Simplifying Algebraic Expressions A Step-by-Step Guide

Hey guys! Let's dive into some algebra today and break down how to simplify the expression (x² + 3x + 7) - (-6x² - 2x + 5). It might look a bit intimidating at first, but don't worry, we'll go through it step by step, making sure it's super clear and easy to understand. We're going to turn this mathematical puzzle into a piece of cake. Think of it as unlocking a secret code, where each step reveals a little more until we get to the satisfying final answer. So, grab your pencils and let’s get started!

Understanding the Basics of Algebraic Expressions

Before we even think about tackling this specific problem, it's super important to make sure we're all on the same page when it comes to the basic building blocks of algebra. So, what exactly is an algebraic expression? Well, simply put, it's a combination of variables (those sneaky letters like 'x' and 'y'), constants (just regular numbers), and mathematical operations (like adding, subtracting, multiplying, and dividing). Think of it like a mathematical phrase – it expresses a relationship between different quantities.

Now, let's zoom in on the individual components. Variables are like placeholders – they represent unknown values that can change. They're the mystery ingredients in our mathematical recipe. Constants, on the other hand, are the known ingredients – they stay the same no matter what. And the operations are the instructions for how to combine these ingredients. We're talking addition (+), subtraction (-), multiplication (*), and division (/). Mastering these operations, especially when dealing with negative numbers, is crucial for simplifying expressions correctly. Remember, subtracting a negative is the same as adding a positive, and vice versa. This little trick is a lifesaver when you're working through problems like the one we're about to tackle.

Another key concept to wrap your head around is the idea of like terms. These are terms that have the same variable raised to the same power. For example, 3x² and -5x² are like terms because they both have x² in them. But 3x² and 3x are not like terms because one has x² and the other just has x. Why does this matter? Because we can only combine like terms! It’s like saying you can only add apples to apples, not apples to oranges. When you're simplifying, you'll be on the lookout for these like terms so you can group them together and make the expression cleaner and easier to work with. Understanding like terms is the secret sauce to simplifying algebraic expressions efficiently and accurately.

Step 1: Distributing the Negative Sign

Okay, so we've got our expression: (x² + 3x + 7) - (-6x² - 2x + 5). The very first thing we need to do is deal with that pesky negative sign sitting outside the parentheses. This is a crucial step, and it's where a lot of folks can make mistakes if they rush through it. What we're actually doing here is distributing the negative sign – essentially multiplying every term inside the second set of parentheses by -1. Think of it like the negative sign is a little ninja that sneaks in and changes the signs of everything inside the house (the parentheses).

So, how does this work in practice? Well, let's take it term by term. We've got - (-6x²). When we multiply -1 by -6x², we get +6x². Remember, a negative times a negative equals a positive! Next up is - (-2x). Again, multiplying -1 by -2x gives us +2x. And finally, we have - (+5). This one's a bit more straightforward: -1 multiplied by +5 is simply -5. See? The ninja has been busy changing signs!

Now, let's rewrite our expression with the negative sign distributed. We started with (x² + 3x + 7) - (-6x² - 2x + 5), and after distributing the negative sign, we're left with: x² + 3x + 7 + 6x² + 2x - 5. Notice how all the signs inside the second set of parentheses have flipped? The -6x² became +6x², the -2x became +2x, and the +5 became -5. This is the heart of this step – making sure those signs are correct. If you get the signs wrong here, the rest of the simplification will be off. So, double-check your work and make sure you've distributed that negative sign like a pro. This step is like laying the foundation for a building; if it's not solid, the whole thing could crumble!

Step 2: Combining Like Terms

Alright, now that we've successfully distributed the negative sign, we're ready for the next exciting step: combining like terms! Remember what we talked about earlier – like terms are those that have the same variable raised to the same power. It's like sorting your laundry – you put the socks with the socks, the shirts with the shirts, and so on. In our expression, x² + 3x + 7 + 6x² + 2x - 5, we need to identify the terms that are similar so we can combine them.

