Solve $\sqrt{3x-1}-2=0$: A Step-by-Step Guide

Welcome, math enthusiasts! Today, we're diving into the world of algebraic equations, specifically tackling a problem that involves a square root. Our mission, should we choose to accept it, is to solve the equation $\sqrt{3x-1}-2=0$. This might look a little intimidating at first glance, with that radical symbol staring back at you, but fear not! We'll break it down into manageable steps, making sure you understand each part of the process. Solving equations like this is a fundamental skill in mathematics, opening doors to more complex problems in algebra, calculus, and beyond. It's not just about finding a number; it's about understanding the logic and the properties of numbers and operations. So, grab your thinking caps, and let's embark on this mathematical journey together. We'll explore the techniques needed to isolate the variable 'x' and arrive at the correct solution. This process involves understanding inverse operations – what's the opposite of taking a square root? We'll also touch upon the importance of checking our answers, a crucial step when dealing with equations that have square roots, as extraneous solutions can sometimes creep in. Get ready to sharpen your problem-solving skills and gain a deeper appreciation for the elegance of algebraic manipulation.

Understanding the Equation: Isolating the Square Root

The very first step in our quest to solve the equation $\sqrt{3x-1}-2=0$ is to make things simpler. Right now, the square root term is sitting there, but it's not quite alone. We have that '-2' hanging out with it. To get closer to finding 'x', we need to isolate the square root. Think of it like unwrapping a present – you want to get to the main gift, right? In this case, the 'gift' is the $\sqrt{3x-1}$ part. To isolate it, we need to perform the opposite operation of subtraction, which is addition. So, we'll add 2 to both sides of the equation. This is a golden rule in algebra: whatever you do to one side of the equation, you must do to the other side to maintain the balance.

Mathematically, this looks like this:

$\sqrt{3x-1}-2 + 2 = 0 + 2$

When we simplify this, the '-2' and '+2' on the left side cancel each other out, and '0 + 2' on the right side simply becomes 2. This leaves us with a much cleaner equation:

$\sqrt{3x-1} = 2$

See? Much better! Now, the square root term is all by itself on one side. This is a significant milestone in solving our equation. It's like reaching a well-lit path after navigating through some tricky terrain. This isolation is key because the next step involves dealing directly with that square root. If we had tried to square both sides before isolating the square root, we'd end up with a more complicated expression involving the '-2' term, which would make the solving process much harder and prone to errors. So, remember this strategy: always isolate the radical term first whenever you encounter an equation of this type. It's a foundational technique that will serve you well in many future mathematical endeavors. This simplified form brings us closer to uncovering the value of 'x' that satisfies the original condition.

Eliminating the Square Root: Squaring Both Sides

Now that we've successfully isolated the square root in our equation, $\\\sqrt{3x-1} = 2$, it's time to tackle that radical symbol head-on. The inverse operation of a square root is squaring. When you square a square root, they cancel each other out, leaving you with just the expression that was inside the radical. So, to get rid of the square root, we're going to square both sides of the equation. Again, the principle of balance is crucial here – what we do to one side, we must do to the other.

Let's see how this plays out:

$(\sqrt{3x-1})^2 = (2)^2$

On the left side, squaring the square root of $(3x-1)$ simply gives us $(3x-1)$. On the right side, squaring 2 means multiplying 2 by itself, which equals 4. So, our equation transforms into:

$3x-1 = 4$

This is fantastic news! We've moved from an equation with a square root to a simple linear equation. This is a testament to the power of using inverse operations. We've effectively 'undone' the square root operation. This step is critical because it allows us to proceed with standard algebraic techniques to solve for 'x'. Without eliminating the square root, we would be stuck. The process of squaring both sides is a common and powerful technique, but it's also important to be aware that it can sometimes introduce what are called 'extraneous solutions.' We'll talk more about that in a bit, but for now, celebrate this victory! We've simplified the problem considerably and are now on the verge of finding our final answer. The journey from a radical equation to a linear one highlights the systematic approach required in mathematics.

