Solve: Square Root Of A Number Plus Two Equals -5

Hey guys! Let's dive into the fascinating world of symbolic language and equation solving, focusing on a classic example: "The square root of a number increased by two is equal to minus five." Sounds a bit cryptic, right? But don't worry, we'll break it down step by step, transforming those words into mathematical symbols and then cracking the solution. Our journey will not only solve this particular problem but also equip you with the skills to tackle similar challenges. We'll explore the fundamental concepts behind square roots, algebraic manipulation, and the importance of checking your solutions. So, grab your thinking caps, and let's embark on this mathematical adventure!

Translating Words into Symbols: The Key to Unlocking Equations

In this section, we're going to focus on the art of translation. Think of it like learning a new language, but instead of French or Spanish, we're learning the language of mathematics. Our key phrase is: "The square root of a number increased by two is equal to minus five." The first step is identifying the unknowns. What are we trying to find? The answer is "a number." In algebra, we often represent unknowns with variables, and the most common one is 'x'. So, let's replace "a number" with 'x'. Now, let's tackle the phrase "the square root of a number." The square root is a mathematical operation symbolized by the radical symbol (√). Therefore, "the square root of x" is written as √x. Next, we have "increased by two." In mathematical terms, "increased by" signifies addition. So, we're adding 2 to the square root of x, giving us √x + 2. Finally, we encounter "is equal to minus five." The phrase "is equal to" is represented by the equals sign (=), and "minus five" is simply -5. Putting it all together, we have successfully translated the verbal expression into a symbolic equation: √x + 2 = -5. This equation is the heart of the problem, and now we can use our algebraic skills to solve for the unknown 'x'. Remember, this process of translation is crucial in mathematics. It allows us to take real-world problems, express them in a concise and precise way, and then apply mathematical tools to find solutions. Understanding the relationship between words and symbols is the first step towards mastering algebra.

Breaking Down the Equation: A Step-by-Step Approach

Now that we've successfully translated the problem into the equation √x + 2 = -5, it's time to solve it! Solving equations is like a puzzle, where we need to isolate the variable (in this case, 'x') on one side of the equation. To do this, we'll use the principles of algebraic manipulation, which essentially involve performing the same operations on both sides of the equation to maintain balance. Our first goal is to isolate the square root term (√x). Notice that we have a '+ 2' on the left side of the equation. To get rid of it, we need to perform the inverse operation, which is subtraction. We subtract 2 from both sides of the equation: √x + 2 - 2 = -5 - 2. This simplifies to √x = -7. Great! We've isolated the square root. Now, to get rid of the square root, we need to perform its inverse operation, which is squaring. We square both sides of the equation: (√x)² = (-7)². This gives us x = 49. So, we've found a potential solution: x = 49. However, it's crucial to remember that when dealing with square roots, we need to check our solution to make sure it's valid. Why? Because squaring both sides of an equation can sometimes introduce extraneous solutions, which are solutions that satisfy the transformed equation but not the original equation. In the next section, we'll delve into the important step of checking our solution.

The Importance of Verification: Checking for Extraneous Solutions

Alright, we've arrived at a potential solution, x = 49, but don't pop the champagne just yet! In the world of square root equations, it's absolutely crucial to verify our answers. This is because squaring both sides of an equation can sometimes lead to extraneous solutions – those sneaky numbers that seem to work but actually don't when you plug them back into the original equation. So, how do we check if x = 49 is a legitimate solution? We substitute it back into our original equation: √x + 2 = -5. Replacing x with 49, we get √(49) + 2 = -5. Now, let's simplify. The square root of 49 is 7, so we have 7 + 2 = -5. This simplifies to 9 = -5. Wait a minute... 9 does not equal -5! This tells us that x = 49 is an extraneous solution. It's a false alarm! So, what does this mean? It means that our original equation, √x + 2 = -5, has no real solutions. There is no real number that, when its square root is increased by 2, will equal -5. This might seem disappointing, but it's a valuable lesson. Not all equations have solutions, and checking our answers is vital to avoid falling into the trap of extraneous solutions. So, remember, always verify your solutions, especially when dealing with square roots and other radical expressions. It's the mark of a true mathematical detective!

Why No Solution? Understanding the Nature of Square Roots

So, we discovered that the equation √x + 2 = -5 has no real solutions. But why is that? Let's delve deeper into the nature of square roots to understand what's happening here. The square root of a number is a value that, when multiplied by itself, gives you the original number. For example, the square root of 9 is 3 because 3 * 3 = 9. Now, here's the key point: the square root of a non-negative number is always non-negative. In other words, the principal square root (the one we usually consider) of a positive number is always positive, and the square root of 0 is 0. We don't typically deal with square roots of negative numbers in the realm of real numbers (that's where complex numbers come in, but that's a topic for another day!). So, when we look at our equation √x + 2 = -5, we see that the square root of x, which is √x, must be equal to -7 (after we subtract 2 from both sides). But as we just discussed, the square root of a real number cannot be negative. This is the fundamental reason why our equation has no real solution. The square root term (√x) can never be equal to -7, no matter what value we substitute for x. This understanding of the properties of square roots is crucial for solving equations and interpreting the results. It helps us recognize when a solution is impossible and avoid wasting time trying to find one. So, remember, the square root of a non-negative real number is always non-negative, and this principle plays a vital role in solving equations involving radicals.

Key Takeaways: Mastering Symbolic Language and Equation Solving

We've journeyed through translating a verbal expression into a symbolic equation, solving it, and verifying our solution. Let's recap the key takeaways from this mathematical exploration. First, we learned the importance of translating words into symbols. This is the foundation of algebra, allowing us to express problems concisely and precisely. We saw how to represent unknowns with variables, operations with mathematical symbols, and relationships with equations. Second, we practiced the art of equation solving. We used algebraic manipulation to isolate the variable and find a potential solution. Remember, the key is to perform the same operations on both sides of the equation to maintain balance. Third, and perhaps most importantly, we emphasized the necessity of verification. Checking our solutions is crucial, especially when dealing with square roots and other radical expressions. We learned about extraneous solutions and how they can arise when squaring both sides of an equation. Finally, we explored the nature of square roots and why certain equations may have no real solutions. Understanding that the square root of a non-negative number is always non-negative is essential for interpreting results and avoiding errors. By mastering these key concepts, you'll be well-equipped to tackle a wide range of algebraic problems. So, keep practicing, keep exploring, and keep those mathematical gears turning!

Practice Problems: Sharpen Your Skills

Now that we've covered the theory and worked through an example, it's time to put your newfound skills to the test! Practice is key to mastering any mathematical concept, so let's tackle a few more problems similar to the one we just solved. These practice problems will give you the opportunity to hone your translation skills, practice algebraic manipulation, and, most importantly, remember to verify your solutions. Here are a few problems to get you started:

1. The square root of a number decreased by 3 is equal to 4.
2. Twice the square root of a number is equal to 10.
3. The square root of a number plus 5 is equal to 2.
4. The square root of (x + 1) is equal to 3.
5. Three times the square root of (x - 2) is equal to 9.

For each problem, follow the steps we discussed earlier: translate the words into an equation, solve the equation using algebraic manipulation, and then, the most important step, verify your solution by plugging it back into the original equation. Don't be discouraged if you encounter extraneous solutions – they're a natural part of the process, and recognizing them is a valuable skill. Remember, the goal is not just to find an answer but to understand the process and the underlying concepts. So, grab a pen and paper, and let's get practicing! The more you practice, the more confident and proficient you'll become in solving equations involving square roots and other mathematical challenges.