our expression, we have terms with x^2, terms with x, and constant terms (those without any variables). Let's start with the x^2 terms. We have 2x^2 and -7x^2. Combining these gives us 2x^2 - 7x^2 = -5x^2. Next, let's look at the terms with x. We have -12x and -13x. Combining these gives us -12x - 13x = -25x. Finally, we have our constant term, which is +3. Since there are no other constant terms, we simply carry it over. Now, let's put it all together. We have -5x^2 from the x^2 terms, -25x from the x terms, and +3 as our constant term. This gives us the simplified expression -5x^2 - 25x + 3. And there you have it! We've successfully combined like terms and simplified our expression. This step is all about careful addition and subtraction, making sure we're only combining terms that truly belong together. With this accomplished, we’re one step closer to our final answer. Let's take a moment to appreciate how much simpler the expression looks now!
Final Simplified Form
After our meticulous journey through distribution and combining like terms, we've arrived at the final simplified form of our expression. Remember, we started with 2x(x-6)-7x^2-(13x-3), and through careful steps, we transformed it into a much cleaner and easier-to-understand form. So, what's the final verdict? The simplest form of the expression is -5x^2 - 25x + 3. This is our final answer, the result of our hard work and attention to detail. It's a testament to the power of algebraic simplification – taking a complex expression and boiling it down to its essence. This final form is not only simpler but also makes it easier to work with in further mathematical operations or problem-solving scenarios. So, let's give ourselves a pat on the back for a job well done! We've successfully navigated the twists and turns of algebraic simplification and emerged victorious with a clear and concise answer. But our journey doesn't end here; let’s take a moment to reflect on the process and reinforce our understanding.
Reflecting on the Process and Key Takeaways
Now that we've successfully simplified the expression 2x(x-6)-7x^2-(13x-3) to -5x^2 - 25x + 3, it's a great time to reflect on the process and highlight some key takeaways. Simplifying algebraic expressions isn't just about following steps; it's about understanding the underlying principles and developing a systematic approach. One of the most important lessons is the significance of the order of operations (PEMDAS). We saw how crucial it was to distribute terms correctly and then combine like terms. Another key takeaway is the importance of attention to detail, especially when dealing with negative signs. A small mistake in sign can throw off the entire solution. We also learned the value of breaking down a complex problem into smaller, manageable steps. By tackling each part of the expression methodically, we were able to simplify it without feeling overwhelmed. This approach can be applied to many other areas of math and problem-solving in general. Finally, we reinforced the concept of like terms and why they can be combined. Understanding this concept is fundamental to simplifying algebraic expressions. So, as we conclude this guide, remember these key takeaways. They'll not only help you simplify expressions but also build a stronger foundation in algebra. With practice and a clear understanding of these principles, you'll be simplifying expressions like a pro in no time!
Choosing the Correct Option
Now that we've simplified the expression 2x(x-6)-7x^2-(13x-3) and arrived at the final form -5x^2 - 25x + 3, it's time to connect our solution to the options provided. This is an important step because it ensures that we not only understand the simplification process but also know how to apply it in a multiple-choice context. Let's revisit the options:
A. 5x^2 - 25x + 3 B. -5x^2 - 25x + 3 C. -5x^2 + x + 3 D. -5x^2 - 25x - 3
Comparing our simplified expression, -5x^2 - 25x + 3, with the options, it's clear that option B matches our solution perfectly. Options A, C, and D have different coefficients or signs, making them incorrect. This step highlights the importance of accuracy throughout the simplification process. A single mistake can lead to an incorrect final form, which might match one of the distractor options. Therefore, it's always a good idea to double-check your work, especially in a multiple-choice setting. Choosing the correct option is the final validation of our simplification efforts. It confirms that we've not only understood the process but also executed it flawlessly. So, congratulations! We've successfully simplified the expression and identified the correct option. Our journey through this problem is complete!
Conclusion: Mastering Algebraic Simplification
Alright, folks, we've reached the end of our journey to simplify the expression 2x(x-6)-7x^2-(13x-3). We started with a seemingly complex expression, navigated through the intricacies of distribution and combining like terms, and emerged victorious with the simplest form: -5x^2 - 25x + 3. But this wasn't just about solving one problem; it was about mastering the art of algebraic simplification. We've learned the importance of the order of operations, the power of breaking down problems into smaller steps, and the necessity of careful attention to detail. We've also reinforced the fundamental concept of like terms and how to combine them effectively. These skills are not just applicable to this specific problem but are essential tools in your mathematical arsenal. They'll help you tackle a wide range of algebraic challenges with confidence and precision. So, as you continue your mathematical journey, remember the lessons we've learned here. Practice regularly, stay curious, and never shy away from a challenging problem. With dedication and the right approach, you can conquer any algebraic expression that comes your way. Keep simplifying, keep learning, and most importantly, keep enjoying the beauty of mathematics!