Solving the Linear Equation for 'x'

We're in the home stretch! We've simplified our equation to $3x-1 = 4$, and now it's just a matter of isolating 'x' using basic algebraic steps. Remember, the goal is to get 'x' all by itself on one side of the equation. We've already dealt with the square root, so now we're working with a standard linear equation. First, let's get rid of that '-1'. Just like before, we'll use the inverse operation, which is addition. Add 1 to both sides of the equation to maintain balance:

$3x - 1 + 1 = 4 + 1$

This simplifies to:

$3x = 5$

Almost there! Now, 'x' is being multiplied by 3. The inverse operation of multiplication is division. So, to isolate 'x', we'll divide both sides of the equation by 3:

$\frac{3x}{3} = \frac{5}{3}$

This gives us our solution:

$x = \frac{5}{3}$

Congratulations! You've successfully navigated the steps to solve the equation. We started with a square root and ended up with a numerical value for 'x'. This process demonstrates how algebraic manipulation, using inverse operations, allows us to unravel complex-looking equations. The ability to move from a radical equation to a linear one and then solve for the variable is a core skill. It showcases the logical progression and systematic approach that mathematics demands. This solution, $x = 5/3$, is what we believe to be the answer, but in the context of radical equations, there's one final, crucial step: verification.

The Crucial Step: Verifying the Solution

In many types of equations, simply finding a value for 'x' is the end of the story. However, when we deal with equations involving square roots (or other even roots), it's absolutely essential to verify our solution. Why? Because the process of squaring both sides of an equation can sometimes introduce extraneous solutions. These are solutions that work in the simplified equation but do not satisfy the original equation. It's like finding a key that fits a lock but doesn't actually open it! So, we must plug our found value of $x = \frac{5}{3}$ back into the original equation: $\sqrt{3x-1}-2=0$ and see if it holds true.

Let's substitute $x = \frac{5}{3}$ into the original equation:

$\sqrt{3(\frac{5}{3})-1}-2$

First, let's calculate the term inside the square root:

$3(\frac{5}{3}) = \frac{15}{3} = 5$

So, the expression inside the square root becomes:

$5 - 1 = 4$

Now, we have:

$\sqrt{4}-2$

The square root of 4 is 2. So, the expression simplifies to:

$2 - 2$

And finally, $2 - 2 = 0$.

Since $0 = 0$, our solution $x = \frac{5}{3}$ is correct! It satisfies the original equation. This verification step is non-negotiable when working with radical equations. It ensures that the answer we've arrived at is not just a mathematical artifact of our solving process but the actual correct answer to the problem posed. It reinforces the principle that mathematical solutions must be consistent with the initial conditions of the problem. This thoroughness is what makes mathematical reasoning so powerful and reliable. Always remember to check your work, especially with square roots!

Conclusion: Mastering Radical Equations

We've successfully navigated the journey to solve the equation $\sqrt{3x-1}-2=0$, and hopefully, you feel more confident in your ability to tackle similar problems. We started by isolating the radical term, a crucial first step that simplifies the equation. Then, we employed the power of inverse operations by squaring both sides to eliminate the square root, transforming the problem into a manageable linear equation. Finally, we solved for 'x' and, most importantly, verified our solution by plugging it back into the original equation. This verification step is a critical safeguard against extraneous solutions, ensuring the accuracy of our answer.

Mastering equations with square roots involves understanding the properties of radicals and inverse operations. It's a process that requires patience, attention to detail, and a systematic approach. Each step builds upon the last, leading you logically to the solution. Remember the key takeaways:

1. Isolate the radical: Always get the square root term by itself before squaring.
2. Square both sides: This is the primary method to eliminate the square root.
3. Solve the resulting equation: This will likely be a simpler algebraic equation.
4. Verify your solution: Plug your answer back into the original equation to confirm it's correct.

These principles are applicable to a wide range of algebraic problems and form a strong foundation for more advanced mathematics. Keep practicing, and don't shy away from challenges. The more you work through problems like this, the more intuitive these steps will become.

For further exploration into the fascinating world of algebra and equation solving, you might find resources from Khan Academy to be incredibly helpful. They offer a vast array of free lessons and practice exercises that cover these topics and much more.