Solving The First-Degree Equation: A Step-by-Step Guide

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Hey guys! Let's dive into solving the first-degree equation: 3x² - 2(2x² + 5) = 4. This might look a bit intimidating at first glance, but trust me, breaking it down step by step makes it totally manageable. We'll walk through the process together, making sure we understand each part. Ready? Let's get started!

Understanding the Basics: What's a First-Degree Equation?

Before we jump in, let's quickly recap what a first-degree equation actually is. Basically, it's an equation where the highest power of the variable (in our case, 'x') is 1. In simple terms, it means the variable isn't raised to a power like 2 (x²), 3 (x³), or anything higher. Our equation 3x² - 2(2x² + 5) = 4 might seem tricky because of the x² terms, but as we go through it, we’ll see how it simplifies down to a standard first-degree equation. The key here is to simplify and combine like terms. First-degree equations are super important in math; they pop up everywhere, from simple everyday problems to complex scientific calculations. Mastering how to solve them is a fundamental skill. So, take a deep breath, and let's get into the step-by-step method. This is a game changer, and you will learn how to approach more complex problems with confidence.

Now, let's clarify something. In our initial equation, we have x², which might lead us to believe it isn't a first-degree equation. However, after we simplify it, the x² terms will cancel out, leaving us with a linear equation, that is, a first-degree equation, where the highest power of x will indeed be 1. This simplification process is critical, and we will do it with the proper steps. This is the crucial part; understanding this nuance will help you solve many similar problems. So keep focused, and let's get into the nitty-gritty of solving this equation. Remember, practice makes perfect, and with each problem you solve, your confidence will soar. Let's make this equation our friend, not our foe!

Step-by-Step Solution: Breaking Down the Equation

Alright, let’s get our hands dirty and start solving the equation 3x² - 2(2x² + 5) = 4. We will break down each step so that you don't miss anything. The goal here is to isolate x. Here is how to do it, step-by-step:

  1. Distribute: First, we need to deal with the parentheses. Multiply the -2 by each term inside the parentheses: 3x² - 2(2x²) - 2*(5) = 4*. This simplifies to 3x² - 4x² - 10 = 4.

  2. Combine Like Terms: Next, combine the x² terms: 3x² - 4x² = -x². So, our equation now looks like this: -x² - 10 = 4.

  3. Isolate the Variable: Add 10 to both sides of the equation to get rid of the -10 on the left side: -x² - 10 + 10 = 4 + 10. This simplifies to -x² = 14.

  4. Solve for x²: Multiply both sides by -1: (-1)(-x²) = 14*(-1)*, which gives us x² = -14.

  5. Find the Value of x: This step is where things get interesting. To find x, we need to take the square root of both sides. However, the square root of a negative number isn’t a real number. This means that x is an imaginary number and there is no real solution for x. The equation does not have a real solution.

See? We've successfully simplified the equation to its core components. Following these steps consistently will help you solve a bunch of similar problems, and also give you a strong foundation in algebra. It is like a puzzle, guys, and we just fit the pieces together. With a little practice, you'll be solving these equations like a pro!

Why Understanding Each Step Matters

So, you've seen how we tackled the equation 3x² - 2(2x² + 5) = 4 step-by-step. But why is it so important to understand each step? Well, breaking down a problem allows us to catch any errors and understand the logic behind each mathematical operation. For example, if we forgot to distribute the -2 correctly, we'd end up with a completely different (and incorrect) answer. By taking it one step at a time, we make sure that our methods are accurate and our solutions are reliable. Furthermore, understanding each step gives you the power to tackle even more complex problems. It builds a solid foundation upon which you can construct more advanced mathematical skills. It also helps in problem-solving in general. Each step is a building block, and by understanding them, you're not just solving an equation; you are building critical thinking skills that are useful in many aspects of your life. Every time you solve an equation, you're essentially sharpening your ability to break down complex issues into more manageable parts. This skill is gold, guys!

In our particular problem, because x² = -14, it leads to an imaginary number. Recognizing this early on prevents us from wasting time trying to find a real solution that doesn't exist. This understanding comes from meticulously following each step, understanding what each operation means, and knowing the rules of algebra inside and out. It’s like knowing the rules of a game before playing it; you’ll be much better at it. Mastering each step ensures you become more confident in your abilities. It gives you the power to apply these concepts in different contexts, making learning even more rewarding. It is not just about getting to the correct answer; it is about how you get there. Understanding this step by step can create a huge impact!

Common Mistakes and How to Avoid Them

Alright, let's talk about some common pitfalls when solving equations like 3x² - 2(2x² + 5) = 4. Being aware of these mistakes helps you avoid them in the first place, and also saves you time and frustration. Let's make sure we steer clear of these traps, okay?

  1. Forgetting to Distribute Correctly: This is a classic! It means not multiplying the -2 by both terms inside the parentheses. For example, some folks might do this: 3x² - 2(2x² + 5) = 3x² - 4x² + 5. Make sure you remember to multiply the -2 by both 2x² and 5. Always double-check this step!

  2. Incorrectly Combining Like Terms: This might be accidentally adding unlike terms, which can lead to a wrong answer. Be sure to group x² terms together and constant numbers together. It is like mixing apples and oranges. Keep them separate unless you can find a common factor.

  3. Making Sign Errors: Watch those pluses and minuses! A small mistake with the signs can make a huge difference in the outcome. Always be careful to carry the signs correctly through each step. It is easy to overlook a minus sign, so take extra care when dealing with negative numbers.

  4. Not Knowing When to Stop: In our case, realizing that the square root of a negative number isn’t a real number is crucial. If you keep going down the wrong path, you'll waste time. Knowing the limits of real numbers helps you identify when there's no real solution, saving you the headache.

  5. Forgetting the Order of Operations: This is also very common. Always remember PEMDAS or BODMAS (Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction). Perform operations in the correct sequence to get the right answer.

Avoiding these mistakes comes down to carefulness and practice. Always double-check your work, and don't be afraid to redo a step if something feels off. With time and effort, you'll become more proficient at avoiding these common errors, and your confidence will soar!

Conclusion: Mastering the Equation

Well, that was fun, wasn't it? We successfully navigated the equation 3x² - 2(2x² + 5) = 4! We've seen how to break it down, step by step, and how to avoid the most common mistakes. Remember, the journey of solving this equation is more important than the final solution. The objective here is that you understand the process and can apply it to similar problems. This isn't just about getting an answer; it's about developing essential problem-solving skills, it enhances your critical thinking, and prepares you to tackle the future with confidence.

Solving equations can be a fantastic way to sharpen your mind and build a solid foundation in mathematics. So keep practicing, keep learning, and keep asking questions. And remember, every equation you solve makes you a little bit stronger in the world of mathematics. Keep up the amazing work! You’ve got this!