Solving $\sqrt{7x+2}=x+2$ A Step-by-Step Guide

Hey guys! 👋 Have you ever stumbled upon an equation that looks like it's straight out of a math puzzle? You know, the kind with those sneaky square roots? Well, you're not alone! Radical equations, especially those involving square roots, can seem daunting at first. But don't worry, we're going to break down the process step by step. In this article, we're going to dive deep into solving an equation with a square root. Our mission? To find every single possible value of x that makes the equation true. So, grab your thinking caps, and let's get started!

The Challenge: $\sqrt{7x + 2} = x + 2$

Let's face it, equations with square roots can look a bit intimidating, right? You see that radical symbol, and it's like a little flag waving, saying, "Hey, this might be tricky!" But trust me, with the right approach, you can totally conquer these problems. We're going to tackle the equation $\sqrt{7x + 2} = x + 2$. This equation is a classic example of a radical equation, where the variable x is hiding inside a square root. Our goal is to isolate x and figure out all the values that make this equation a true statement. Think of it like solving a puzzle – each step gets us closer to the final solution. We'll be using some key algebraic techniques, like squaring both sides of the equation, but we'll also need to be careful about something called extraneous solutions. These are sneaky little values that might pop up during our solving process but don't actually work when we plug them back into the original equation. So, it's like finding a piece of the puzzle that looks like it fits but ultimately doesn't. We'll learn how to identify and discard these imposters to make sure we have the correct solutions. So, buckle up, and let's get ready to unravel this mathematical mystery!

Step 1: Squaring Both Sides – Unmasking the Variable

The first big move when you're facing a square root in an equation is to get rid of that radical sign. How do we do that? By squaring! Squaring both sides of the equation is like the magic spell that unlocks the variable hidden inside the square root. Remember, whatever you do to one side of the equation, you've got to do to the other to keep things balanced. It's like a seesaw – if you add weight to one side, you need to add the same weight to the other to keep it level. So, let's take our equation, $\sqrt{7x + 2} = x + 2$, and square both sides. On the left side, the square root and the square cancel each other out, leaving us with just the expression inside the radical, which is 7x + 2. On the right side, we have to square the entire expression (x + 2). This means multiplying (x + 2) by itself. You might remember this from algebra as the FOIL method (First, Outer, Inner, Last) or as expanding a binomial. Either way, it's crucial to get this part right. When we square (x + 2), we get x² + 4x + 4. So, after squaring both sides, our equation transforms from a radical equation into a more familiar quadratic equation. This is a big step forward! We've taken the equation from something that looked tricky to something we know how to handle. Our new equation is 7x + 2 = x² + 4x + 4. Now, we're on our way to finding those values of x!

Step 2: Rearranging into a Quadratic Equation – Getting Everything in Order

Alright, we've successfully squared both sides of our equation, and now we're looking at 7x + 2 = x² + 4x + 4. This is fantastic progress, but to really solve for x, we need to get this equation into a specific form: the standard quadratic form. You might remember this as ax² + bx + c = 0, where a, b, and c are just numbers. The reason we want this form is because it sets us up perfectly to use methods like factoring or the quadratic formula to find the solutions. Think of it like organizing your tools before starting a project – having everything in its place makes the job much smoother. So, how do we rearrange our equation? Our goal is to get everything on one side, leaving zero on the other side. To do this, we'll subtract 7x and 2 from both sides of the equation. This keeps the equation balanced, just like we talked about before. When we subtract 7x from both sides, we're left with -7x on the left and we need to combine it with the 4x on the right. Similarly, subtracting 2 from both sides means we have -2 on the left and we combine it with the +4 on the right. After doing this, the left side becomes zero (which is exactly what we wanted!), and the right side is a simplified quadratic expression. So, after all the rearranging, our equation transforms into 0 = x² - 3x + 2. Now, we have a classic quadratic equation ready to be solved. We're one step closer to uncovering the values of x that make our original equation true!

Step 3: Solving the Quadratic Equation – Unlocking the Values of x

Now that we've got our equation in the beautiful standard quadratic form, 0 = x² - 3x + 2, it's time to actually solve for x. There are a couple of ways we can tackle this, and the first one we'll explore is factoring. Factoring is like reverse multiplication – we're trying to find two expressions that, when multiplied together, give us our quadratic expression. If you can spot the factors, it's often the quickest way to solve a quadratic equation. So, let's look at x² - 3x + 2. We need to find two numbers that multiply to give us 2 (the constant term) and add up to give us -3 (the coefficient of the x term). After a little thought, you might realize that -1 and -2 fit the bill perfectly. -1 multiplied by -2 is indeed 2, and -1 plus -2 is -3. So, we can factor our quadratic expression as (x - 1)(x - 2) = 0. Now, here's the magic part: if the product of two things is zero, then at least one of them must be zero. It's like saying, "If I have two bags of marbles, and when I count all the marbles together I have zero, then at least one of the bags must be empty." So, either (x - 1) = 0 or (x - 2) = 0. Solving these two simple equations gives us our potential solutions for x. If x - 1 = 0, then x = 1. And if x - 2 = 0, then x = 2. So, we have two possible values for x: 1 and 2. But hold on a second! We're not done yet. Remember those sneaky extraneous solutions we talked about earlier? We need to check if these values actually work in the original equation.

Step 4: Checking for Extraneous Solutions – The Reality Check

Okay, we've found two potential solutions for x: 1 and 2. We solved our quadratic equation like champs, but here's the thing about radical equations – sometimes, the process of squaring both sides can introduce solutions that don't actually work in the original equation. These are those pesky extraneous solutions we mentioned earlier. They're like the gatecrashers at a party – they weren't invited, and they don't belong there. So, how do we kick them out? By checking our solutions! We need to plug each potential solution back into the original equation, $\sqrt{7x + 2} = x + 2$, and see if it makes the equation true. Let's start with x = 1. We substitute 1 for x in the original equation: $\sqrt{7(1) + 2} = 1 + 2$. This simplifies to $\sqrt{9} = 3$, which is true because the square root of 9 is indeed 3. So, x = 1 is a valid solution – it's one of the invited guests! Now, let's check x = 2. We substitute 2 for x: $\sqrt{7(2) + 2} = 2 + 2$. This simplifies to $\sqrt{16} = 4$, which is also true because the square root of 16 is 4. So, x = 2 is also a valid solution – another welcome guest! In this case, both of our potential solutions passed the test. But it's crucial to always perform this check because sometimes, you'll find that one (or even both!) of your solutions is extraneous and needs to be discarded. It's like having a detective's eye, making sure everything adds up before declaring the case closed.

The Grand Finale: The Solution Set

We've reached the end of our mathematical journey! 🎉 We started with a radical equation that looked a bit intimidating, and now we've successfully navigated through the steps to find all possible values of x. We squared both sides, rearranged into a quadratic equation, solved for x, and, most importantly, we checked for extraneous solutions to make sure our answers were the real deal. So, what's the final verdict? Our solutions are x = 1 and x = 2. Both of these values make the original equation, $\sqrt{7x + 2} = x + 2$, true. We can express our solution in a neat little set notation: {1, 2}. This set notation is just a way of saying, "These are the numbers that work!" Think of it like the treasure chest at the end of a treasure hunt – inside, you find the solutions you've been searching for. So, there you have it! We've conquered this radical equation, and you've added another tool to your math-solving arsenal. Remember, the key to solving these types of equations is to take it step by step, be careful with your algebra, and always, always check for extraneous solutions. You've got this! 💪