Solving X = -(-6) = ±√((-6)² - 4(1)(-27)) A Detailed Mathematical Analysis

Hey math enthusiasts! Today, let's dive deep into solving the equation x = -(-6) = ±√((-6)² - 4(1)(-27)). This equation might seem a bit complex at first glance, but we'll break it down step by step to understand each component and how they interact. Our goal is to provide a clear, detailed mathematical analysis that not only solves the equation but also enhances your understanding of the underlying mathematical principles. We'll cover everything from basic arithmetic operations to square roots and the significance of the ± sign. So, grab your calculators and let's get started!

Understanding the Basics

Before we tackle the main equation, it's crucial to ensure we're solid on the foundational concepts. At the heart of this problem lies the arithmetic operations, so let’s break it down. First, we need to remember the rules for dealing with negative numbers, squares, and square roots. A negative number multiplied by a negative number yields a positive number, and the square of any number (negative or positive) is always positive. When we encounter a square root, we need to consider both the positive and negative solutions, which is why the ± symbol is so important. Now, let's delve deeper into each component of the equation and demystify it for everyone.

In this equation, you'll notice several components that require careful attention. The first part, -(-6), involves dealing with double negatives, which is a fundamental concept in arithmetic. The second part, ±√((-6)² - 4(1)(-27)), involves a square root operation, which introduces both positive and negative solutions due to the nature of squaring. The expression inside the square root includes squares, multiplication, and subtraction, each of which must be performed in the correct order. Understanding the order of operations (PEMDAS/BODMAS) is crucial here to avoid errors. So, we need to follow these rules strictly to arrive at the correct solution.

Another critical aspect to understand is the ± symbol. This symbol signifies that there are two possible solutions to the equation: one where the square root is taken as positive and another where it is taken as negative. This arises because both a positive number and its negative counterpart, when squared, will yield the same positive result. For instance, both 3 and -3, when squared, result in 9. Thus, when taking the square root of 9, we must consider both 3 and -3 as valid solutions. This understanding is particularly important in quadratic equations and other problems involving square roots, ensuring we capture all possible solutions. This careful consideration ensures that our analysis is comprehensive and accurate.

Analyzing x = -(-6)

Let's begin by unraveling the simpler part of the equation: x = -(-6). This component is all about mastering negative numbers, guys! You've probably heard the saying, “two negatives make a positive,” and that's precisely what we're dealing with here. When you have a negative sign outside parentheses containing another negative number, it effectively cancels out the negative inside. It’s like saying the opposite of a negative, which naturally leads us to a positive. This is a core concept in basic algebra and is crucial for simplifying expressions and equations. So, let’s break this down step by step to make sure we understand it completely.

To simplify -(-6), think of it as multiplying -1 by -6. The rule for multiplying negative numbers is that a negative times a negative equals a positive. So, -1 multiplied by -6 gives us a positive 6. Therefore, the equation x = -(-6) simplifies to x = 6. This might seem straightforward, but it’s a foundational step in solving more complex equations. Getting this basic principle down pat is super important for any math problem you'll encounter later on. Plus, understanding this will make the rest of the equation much easier to handle.

Now that we've established that x = 6 from this part of the equation, we have a clear value to work with. This value will be crucial when we compare it with the solutions derived from the other part of the equation, which involves the square root. By isolating and solving this part first, we’ve made the overall problem much more manageable. It’s always a good strategy in math to break down complex problems into smaller, more digestible parts. This approach not only simplifies the calculations but also reduces the chances of making errors. So, remember to always look for opportunities to simplify before diving into the more complicated aspects of a problem. This makes the whole process much less intimidating and more fun!

Deconstructing ±√((-6)² - 4(1)(-27))

Now, let's tackle the more intricate part of the equation: ±√((-6)² - 4(1)(-27)). This section involves several mathematical operations, including squaring, multiplication, subtraction, and taking the square root. Remember PEMDAS/BODMAS, guys? Parentheses/Brackets first, then Exponents/Orders, Multiplication and Division (from left to right), and finally Addition and Subtraction (from left to right). Following the correct order of operations is absolutely key to solving this accurately. So, let's break it down piece by piece to make sure we're on the right track.

First, we need to address the expression inside the square root. Within the parentheses, we have (-6)², which means -6 multiplied by -6. As we discussed earlier, a negative times a negative equals a positive, so (-6)² equals 36. Next, we have -4(1)(-27), which involves multiplying three numbers together. Multiplying -4 by 1 gives us -4, and then multiplying -4 by -27 gives us a positive 108 because, again, a negative times a negative is a positive. Now we can rewrite the expression inside the square root as 36 + 108.

Next, we add 36 and 108, which equals 144. So, now we have ±√144. Taking the square root of 144 means finding a number that, when multiplied by itself, equals 144. The square root of 144 is 12, but remember the ± symbol! This indicates that we have two possible solutions: +12 and -12. This is because 12 * 12 = 144 and (-12) * (-12) = 144. The ± symbol is super important here because it tells us to consider both possibilities to get a complete solution. Without it, we’d be missing half the answer! So always keep an eye out for that little symbol.

Comparing the Solutions and Final Analysis

Alright, we’ve arrived at the crucial point where we compare the solutions we found from both parts of the equation. From the first part, x = -(-6), we determined that x = 6. From the second part, ±√((-6)² - 4(1)(-27)), we found two solutions: +12 and -12. Now, we need to see if these solutions align or if there’s a discrepancy. This comparison will help us understand the full scope of the equation and its implications. So, let’s dive into it and make sense of what we’ve got!

When we compare the solutions, we see that x = 6 from the first part does not match the solutions ±12 from the second part. This discrepancy is significant because it tells us that the equation as a whole is inconsistent. In other words, there is no single value of x that satisfies the entire equation simultaneously. This can happen in mathematics for various reasons, such as contradictory conditions or misinterpretations of the equation's structure. Spotting these inconsistencies is a key skill in problem-solving and critical thinking. So, let's dig a little deeper into why this might be the case.

This inconsistency highlights the importance of careful analysis in mathematics. It’s not enough to just find solutions; we must also verify that those solutions make sense within the context of the entire equation. In this case, the two parts of the equation present conflicting requirements for the value of x. The first part clearly states that x equals 6, while the second part implies that x could be either +12 or -12. Since 6 is neither 12 nor -12, there’s a clear mismatch. This kind of analysis helps us catch errors and understand the nuances of mathematical problems. It also reminds us that sometimes, equations don't have a solution, which is a perfectly valid outcome. Keep these things in mind as you tackle future math challenges!

Final Thoughts

In conclusion, guys, solving the equation x = -(-6) = ±√((-6)² - 4(1)(-27)) involves a detailed step-by-step analysis of both sides. We found that x = -(-6) simplifies to x = 6, while ±√((-6)² - 4(1)(-27)) yields solutions of +12 and -12. The discrepancy between these solutions indicates that the equation is inconsistent and does not have a single value of x that satisfies it. This exercise reinforces the importance of understanding basic arithmetic operations, following the order of operations, and carefully comparing solutions. Keep practicing, and you'll become math pros in no time! Remember, math is a journey, and every problem you solve makes you stronger. So, keep challenging yourselves and enjoy the process!