Solve Sqrt X+14 = X+8 A Step By Step Guide

Hey math enthusiasts! Today, we're diving deep into the fascinating world of radical equations. Specifically, we're going to tackle the equation $\sqrt{x+14}=x+8$. This equation might seem intimidating at first glance, but don't worry, we'll break it down step-by-step, making sure you understand every twist and turn. So, buckle up and get ready to sharpen those problem-solving skills!

The Art of Solving Radical Equations

Before we jump into the specifics of our equation, let's talk strategy. Solving radical equations is like navigating a maze – you need a clear plan to reach the exit. The core idea is to isolate the radical and then eliminate it by raising both sides of the equation to the appropriate power. But remember, this process can sometimes introduce extraneous solutions, which are solutions that satisfy the transformed equation but not the original one. So, checking your answers is super crucial in this game!

Step 1: Isolating the Radical

The first step in solving our radical equation, $\sqrt{x+14}=x+8$, is to isolate the radical term. In this case, the square root is already isolated on the left side of the equation. That's one less step for us! This isolation is key because it sets us up to eliminate the radical in the next stage. Think of it like preparing your ingredients before you start cooking – you need everything in its place before the real magic happens. When dealing with more complex equations, you might need to perform algebraic manipulations like adding, subtracting, multiplying, or dividing to get the radical term by itself. But for our equation, we're already off to a great start.

Step 2: Eliminating the Radical

Now comes the exciting part: getting rid of that pesky square root! To do this, we'll square both sides of the equation. Squaring both sides is a valid operation because if two quantities are equal, then their squares are also equal. This is a fundamental principle in algebra that allows us to transform the equation into a more manageable form. Squaring the left side, $(\sqrt{x+14})^2$, simply gives us $x+14$. On the right side, we have $(x+8)^2$, which expands to $x^2 + 16x + 64$. So, our equation now looks like this: $x+14 = x^2 + 16x + 64$. We've successfully eliminated the radical, but we've also created a quadratic equation. Don't worry, we know how to handle those!

Step 3: Solving the Quadratic Equation

We've transformed our radical equation into a quadratic equation: $x+14 = x^2 + 16x + 64$. To solve this, we first need to rearrange it into the standard quadratic form, which is $ax^2 + bx + c = 0$. Subtracting $x$ and 14 from both sides, we get $0 = x^2 + 15x + 50$. Now we have a classic quadratic equation ready to be solved. There are several ways to solve quadratic equations, including factoring, completing the square, and using the quadratic formula. In this case, factoring seems like the easiest approach. We're looking for two numbers that multiply to 50 and add up to 15. Those numbers are 5 and 10! So, we can factor the quadratic as $(x+5)(x+10) = 0$. This gives us two potential solutions: $x = -5$ and $x = -10$.

Step 4: The Crucial Check for Extraneous Solutions

This is the most important step! Remember how we talked about extraneous solutions? These are solutions that arise from the process of solving the equation but don't actually satisfy the original equation. Squaring both sides can sometimes introduce these pesky false solutions. So, we need to plug each of our potential solutions, $x = -5$ and $x = -10$, back into the original equation, $\sqrt{x+14}=x+8$, to see if they work.

Let's start with $x = -5$. Plugging it in, we get $\sqrt{-5+14} = -5+8$, which simplifies to $\sqrt{9} = 3$. Since 3 = 3, $x = -5$ is a valid solution. Phew!

Now let's check $x = -10$. Plugging it in, we get $\sqrt{-10+14} = -10+8$, which simplifies to $\sqrt{4} = -2$. This gives us 2 = -2, which is definitely not true. So, $x = -10$ is an extraneous solution. It's a false alarm! This highlights why checking is so vital when solving radical equations.

Final Answer: The Solution to $\sqrt{x+14}=x+8$

After all our hard work, we've arrived at the solution! The only valid solution to the equation $\sqrt{x+14}=x+8$ is $x = -5$. We successfully navigated the maze of the radical equation, avoided the trap of extraneous solutions, and emerged victorious! Remember, the key to solving these types of equations is to isolate the radical, eliminate it by raising both sides to the appropriate power, solve the resulting equation, and, most importantly, check your answers. Keep practicing, and you'll become a radical equation master in no time!

Let's make sure the question is crystal clear. The original question was: "Solve and check: $\sqrtx+14}=x+8$ Enter any solution as an integer or reduced fraction. Separate with commas if there is more than one solution. If there are no solutions, click the No Solution button." To make it even easier to understand, we can rephrase it as "Find the solution(s) to the equation $\sqrt{x+14=x+8$. Express your answer(s) as integers or reduced fractions, separated by commas if there are multiple solutions. If no solutions exist, indicate 'No Solution'. Remember to check your solutions!"

Hey guys! Ever feel like math problems are trying to trick you? Well, today we're tackling one that might seem a little tricky at first: $\sqrt{x+14}=x+8$. But don't sweat it! We're going to break it down step-by-step, so you can totally conquer this equation. Think of it like unlocking a secret level in your favorite game – you just need the right strategy!

Why Radical Equations Can Feel Like a Puzzle

So, what's the deal with these radical equations? The main thing is that little square root symbol – it's like a locked door! To solve the equation, we need to figure out how to open that door. But here's a heads-up: sometimes, the process of opening the door can lead to extra