Solving -2√x + 4 = -12 A Step-by-Step Guide To Finding X

by ADMIN 57 views

Hey guys! Today, we're diving into the exciting world of algebra to solve the equation -2√x + 4 = -12. This might look a bit intimidating at first, but trust me, with a step-by-step approach, we'll crack it in no time. Think of it like a puzzle – each step is a piece that fits perfectly to reveal the final solution. So, grab your thinking caps, and let's get started!

Understanding the Problem

Before we jump into solving, let's make sure we really understand what the equation is asking us. In simple terms, we need to find the value of 'x' that makes the equation -2√x + 4 = -12 true. The equation involves a square root (√x), which means we're looking for a number that, when its square root is multiplied by -2 and then added to 4, gives us -12. Equations like these are super common in math and have tons of real-world uses, from calculating distances to designing structures.

Breaking Down the Equation

Our equation has a few key parts: the square root (√x), the variable 'x', and some constants (-2, 4, and -12). The square root is like a hidden box – we need to figure out what number is inside. The variable 'x' is our mystery number, and the constants are the known values that help us solve for 'x'. Think of it as a balancing act; we need to manipulate the equation while keeping both sides equal. To effectively solve this equation, we're going to isolate the square root term first. This involves getting the term with the square root by itself on one side of the equation. This is a crucial step because once the square root is isolated, we can easily get rid of it by squaring both sides. Remember, the goal is to peel away the layers around 'x' until we reveal its true value. This process is similar to solving any algebraic equation, where we use inverse operations to undo what's being done to the variable. So, with a clear plan in mind, let's dive into the actual steps of solving this equation!

Step-by-Step Solution

Alright, let's break down how to solve -2√x + 4 = -12 step-by-step. This is where the fun begins, guys! We'll go through each step slowly and explain the logic behind it. Don't worry if it seems tricky at first; with practice, you'll become a pro at solving these types of equations.

Step 1: Isolate the Square Root Term

The first thing we need to do is get the square root term (-2√x) all by itself on one side of the equation. To do this, we'll subtract 4 from both sides of the equation. This is like moving a piece in a game of chess – we're making a strategic move to simplify the equation. When we subtract 4 from both sides, we get: -2√x + 4 - 4 = -12 - 4, which simplifies to -2√x = -16. See? We're already making progress! We've managed to isolate the term with the square root. This is a super important step because now we can focus on getting rid of that pesky square root symbol. Think of it as clearing the path so we can see what 'x' really is.

Step 2: Divide to Simplify

Now that we have -2√x = -16, we want to get the square root by itself without any coefficients. In other words, we need to get rid of that -2 that's multiplying the square root. To do this, we'll divide both sides of the equation by -2. Remember, whatever we do to one side, we have to do to the other – it's all about keeping the equation balanced. When we divide both sides by -2, we get: (-2√x) / -2 = -16 / -2, which simplifies to √x = 8. Awesome! We've isolated the square root. This means we're one step closer to finding 'x'. The equation is getting simpler and simpler, making it easier for us to see the solution. It's like peeling an onion – we're removing the layers one by one until we get to the core.

Step 3: Square Both Sides

We've got √x = 8, and now we need to get rid of that square root symbol. The trick to doing this is to square both sides of the equation. Squaring is the inverse operation of taking the square root, so they cancel each other out. When we square both sides, we get: (√x)² = 8², which simplifies to x = 64. Boom! We've found our solution. The square root is gone, and we have 'x' all by itself. This is the moment we've been working towards. It's like reaching the end of a treasure hunt – the prize is ours!

Checking the Solution

Before we celebrate too much, it's always a good idea to check our solution. This is like double-checking your work on a test – it ensures we haven't made any mistakes along the way. To check our solution, we'll plug x = 64 back into the original equation and see if it holds true.

Plugging the Value Back In

Our original equation is -2√x + 4 = -12. Let's substitute x = 64 into this equation: -2√64 + 4 = -12. Now, we need to simplify. The square root of 64 is 8, so we have: -2 * 8 + 4 = -12. Multiplying -2 by 8 gives us -16, so the equation becomes: -16 + 4 = -12. Finally, adding -16 and 4 gives us -12, which means the equation is: -12 = -12. Hooray! Our solution checks out. Both sides of the equation are equal, which means x = 64 is indeed the correct answer. This step is super important because it gives us confidence in our solution. It's like having a map to guide you – checking our answer ensures we're on the right path.

The Correct Answer

So, after all that work, we've arrived at the correct answer! The solution to the equation -2√x + 4 = -12 is x = 64. That means the correct choice is D. 64. Awesome job, guys! We tackled a tricky equation and came out victorious. Remember, practice makes perfect. The more you solve equations like this, the easier they'll become. It's like learning a new language – the more you practice, the more fluent you become. So, keep up the great work, and you'll be solving algebraic equations like a pro in no time!

Why Other Options Are Incorrect

It's also helpful to understand why the other answer choices are incorrect. This can give us a deeper understanding of the equation and the solution process. Let's take a look at why options A, B, and C are not the correct answers.

Option A: 4

If we plug x = 4 into the original equation, we get: -2√4 + 4 = -2 * 2 + 4 = -4 + 4 = 0, which is not equal to -12. So, 4 is not the correct solution. This shows us that simply guessing a number won't work; we need to follow the correct steps to solve the equation.

Option B: 8

Let's try x = 8: -2√8 + 4. The square root of 8 is not a whole number, but it's approximately 2.83. So, we have: -2 * 2.83 + 4 ≈ -5.66 + 4 ≈ -1.66, which is also not equal to -12. This demonstrates the importance of accuracy in our calculations. Even a small difference can lead to an incorrect solution.

Option C: 16

Now, let's try x = 16: -2√16 + 4 = -2 * 4 + 4 = -8 + 4 = -4, which is still not equal to -12. This reinforces the idea that we need to isolate the variable and use inverse operations to find the correct solution. Plugging in random numbers might seem like a shortcut, but it rarely leads to the right answer.

Understanding the Errors

By understanding why these options are incorrect, we can see the importance of each step in the solution process. We need to isolate the square root, divide to simplify, square both sides, and check our solution. Skipping any of these steps or making a mistake in our calculations can lead to the wrong answer. It's like building a house – each step is crucial, and if you skip one, the whole structure might collapse. So, let's always remember to be careful and methodical when solving equations!

Conclusion: Mastering Algebraic Equations

So, there you have it, guys! We've successfully solved the equation -2√x + 4 = -12, and we've learned a lot along the way. We started by understanding the problem, then we broke it down into manageable steps, and finally, we checked our solution to make sure it was correct. Remember, solving algebraic equations is like learning a new skill – it takes time and practice, but it's totally worth it. By mastering these skills, you'll be able to tackle more complex problems and apply them to real-world situations. Keep practicing, and you'll become a math whiz in no time!

Key Takeaways

  • Isolate the square root term: This is the first crucial step in solving equations with square roots.
  • Divide to simplify: Getting rid of coefficients makes the equation easier to solve.
  • Square both sides: This eliminates the square root and allows us to solve for 'x'.
  • Check your solution: Always plug your answer back into the original equation to make sure it's correct.

Keep Practicing!

The best way to get better at math is to practice regularly. Try solving similar equations and challenge yourself with more complex problems. Math is like a muscle – the more you use it, the stronger it gets. So, keep flexing those math muscles, and you'll be amazed at what you can achieve. And remember, it's okay to make mistakes – they're a part of the learning process. Just keep trying, and you'll eventually get there. Happy solving, guys!