Solving The Equation: √x+76 = X+4 - A Step-by-Step Guide

Hey guys! Let's dive into solving a radical equation today. We're tackling the equation √x+76 = x+4. This type of problem might seem intimidating at first, but don't worry! We'll break it down step by step so it becomes super clear. Understanding how to solve these equations is a fundamental skill in algebra, and it pops up in various areas of math and even in real-world applications. So, grab your thinking caps, and let's get started!

Understanding Radical Equations

Before we jump into the nitty-gritty of this particular equation, let's quickly chat about what radical equations are in general. Essentially, a radical equation is any equation where the variable (in our case, x) is stuck inside a radical symbol, most commonly a square root. The trick with these equations is that we need to isolate the radical and then get rid of it. The most common way to eliminate a square root is by squaring both sides of the equation. This is the core strategy we'll be using here, but there are a few things we need to keep in mind to avoid pitfalls, such as extraneous solutions. Now, why is understanding this important? Well, radical equations show up in many different contexts, from physics problems involving projectile motion to engineering calculations dealing with areas and volumes. By mastering these equations, you're not just learning a math skill; you're building a foundation for tackling more complex problems in various fields. Plus, it's a great exercise for your algebraic manipulation skills, which is always a good thing.

Step 1: Isolate the Radical

Our first move is to get the square root part all by itself on one side of the equation. Looking at our equation, √x+76 = x+4, we can see that the radical, √x+76, is already isolated on the left side! How cool is that? Sometimes, you might need to add, subtract, multiply, or divide to get the radical term alone, but in this case, we're one step ahead. This is a crucial step because it sets us up to eliminate the square root in the next step. If the radical isn't isolated, you'll end up with a much messier situation when you square both sides, trust me! So, always double-check that radical is by its lonesome before proceeding. Now, why is this isolation step so vital? Think of it like preparing the canvas before you start painting. You need a clean, clear surface to work with, right? Similarly, isolating the radical sets the stage for a straightforward algebraic maneuver without unnecessary complications. It's all about making our lives easier and our math cleaner.

Step 2: Square Both Sides

Now comes the fun part – getting rid of that pesky square root! To do this, we square both sides of the equation. Remember, whatever you do to one side of the equation, you must do to the other to keep things balanced. So, we have (√x+76)² = (x+4)². Squaring the square root on the left side simply cancels it out, leaving us with x+76. On the right side, we need to square the binomial (x+4). This means we're doing (x+4)(x+4), which expands to x² + 8x + 16. So, our equation now looks like this: x+76 = x² + 8x + 16. This transformation is a pivotal moment because we've successfully eliminated the radical and turned our equation into a quadratic equation. Why is this so significant? Well, quadratic equations are something we know how to handle! We have a bunch of tools in our algebraic toolbox for solving them, like factoring, completing the square, or using the quadratic formula. By squaring both sides, we've essentially translated our original problem into a more familiar and manageable format. It's like turning a complex puzzle into a set of simpler pieces that we can put together.

Step 3: Rearrange into a Quadratic Equation

Okay, we've got x+76 = x² + 8x + 16, which is a good start, but it's not quite in the standard form for a quadratic equation. To make it easier to solve, we want to rearrange everything so that one side is equal to zero. This standard form is ax² + bx + c = 0. To get there, let's subtract x and 76 from both sides of our equation. This gives us 0 = x² + 7x - 60. Now we have a classic quadratic equation! Why is this form so important? Because it allows us to easily apply various methods for solving quadratic equations. Factoring, for example, becomes much more straightforward when the equation is set to zero. Similarly, the quadratic formula is specifically designed to work with equations in this standard form. By rearranging our equation, we're setting ourselves up for success in the next step. Think of it like organizing your ingredients before you start cooking; having everything in its place makes the whole process smoother and more efficient.

Step 4: Solve the Quadratic Equation

Now we're at the heart of the problem: solving the quadratic equation x² + 7x - 60 = 0. There are a couple of ways we can tackle this. One common method is factoring. We're looking for two numbers that multiply to -60 and add up to 7. After a bit of thought, we can find that 12 and -5 fit the bill perfectly. So, we can factor the quadratic as (x + 12)(x - 5) = 0. This means either (x + 12) = 0 or (x - 5) = 0. Solving these gives us two potential solutions: x = -12 and x = 5. Factoring is a fantastic technique when it works because it's often the quickest way to solve a quadratic. But what if you can't easily factor the equation? That's where the quadratic formula comes in handy. It's a bit more involved, but it works every time. The quadratic formula is x = [-b ± √(b² - 4ac)] / 2a. In our equation, a = 1, b = 7, and c = -60. Plugging these values into the formula will also give you the solutions x = -12 and x = 5. So, we've got two candidates for our solution, but we're not quite done yet. We need to remember something crucial about radical equations: extraneous solutions.

Step 5: Check for Extraneous Solutions

This is a super important step that we can't skip! When we squared both sides of the original equation, we potentially introduced extraneous solutions. These are solutions that satisfy the transformed equation (the quadratic) but don't actually work in the original radical equation. So, we need to check both of our potential solutions, x = -12 and x = 5, in the equation √x+76 = x+4. Let's start with x = -12. Plugging it in, we get √(-12 + 76) = -12 + 4, which simplifies to √64 = -8. That's 8 = -8, which is definitely not true. So, x = -12 is an extraneous solution. Now let's check x = 5. Plugging it in, we get √(5 + 76) = 5 + 4, which simplifies to √81 = 9. That's 9 = 9, which is true! So, x = 5 is a valid solution. Why is this checking step so critical? Because squaring both sides can sometimes create solutions that don't really exist in the context of the original equation. It's like adding extra pieces to a puzzle that don't actually fit. By checking our solutions, we're ensuring that we only keep the ones that truly work. It's a bit like quality control in a manufacturing process; we're making sure our final product is correct and reliable.

Final Answer

After all that work, we've arrived at our final answer! The only valid solution to the equation √x+76 = x+4 is x = 5. We started by isolating the radical, then squared both sides to get rid of it, which led us to a quadratic equation. We solved the quadratic by factoring and found two potential solutions. But, we remembered the crucial step of checking for extraneous solutions. By plugging both potential solutions back into the original equation, we discovered that only x = 5 works. So, there you have it! Solving radical equations involves a few key steps, and checking your answers is always a must. With practice, you'll become a pro at these types of problems. Keep up the great work, and remember, math is like a muscle – the more you use it, the stronger it gets!