Simplify Algebra: Expand X(4x-2)+x(3x+7) Step-by-Step

Hey everyone! Ever stared at an algebraic expression like x(4x - 2) + x(3x + 7) and thought, "Whoa, where do I even begin?" You're definitely not alone! Algebraic simplification might look a bit intimidating at first glance, but I promise you, it's one of the most fundamental and satisfying skills you can learn in mathematics. It's like being a detective, uncovering the hidden, simpler truth behind a complex facade. Today, we're going to dive deep into exactly how to expand and fully simplify expressions just like our example. We'll break down every single step, making sure you understand the why behind the how. So, grab a cup of coffee, get comfy, and let's unravel this algebraic mystery together, shall we? You're going to feel like a math wizard by the end of this, I guarantee it!

Understanding the Basics: What Are Algebraic Expressions, Anyway?

Alright, guys, before we jump into the nitty-gritty of simplifying x(4x - 2) + x(3x + 7), let's make sure we're all on the same page about what an algebraic expression actually is. Think of it like this: it's a mathematical phrase that can contain numbers (which we call constants), letters (our famous variables like 'x', 'y', or 'a'), and mathematical operations (like addition, subtraction, multiplication, and division). What makes an expression different from an equation is that it doesn't have an equals sign, so we're not solving for 'x'; instead, we're just making it look neater or more manageable. Each part of an expression separated by a plus or minus sign is called a term. For instance, in 4x - 2, 4x is one term and -2 is another. The number multiplying the variable (like the '4' in 4x) is known as the coefficient. Understanding these basic building blocks is super important because when we simplify, we're essentially just reorganizing these terms in a more efficient way.

Now, you might be asking, "Why bother simplifying? Can't I just leave it as it is?" And that, my friends, is a fantastic question! The truth is, simplifying expressions is incredibly valuable for several reasons. First, it makes calculations much easier. Imagine trying to plug in a value for 'x' into a long, messy expression versus a short, crisp one. The simplified version is going to save you tons of time and reduce the chances of making errors. Second, it helps us understand the underlying relationships more clearly. A simplified expression often reveals patterns or structures that were obscured by all the extra clutter. Third, it's a foundational skill for more advanced algebra, calculus, physics, engineering, and pretty much any field that uses mathematics to model the real world. Think of it like tidying up your room; everything is easier to find and work with once it's organized. So, when we tackle an expression like x(4x - 2) + x(3x + 7), our goal is to transform it into its most elegant and condensed form, ready for whatever comes next. The main tools in our simplification toolkit today will be the distributive property and combining like terms, which we'll explore in detail. These concepts aren't just abstract math ideas; they're practical tools that make complex problems much more approachable. Trust me, once you get the hang of these, you'll start seeing the beauty in algebraic expressions!

The Core Skill: Mastering the Distributive Property

Alright, team, let's talk about the absolute superpower of algebraic simplification: the distributive property. If you want to conquer expressions like x(4x - 2) + x(3x + 7), this is your main weapon. Simply put, the distributive property tells us how to handle multiplication when there are terms inside parentheses. It states that if you have a number or a variable (let's call it 'a') multiplied by a sum or difference of other terms inside parentheses (like b + c), you multiply 'a' by each term inside the parentheses. So, a(b + c) becomes ab + ac. Similarly, a(b - c) transforms into ab - ac. It's like you're distributing 'a' to everyone at the party inside the parentheses! Each term gets a share of the multiplication. It's a fair system, right?

Let's look at a simpler example to really nail this down before we hit our main one. Imagine you have 3(y + 5). Here, '3' is our 'a', 'y' is 'b', and '5' is 'c'. Applying the distributive property, we multiply 3 by y and 3 by 5. So, 3 * y gives us 3y, and 3 * 5 gives us 15. Putting it together, 3(y + 5) simplifies to 3y + 15. See? Not too shabby! What if we have negative signs? No problem! Let's try 2(4z - 7). We distribute the 2 to 4z and also to -7. So, 2 * 4z is 8z, and 2 * -7 is -14. Thus, 2(4z - 7) becomes 8z - 14. Always be careful with those negative signs, guys – they're sneaky and can trip you up if you're not paying attention. A common mistake is forgetting to multiply the outside term by every term inside the parentheses, or messing up the signs. Remember, a negative multiplied by a negative gives a positive, and a negative multiplied by a positive gives a negative. If you keep these rules in mind, you'll be golden.

