Solve For X: Sqrt(x) + 5 = Sqrt(x + 45)

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Hey math whizzes and number crunchers, gather 'round! Today, we're diving deep into the exciting world of algebraic equations, specifically tackling a problem that involves those tricky square roots. We're on a mission to find the value of $x$ in the equation $\sqrt{x}+5=\sqrt{x+45}$. This isn't just about getting a number; it's about understanding the process, the logic, and how to manipulate these equations to reveal the hidden value of $x$. We'll break down each step, making sure that by the end, you'll feel confident in your ability to solve similar problems. So, grab your calculators, sharpen your pencils, and let's get started on this mathematical adventure!

Understanding the Equation and Strategy

Alright guys, let's first take a good, hard look at the equation we're dealing with: $\sqrt{x}+5=\sqrt{x+45}$. Our primary goal here is to isolate $x$. However, we have square roots staring us down on both sides, which makes direct isolation a bit of a challenge. The most effective strategy when dealing with equations involving square roots is to get rid of those pesky roots by squaring both sides of the equation. But here's the catch: we can only do this easily if we have a single square root term on each side. In our current equation, we have $\sqrt{x}$ and $5$ on the left side. So, the first crucial step is to rearrange the equation to have one square root on each side. We can achieve this by subtracting $5$ from both sides. This gives us $\sqrt{x} = \sqrt{x+45} - 5$. Now, this looks a little more manageable. With a single square root term on each side, we're ready to square both sides to eliminate the square roots. This step is key to moving forward and ultimately solving for $x$. Remember, squaring both sides can sometimes introduce extraneous solutions, so we'll need to be vigilant and check our final answer(s) in the original equation. This pre-planning and strategy session is super important in mathematics; it’s like having a roadmap before you start a long journey. We're not just randomly trying things; we have a plan, and that plan involves strategically eliminating the square roots to get closer to our answer.

Step 1: Isolating the Square Root

So, to kick things off, we need to get one of the square root terms by itself on one side of the equation. Our original equation is $\sqrt{x}+5=\sqrt{x+45}$. The goal is to have something like $\text{square root term} = ext{everything else}$. Looking at the left side, we have $\sqrt{x}$ and a $+5$. To get $\sqrt{x}$ alone, we simply need to move that $+5$ over to the other side. How do we do that? By performing the opposite operation! So, we subtract $5$ from both sides of the equation. This gives us:

$\sqrt{x} = \sqrt{x+45} - 5$

Boom! Just like that, we've successfully isolated one of the square root terms. This is a critical move because it sets us up perfectly for the next major step: squaring both sides. Without this isolation, trying to square the original equation would lead to a messier expansion involving $(\sqrt{x}+5)^2$, which would still leave us with a square root term. By isolating $\sqrt{x}$ first, we ensure that when we square both sides, the left side becomes a simple $x$, and the right side, while still needing expansion, will have its primary square root term removed directly. This strategy is all about making the process as clean and straightforward as possible. Think of it as clearing the path before you start building. We dealt with the addition on the left, and now the stage is set for demolition – the demolition of those square roots!

Step 2: Squaring Both Sides

Now that we have our equation in the form $\sqrt{x} = \sqrt{x+45} - 5$, it's time for the main event: squaring both sides. This is our primary weapon for eliminating square roots. When you square a square root, they cancel each other out. So, let's apply this to both sides of our equation:

$(\sqrt{x})^2 = (\sqrt{x+45} - 5)^2$

On the left side, it's straightforward: $(\sqrt{x})^2 = x$. Easy peasy!

Now, the right side requires a bit more attention. We need to expand $(\sqrt{x+45} - 5)^2$. Remember the formula for squaring a binomial: $(a-b)^2 = a^2 - 2ab + b^2$. In our case, $a = \sqrt{x+45}$ and $b = 5$. Let's plug these values in:

$a^2 = (\sqrt{x+45})^2 = x+45$

$2ab = 2 \times \sqrt{x+45} \times 5 = 10\sqrt{x+45}$

$b^2 = 5^2 = 25$

Putting it all together, the expansion of $(\sqrt{x+45} - 5)^2$ is $(x+45) - 10\sqrt{x+45} + 25$.

