Solving Radical Equations: Find The Value Of $u$

$\sqrt{-u+11}=u+1$

(If there is more than one solution, separate them with commas.)

Solving the Radical Equation for $u$: A Step-by-Step Guide

Hey math enthusiasts! Today, we're diving into the world of radical equations to solve for $u$, where $u$ is a real number. Our equation is $\sqrt{-u+11}=u+1$. Don't worry, it might look a bit intimidating at first, but trust me, we'll break it down into manageable steps. The goal here is to find the values of $u$ that make this equation true. Remember, a solution to an equation is simply a value (or values) that, when plugged back into the original equation, makes both sides equal. So, let's roll up our sleeves and get started. We'll go through this systematically, ensuring that every step is clear and easy to follow. We'll focus on getting rid of that pesky square root and isolating $u$. By the end of this journey, you'll be equipped with the skills to confidently tackle similar problems. Let's do this!

The Core Strategy: Isolating and Eliminating the Radical

Okay guys, the primary strategy for solving this type of equation is pretty straightforward: we want to isolate the radical (the square root) and then eliminate it. To do that, the first step involves squaring both sides of the equation. This is like saying, "If two things are equal, then their squares are also equal." However, squaring both sides can sometimes introduce extraneous solutions, which are solutions that satisfy the squared equation but not the original one. So, we'll need to check our answers later to make sure they're valid. It's like a detective making sure the clues fit the crime. First things first, let's get rid of that square root. Squaring both sides of our equation $\sqrt{-u+11}=u+1$, we get $(-u + 11) = (u + 1)^2$. Now, this is where the fun begins. Remember, squaring $(u + 1)$ means multiplying $(u + 1)$ by itself. Let's expand the right side to make it a bit more friendly. This is where we'll encounter a quadratic equation – the next level of problem-solving. Remember, it’s super important to be careful with your arithmetic here; one small mistake can lead you astray. Always double-check your work!

Expanding and Simplifying the Equation

Alright, now that we've squared both sides of the equation, it's time to expand and simplify. This step involves a little bit of algebra, but nothing we can't handle. So, $(-u + 11) = (u + 1)^2$ becomes $-u + 11 = u^2 + 2u + 1$ after expanding the right side using the FOIL method (First, Outer, Inner, Last) or simply by recognizing that $(u+1)^2$ is $(u+1)(u+1)$. Now, we've got a quadratic equation on our hands. To solve a quadratic, we generally want to set the equation to zero and then either factor it or use the quadratic formula. Let's rearrange the equation by moving all terms to one side to set it equal to zero. We'll add $u$ and subtract $11$ from both sides, resulting in $0 = u^2 + 3u - 10$. Always make sure you’ve combined like terms and that everything is in the right order before moving on. It is crucial to avoid errors here because the outcome relies on these steps.

Solving the Quadratic Equation: Factoring or Formula?

Now that we have our quadratic equation in the form $0 = u^2 + 3u - 10$, we need to find the values of $u$ that make this equation true. We have two main paths to choose from: factoring or using the quadratic formula. In this case, factoring might be the easier route. We're looking for two numbers that multiply to $-10$ and add up to $3$. Those numbers are $5$ and $-2$. So, we can factor the quadratic as $(u + 5)(u - 2) = 0$. Setting each factor to zero gives us two potential solutions: $u + 5 = 0$ which means $u = -5$, and $u - 2 = 0$ which means $u = 2$. But hey, hold your horses! Remember what we said earlier about extraneous solutions? Now is the time to check if these values actually work in the original equation $\sqrt{-u+11}=u+1$.

Checking for Extraneous Solutions: The Final Test

This is where the rubber meets the road. We've got two potential solutions, $u = -5$ and $u = 2$, but we need to verify if they both actually work in the original equation. Let's start with $u = -5$. Plugging this value into the original equation, we get $\sqrt{-(-5) + 11} = -5 + 1$, which simplifies to $\sqrt{16} = -4$. Since $\sqrt{16}$ is $4$, and $4 eq -4$, $u = -5$ is an extraneous solution and does not work. Now, let's check $u = 2$. Plugging $u = 2$ into the original equation, we get $\sqrt{-2 + 11} = 2 + 1$, which simplifies to $\sqrt{9} = 3$. Since $\sqrt{9}$ is $3$, and $3 = 3$, $u = 2$ is a valid solution. So, after all that hard work, we've found our answer. Always remember to plug your solutions back into the original equation to make sure they're valid. This step is crucial for catching any sneaky extraneous solutions that might have popped up along the way. And there you have it – we've successfully solved for $u$!

The Solution

So, after all the steps, the correct solution for $u$ is only $2$. Therefore, our final answer is 2. Always double-check your answers to make sure they are correct and valid by plugging them back into the original equation, which can help to prevent any confusion.