Now, when our 'a' term is a variable, like 'x' in our main problem, the process is exactly the same, but you also need to remember your exponent rules. For example, x * x isn't just x; it's x^2 (x squared). And x * 4x becomes 4x^2. These little details are super important for getting the correct answer. The distributive property is not just a trick; it's a fundamental principle that allows us to break down complex expressions into simpler, additive parts. It's a cornerstone of algebraic manipulation, enabling us to remove parentheses and prepare expressions for the next step: combining like terms. Understanding and applying this property correctly is the first major hurdle you'll clear on your path to becoming an algebraic pro. So, practice a few examples until it feels second nature, because we're about to apply it twice in our main problem!

Step-by-Step Breakdown: Expanding Our Expression, Piece by Piece

Alright, buckle up, everyone! We've covered the basics and mastered the distributive property. Now it's time to put that knowledge into action and tackle our star expression: x(4x - 2) + x(3x + 7). We're going to break this down methodically, piece by piece, just like a pro. Remember, the key here is to apply the distributive property to each set of parentheses separately, and then we'll deal with combining everything later.

Let's focus on the first part of the expression: x(4x - 2). Here, 'x' is our term outside the parentheses, and inside we have 4x and -2. Following the distributive property, we need to multiply 'x' by 4x AND by -2.

1. Multiply x by 4x: When we multiply x by 4x, we're essentially doing (1 * 4) * (x * x). The numbers multiply, and the variables multiply. So, 1 * 4 is 4, and x * x is x^2 (x squared). Therefore, x * 4x gives us 4x^2.
2. Multiply x by -2: Now, we multiply 'x' by -2. This is pretty straightforward: x * -2 simply becomes -2x. Remember to keep that negative sign!

So, the first part, x(4x - 2), expands to 4x^2 - 2x. See? Not so bad, right? We've successfully removed the first set of parentheses!

Now, let's move on to the second part of our expression: x(3x + 7). This is a similar process. We're distributing 'x' to 3x and also to 7.

1. Multiply x by 3x: Just like before, x * 3x means (1 * 3) * (x * x). This results in 3 * x^2, which is 3x^2.
2. Multiply x by 7: Multiplying 'x' by 7 gives us 7x. It's a positive 7, so it becomes +7x.

Thus, the second part, x(3x + 7), expands to 3x^2 + 7x. Awesome work, guys! We've conquered both sets of parentheses. Now, let's put our expanded pieces back together. Our original expression, x(4x - 2) + x(3x + 7), has now become:

(4x^2 - 2x) + (3x^2 + 7x)

Since we're just adding these two expanded parts, we can simply drop the parentheses. So, the expression after expanding everything looks like this:

4x^2 - 2x + 3x^2 + 7x

This is a huge step forward! We've successfully used the distributive property to get rid of all the multiplication involving parentheses. Now we have a longer, but flatter, expression, which is much easier to work with. The next, and final, step in our simplification journey is to gather up all the like terms. This is where we tidy everything up and condense it into its absolute simplest form. But for now, take a moment to appreciate how we broke down a seemingly complex problem into manageable chunks. You're doing great!

Combining Like Terms: The Grand Finale of Simplification

Alright, folks, we're in the home stretch! We've expanded our expression x(4x - 2) + x(3x + 7) into 4x^2 - 2x + 3x^2 + 7x. Now, it's time for the grand finale: combining like terms. This step is all about tidying up and making our expression as neat and concise as possible. Think of it like sorting laundry – you put all the darks together, all the whites together, and all the colors together. You wouldn't mix a sock with a shirt for storage, would you? The same principle applies here!

So, what exactly are