Simplify $7x^2 + 3 - 5(x^2 - 4)$ An Illustrated Guide

Introduction

Hey guys! Ever stared at a math expression and felt like you were trying to decipher an ancient language? Don't worry, we've all been there. Today, we're going to break down a common type of math problem: simplifying expressions. Specifically, we'll tackle the expression $7x^2 + 3 - 5(x^2 - 4)$. It might look intimidating at first glance, but trust me, with a step-by-step approach, it's totally manageable. Our goal here is not just to get the right answer, but also to understand the underlying principles of simplification. This way, you'll be able to confidently handle similar problems in the future. We'll cover everything from the distributive property to combining like terms, making sure you grasp each concept along the way. So, let's dive in and make math a little less mysterious, shall we?

This expression, $7x^2 + 3 - 5(x^2 - 4)$, involves several key mathematical operations: multiplication, subtraction, and addition, as well as the application of the distributive property. To simplify it effectively, we need to follow the order of operations (often remembered by the acronym PEMDAS/BODMAS) and combine like terms. By meticulously working through each step, we can transform the complex-looking expression into its simplest form. Understanding how to simplify expressions like this is fundamental in algebra and crucial for solving more advanced mathematical problems. We'll also explore common pitfalls and mistakes students often make, so you can avoid them and build a solid foundation in algebraic manipulation. Remember, the key to success in math is practice and a clear understanding of the rules. So, grab a pen and paper, and let's get started on simplifying this expression together! By the end of this guide, you'll not only be able to simplify this particular expression but also have a much better grasp of the general techniques involved in simplifying algebraic expressions.

Step 1: Applying the Distributive Property

Okay, first things first, let's talk about the distributive property. This is a cornerstone of simplifying expressions, and it's super important to get it right. In our expression, $7x^2 + 3 - 5(x^2 - 4)$, we see that $-5$ is multiplied by the entire expression inside the parentheses $(x^2 - 4)$. The distributive property tells us that we need to multiply $-5$ by each term inside the parentheses separately. This means we'll multiply $-5$ by $x^2$ and then $-5$ by $-4$. So, let's break it down:

• $-5 * x^2 = -5x^2$
• $-5 * -4 = 20$

Notice that a negative times a negative gives us a positive. This is a crucial rule to remember! Now, we can rewrite our expression, replacing $-5(x^2 - 4)$ with the result we just got. So, $7x^2 + 3 - 5(x^2 - 4)$ becomes $7x^2 + 3 - 5x^2 + 20$. See how we've expanded the expression by applying the distributive property? This is a big step forward in simplifying it. Ignoring the distributive property or applying it incorrectly is a very common mistake, so make sure you've got this down. It's like the foundation of a house – if it's not solid, the rest of the structure won't be either. Remember, always multiply the term outside the parentheses by each term inside. This seemingly small step makes a huge difference in getting to the correct answer. And always pay close attention to the signs (positive or negative) – they can make or break your calculation. With the distributive property under our belts, we're ready to move on to the next step: combining like terms. This is where we'll tidy up the expression even further and bring it closer to its simplest form.

Understanding the distributive property is crucial not just for this specific problem, but for a wide range of algebraic manipulations. It allows us to handle expressions with parentheses and to break them down into more manageable parts. Think of it as unlocking a secret code within the expression. Without the distributive property, we'd be stuck with the parentheses and unable to simplify further. It's also important to practice this step with various examples to build your confidence. Try different numbers and variables to see how the property works in different scenarios. The more you practice, the more natural and intuitive it will become. And remember, it's not just about memorizing the steps – it's about understanding why the distributive property works. This deeper understanding will help you apply it correctly in even the most complex situations. So, take your time, work through some examples, and make sure you're comfortable with this fundamental concept. It's the key to unlocking the rest of the simplification process.

Step 2: Combining Like Terms

Alright, now that we've used the distributive property, our expression looks like this: $7x^2 + 3 - 5x^2 + 20$. The next step is to combine like terms. What does that even mean, right? Well, “like terms” are terms that have the same variable raised to the same power. In our expression, we have two terms with $x^2$: $7x^2$ and $-5x^2$. We also have two constant terms (numbers without variables): $3$ and $20$.

To combine like terms, we simply add or subtract their coefficients (the numbers in front of the variables). So, let's start with the $x^2$ terms:

• $7x^2 - 5x^2 = (7 - 5)x^2 = 2x^2$

Now, let's combine the constant terms:

• $3 + 20 = 23$

We've successfully combined our like terms! This means we can rewrite our expression as $2x^2 + 23$. See how much simpler it looks now? This is the power of combining like terms – it helps us condense the expression into its most basic form. A common mistake students make is trying to combine terms that aren't alike, like adding an $x^2$ term to a constant term. Remember, you can only combine terms that have the same variable raised to the same power. Think of it like this: you can add apples to apples, but you can't add apples to oranges. The same principle applies to algebraic terms. It's also important to pay attention to the signs. Make sure you're correctly adding or subtracting the coefficients based on the signs in front of the terms. A little bit of carefulness here can prevent a lot of errors. By combining like terms, we're essentially tidying up the expression and making it easier to work with. This is a crucial step in simplifying algebraic expressions and solving equations. It's like organizing your tools before you start a project – it makes the whole process much smoother and more efficient. With practice, you'll be able to quickly identify like terms and combine them without any trouble. So, let's move on to the final simplified form and celebrate our mathematical victory!

Combining like terms is a fundamental skill in algebra, and mastering it will make your mathematical journey much smoother. It's not just about following the rules; it's about understanding the logic behind them. When we combine like terms, we're essentially saying,