Let's start with the x² terms. We have x² and +6x². These are definitely like terms because they both have x raised to the power of 2. To combine them, we simply add their coefficients (the numbers in front of the variables). In this case, we have 1x² (remember, if there's no number written, it's understood to be 1) plus 6x², which gives us 7x². So, we've combined our x² terms into a single, neater term.

Next, let's tackle the x terms. We have +3x and +2x. These are also like terms because they both have x raised to the power of 1 (which we usually don't write explicitly). Adding their coefficients, we get 3 + 2 = 5. So, +3x combined with +2x gives us +5x. We're making progress! The expression is getting simpler and simpler.

Finally, let's look at the constants – the plain old numbers without any variables attached. We have +7 and -5. These are like terms too, because they're both just numbers. Combining them is straightforward: 7 - 5 = 2. So, our constant terms combine to give us +2.

Now, let's put it all together. We combined the x² terms to get 7x², the x terms to get +5x, and the constants to get +2. So, our simplified expression is 7x² + 5x + 2. See? That looks much cleaner and less intimidating than the original expression! Combining like terms is like decluttering your mathematical space – it makes everything easier to see and work with. This step is all about organization and attention to detail. Make sure you've identified all the like terms and combined them correctly, and you'll be well on your way to the final answer.

Step 3: Writing the Simplified Expression

Okay, guys, we've done the heavy lifting! We've distributed the negative sign like algebraic ninjas, and we've combined all the like terms like mathematical Marie Kondos. Now comes the really satisfying part: writing out our final, simplified expression. This is where all our hard work pays off, and we get to see the beautifully streamlined result of our efforts. It’s like finally solving a puzzle and seeing the complete picture – super rewarding!

So, let's recap where we are. After distributing the negative sign in the original expression, (x² + 3x + 7) - (-6x² - 2x + 5), we got x² + 3x + 7 + 6x² + 2x - 5. Then, we identified and combined the like terms: the x² terms, the x terms, and the constants. We found that x² + 6x² = 7x², 3x + 2x = 5x, and 7 - 5 = 2.

Now, all that's left to do is put these simplified terms together in the correct order. In algebra, we typically write expressions in descending order of the exponent of the variable. That means we start with the term that has the highest power of x (in this case, x²), then move on to the term with the next highest power of x (which is x), and finally end with the constant term (the number without any x's). It's like arranging your books on a shelf – you want them in a logical order so you can find them easily.

So, following this convention, we write our simplified expression as 7x² + 5x + 2. And there you have it! That's the final answer. We've taken a seemingly complicated expression and, step by step, transformed it into something much simpler and easier to understand. This final step is all about presentation. Writing the expression in the standard order not only looks neat and tidy but also makes it easier for others to read and understand your work. Plus, it’s just good mathematical etiquette! So, take a moment to admire your handiwork – you've successfully simplified an algebraic expression like a true math whiz!

Common Mistakes to Avoid

Alright, now that we've successfully navigated our way through simplifying this algebraic expression, let's take a moment to talk about some common pitfalls that people often stumble into. Knowing these mistakes beforehand can help you steer clear of them and ensure you get the correct answer every time. Think of it as learning the warning signs on a hiking trail – they help you avoid getting lost or injured.

One of the biggest culprits is messing up the distribution of the negative sign. We talked about how crucial this step is, and it's worth hammering home. Remember, when you have a negative sign outside parentheses, you're essentially multiplying every term inside by -1. This means you need to flip the sign of every single term, not just the first one. Forgetting to distribute the negative sign to all terms is like only putting sunscreen on half your face – you're going to get burned in the spots you missed!

Another common mistake is combining unlike terms. We stressed earlier how important it is to only add or subtract terms that have the same variable raised to the same power. Trying to combine x² with x, or x with a constant, is like trying to mix oil and water – it just doesn't work. Always double-check that the terms you're combining are truly