Solving The Equation 4 + √(v) = √(v + 56) A Step-by-Step Guide

Hey guys! Let's dive into solving this radical equation together. We've got $4 + \sqrt{v} = \sqrt{v + 56}$, and our mission is to find the real number value(s) for v that make this equation true. Solving radical equations might seem tricky, but don't worry, we'll break it down step by step. Remember, the key is to isolate the radicals and then eliminate them by squaring. We'll also need to check our solutions at the end to make sure they're legit, since squaring can sometimes introduce extraneous solutions. So, grab your thinking caps, and let’s get started!

Step 1: Isolate a Radical

First things first, let’s take a look at our equation: $4 + \sqrt{v} = \sqrt{v + 56}$. In this case, we already have the radical terms mostly isolated. We have $\sqrt{v}$ on the left side and $\sqrt{v + 56}$ on the right side. There isn't a single radical term we need to isolate by itself before proceeding. This is a great starting point! Sometimes you might need to add or subtract terms to get a radical alone on one side, but we're already set to move on to the next step. Remember, the goal here is to simplify the equation as much as possible before we start squaring things. So, with our radicals reasonably isolated, we’re ready to move forward and eliminate those square roots. This is where things get interesting, as we'll be squaring both sides of the equation, so pay close attention to the algebra. Getting this step right is crucial for finding the correct solutions, so let's keep our focus sharp and proceed with confidence!

Step 2: Square Both Sides

Now that we have our equation $4 + \sqrt{v} = \sqrt{v + 56}$, it’s time to square both sides. Squaring both sides helps us eliminate the square roots, making the equation easier to solve. When we square the left side, $(4 + \sqrt{v})^2$, we need to remember the formula $(a + b)^2 = a^2 + 2ab + b^2$. So, we have:

$(4 + \sqrt{v})^2 = 4^2 + 2 \cdot 4 \cdot \sqrt{v} + (\sqrt{v})^2 = 16 + 8\sqrt{v} + v$

On the right side, squaring $\sqrt{v + 56}$ simply gives us $v + 56$. So, our equation now looks like this:

$16 + 8\sqrt{v} + v = v + 56$

Notice how squaring both sides has removed one of the square roots, but we still have a radical term remaining. This is perfectly normal, and we'll address it in the next steps. The key takeaway here is that we've made progress by getting rid of one layer of complexity. Squaring both sides can sometimes introduce extraneous solutions, which is why we'll need to check our answers later. But for now, we're on the right track to simplifying this equation and finding the value(s) of v that satisfy it. Let's keep moving forward!

Step 3: Simplify and Isolate the Remaining Radical

Okay, so after squaring both sides, we've arrived at the equation $16 + 8\sqrt{v} + v = v + 56$. Now, let's simplify this and isolate the remaining radical term. Notice that we have a v on both sides of the equation. We can subtract v from both sides to get rid of it:

$16 + 8\sqrt{v} + v - v = v + 56 - v$

This simplifies to:

$16 + 8\sqrt{v} = 56$

Now, let’s isolate the term with the square root. We can do this by subtracting 16 from both sides:

$16 + 8\sqrt{v} - 16 = 56 - 16$

Which gives us:

$8\sqrt{v} = 40$

Great! We're almost there. We have the term with the square root pretty much isolated. To get $\sqrt{v}$ completely by itself, we'll divide both sides by 8:

$\frac{8\sqrt{v}}{8} = \frac{40}{8}$

This simplifies to:

$\sqrt{v} = 5$

Fantastic! We've successfully isolated the square root. Now we're just one step away from finding v. Isolating the radical is a crucial step because it sets us up for the final move: squaring again to eliminate the square root and solve for our variable. This careful manipulation of the equation brings us closer to the solution, so let's keep the momentum going!

Step 4: Square Both Sides Again

Alright, we've reached a significant milestone: $\sqrt{v} = 5$. We’re now in a position where we can easily solve for v. To get rid of the square root, we simply square both sides of the equation. So, let's do it:

$(\sqrt{v})^2 = 5^2$

This simplifies to:

$v = 25$

Boom! We've got a potential solution: $v = 25$. Squaring both sides is a powerful technique for solving radical equations, but it's super important to remember that this step can sometimes introduce solutions that don't actually work in the original equation. These are called extraneous solutions. That's why the next step, checking our solution, is absolutely crucial. We can't just assume that $v = 25$ is the correct answer without verifying it. So, let's hold on to this value and move on to the all-important check to make sure it's the real deal!

Step 5: Check the Solution

Okay, we've arrived at a potential solution: $v = 25$. But before we celebrate, we need to make absolutely sure it works in the original equation. Remember, squaring both sides can sometimes lead to extraneous solutions, so this check is non-negotiable. Let's go back to our original equation:

$4 + \sqrt{v} = \sqrt{v + 56}$

Now, we'll substitute $v = 25$ into the equation and see if both sides are equal:

$4 + \sqrt{25} = \sqrt{25 + 56}$

Let's simplify each side:

$4 + 5 = \sqrt{81}$

$9 = 9$

Woohoo! The left side equals the right side. This means that $v = 25$ is indeed a valid solution. It satisfies our original equation perfectly. Checking our solution is like the final seal of approval, confirming that our hard work has paid off. We've successfully navigated the steps of solving this radical equation, and now we can confidently state our answer.

Final Answer

So, after all our careful steps and checks, we've found the solution to the equation $4 + \sqrt{v} = \sqrt{v + 56}$. The value of v that satisfies this equation is:

$v = 25$

We isolated the radicals, squared both sides (twice!), simplified, and, most importantly, checked our solution. This thorough process ensures that we haven't fallen victim to any extraneous solutions. Solving radical equations can be a rewarding challenge, and you guys nailed it! Remember, the key is to take it one step at a time, be meticulous with your algebra, and always, always check your answers. Great job, and keep up the awesome work!