So, our equation now looks like this:

$x = (x+45) - 10\sqrt{x+45} + 25$

Let's simplify the right side by combining the constant terms: $45 + 25 = 70$.

$x = x + 70 - 10\sqrt{x+45}$

See how we've managed to eliminate one of the square roots? That's a huge win! However, we still have another square root term, $-10\sqrt{x+45}$, lingering in the equation. This means we're not quite done yet. Our next step will involve isolating this remaining square root term so we can eliminate it in a final squaring step. Keep your eyes peeled; we're getting closer to the solution!

Step 3: Isolating the Remaining Square Root

We're at the stage where our equation is $x = x + 70 - 10\sqrt{x+45}$. Our mission, should we choose to accept it, is to get that pesky $-10\sqrt{x+45}$ term all by itself. Let's simplify the equation first. Notice that we have an '$x$' on both sides. If we subtract '$x$' from both sides, they cancel out:

$x - x = x + 70 - 10\sqrt{x+45} - x$

$0 = 70 - 10\sqrt{x+45}$

Now, this is much cleaner! We've got $0 = 70 - 10\sqrt{x+45}$. Our goal is to isolate the square root term, which is $-10\sqrt{x+45}$. Let's move the $70$ to the other side by subtracting $70$ from both sides:

$0 - 70 = 70 - 10\sqrt{x+45} - 70$

$-70 = -10\sqrt{x+45}$

We're almost there! The square root term is now partially isolated, but it's being multiplied by $-10$. To get $\sqrt{x+45}$ alone, we need to divide both sides by $-10$:

$\frac{-70}{-10} = \frac{-10\sqrt{x+45}}{-10}$

$7 = \sqrt{x+45}$

Success! We've finally managed to isolate the remaining square root. This is the moment we've been working towards. Now that we have a single square root term equal to a constant, we can perform our final squaring operation to completely eliminate the square root and solve for $x$.

Step 4: The Final Squaring and Solving for x

We've reached the home stretch, folks! Our equation is now simplified to $7 = \sqrt{x+45}$. To get rid of the square root on the right side, we do what we've done before: square both sides.

$(7)^2 = (\sqrt{x+45})^2$

Squaring the left side is simple: $7^2 = 49$.

Squaring the right side cancels out the square root: $(\sqrt{x+45})^2 = x+45$.

So, our equation transforms into:

$49 = x+45$

This is as simple as it gets! To find the value of $x$, we just need to isolate it. Subtract $45$ from both sides:

$49 - 45 = x + 45 - 45$

$4 = x$

And there you have it! We've found a potential solution: $x=4$. But hold on, remember that little warning about extraneous solutions? We absolutely must check this answer in the original equation to make sure it's valid.

Step 5: Checking for Extraneous Solutions

This step is non-negotiable, guys. Squaring both sides of an equation can sometimes introduce solutions that don't actually work in the original equation. These are called extraneous solutions. We found $x=4$, so let's plug it back into our original equation: $\sqrt{x}+5=\sqrt{x+45}$.

Substitute $x=4$ into the left side:

$\sqrt{4} + 5 = 2 + 5 = 7$

Now, substitute $x=4$ into the right side:

$\sqrt{4+45} = \sqrt{49} = 7$

Since the left side ($7$) equals the right side ($7$), our solution $x=4$ is correct! It is not an extraneous solution. We can all breathe a sigh of relief and celebrate this victory. The value of $x$ that satisfies the equation $\sqrt{x}+5=\sqrt{x+45}$ is indeed $4$.

Conclusion and Answer

So, after carefully navigating through the steps of isolating square roots, squaring both sides, and diligently checking our work, we've arrived at the solution. The value of $x$ in the equation $\sqrt{x}+5=\sqrt{x+45}$ is $4$. This corresponds to option C. Remember, the key takeaways from this problem are the systematic approach to solving radical equations: isolate, square, isolate again if necessary, square again, and always check your solutions. It might seem like a lot of steps, but with practice, these techniques become second nature. Keep practicing, keep exploring, and don't be afraid of those square roots – they're just numbers waiting to be understood! Great job tackling this one, everyone!