Solving The Square Root Equation: A Step-by-Step Guide

Hey math enthusiasts! Today, we're diving into a problem that might look a little intimidating at first glance: c. √x+7+ √√2x = √√x+1. Don't worry, we'll break it down step by step and make sure you understand every move. This type of equation, dealing with square roots and nested square roots, can be tricky. But with a systematic approach, we can conquer it. Let's get started, guys!

Understanding the Problem: The Square Root Puzzle

So, what are we actually dealing with? We've got an equation with several square roots hanging around. Our goal is to find the value (or values) of 'x' that makes this equation true. This involves a few key steps: isolating the square roots, squaring both sides to eliminate them, and then solving the resulting equation. Remember, when dealing with square roots, we need to be extra cautious about potential extraneous solutions – solutions that appear to work but don't when plugged back into the original equation. We'll keep an eye out for those along the way, alright?

First, let's just make sure we are clear on the equation: c. √x+7+ √√2x = √√x+1. Okay, let's take this slowly. The equation contains two nested square roots, which is where things can get a bit confusing. A nested square root is a square root inside of another square root. It is essential to remember that when solving radical equations, we should always check our solutions in the original equation to ensure they're valid and not extraneous. Extraneous solutions arise when we square both sides of an equation, as this process can introduce solutions that don't satisfy the original equation. It's like a sneaky trick, but we'll outsmart it by verifying our answers at the end. Make sure to keep the context that square roots always produce non-negative values. So, any expression under a square root must be greater than or equal to zero. These concepts are very important, guys. We must keep in mind to solve these types of equations.

Isolating the Square Roots: The First Move

The first strategy when faced with this equation is to isolate at least one of the square roots on one side of the equation. This simplifies the process of getting rid of them. Let's start by isolating the term with the simpler square root. Rewriting the original equation: c. √x+7+ √√2x = √√x+1. Because our goal is to isolate one of the square roots on one side of the equation, we can start by subtracting √√2x from both sides. We have to isolate some of the square roots to start simplifying, so we'll start with √x+7 = √√x+1 - √√2x. Remember, the key is to isolate a square root, so we can square both sides and hopefully eliminate some of those pesky radicals. What do you think, guys? Ready to move forward with the next step?

Now, we'll continue with the next step to eliminate the square roots. We will square both sides of the equation. Squaring both sides is a critical step, but it's where we might introduce those extraneous solutions we talked about. Squaring both sides of √x+7 = √√x+1 - √√2x, we get (√x+7)^2 = (√√x+1 - √√2x)^2. This simplifies to x + 7 = (√√x+1)^2 - 2*(√√x+1)(√√2x) + (√√2x)^2. This step gets rid of the outer square roots, but it also creates some new terms. Expand and simplify, keeping a close eye on your algebra skills, and don't forget to double-check each term. It's easy to make a small mistake here, so take your time. This step is very important, because we are eliminating some of the square roots, so pay attention. Next, we will continue to simplify to find a possible value of x.

Squaring Both Sides: Eliminating Square Roots

We squared both sides of the equation, we need to continue simplifying the terms. Squaring both sides is a common technique for solving radical equations, but it’s crucial to remember that this process can sometimes introduce extraneous solutions. Extraneous solutions are values that satisfy the derived equation but do not satisfy the original equation. Thus, we have to keep an eye on this when we find possible solutions for x.

So, as we already squared both sides, now let's simplify them to get x + 7 = (√√x+1)^2 - 2*(√√x+1)(√√2x) + (√√2x)^2. Simplifying each part will give us: x + 7 = (√x + 1) - 2*(√√x+1)(√√2x) + 2x. Now, we want to isolate the remaining square root term: 2*(√√x+1)(√√2x). To do this, rearrange the equation: 2*(√√x+1)(√√2x) = x + 1 - x - 7. Simplifying the equation, we have: 2*(√√x+1)(√√2x) = -6. Now, let's divide both sides by 2: (√√x+1)(√√2x) = -3. Now, we have an equation with nested square roots again. What do you guys think? This is a bit unexpected, but we will solve it. Let's move to the next section.

Further Simplification and Solving

Okay, we're at a point where we have a slightly different equation: (√√x+1)(√√2x) = -3. Remember, the product of two square roots cannot be negative, since square roots always produce non-negative values. Notice that the left side of the equation involves the product of two square roots. The square root of any real number is non-negative, and the product of non-negative numbers is always non-negative. This implies that the product (√√x+1)(√√2x) must be greater than or equal to zero. However, our equation states that the product equals -3. This is a contradiction, meaning there is no real number that can satisfy the original equation. Therefore, the equation has no solution within the set of real numbers.

Checking for Extraneous Solutions: The Final Step (Even Though We Know There's No Solution)

Even though we've determined that there are no solutions, it's good practice to check for potential extraneous solutions. Extraneous solutions are values that satisfy the transformed equation but not the original one. In this case, we need to plug any potential solutions back into the original equation: c. √x+7+ √√2x = √√x+1, and verify that it holds true. Since we found no solutions, there’s nothing to check in this case. But in other problems, this is a very important step to check your final solutions.

Here’s how the check would work if we had a potential solution. Let's pretend we found a possible solution, x = a. We would substitute 'a' for 'x' in the original equation and evaluate both sides. If the left side equals the right side, then 'a' is a valid solution. If the left side does not equal the right side, then 'a' is an extraneous solution and we have to discard it. In our case, because we found no solution in the first place, this step is automatically done, because we already checked at the beginning. Remember, always check your final answers to avoid getting tricked by extraneous solutions. You've got this, guys!

Conclusion: No Real Solutions

So, to recap, after a careful step-by-step process of isolating square roots, squaring both sides, and simplifying, we found that the equation c. √x+7+ √√2x = √√x+1 has no real solutions. This means there is no value of 'x' that satisfies the equation in the real number system. This is a common outcome in radical equations, so don't be discouraged if you encounter it. The key is to be methodical, thorough, and always check your work. And that's all, folks! Hope you've found this guide helpful. Keep practicing, and you'll get the hang of solving these types of problems in no time. If you have any questions, feel free to ask. Keep up the great work in mathematics! See you next time, guys!