First Step To Solve The Equation √(5 + X) - √x = 3
Navigating the realm of radical equations can seem daunting, but with a systematic approach, even the most intricate problems can be解明. This article delves into the first crucial step in solving the equation √(5 + x) - √x = 3, a common type of problem encountered in algebra and beyond. We'll explore why this step is essential and how it sets the stage for a successful solution. Before diving into the specifics, let's first understand the nature of radical equations and the strategies we employ to solve them. Radical equations, characterized by the presence of variables within radical symbols (like square roots, cube roots, etc.), require specific techniques to isolate the variable. Unlike linear or quadratic equations, we can't simply add, subtract, multiply, or divide terms to get the variable by itself. Instead, we need to eliminate the radicals themselves. This is where the concept of squaring (or cubing, etc., depending on the type of radical) both sides of the equation comes into play. However, there's a critical preliminary step that often makes the squaring process much smoother and prevents unnecessary complications. That step, as we'll see, involves strategically rearranging the terms in the equation. Understanding this initial move is paramount to efficiently解明 radical equations and ensuring accurate solutions. So, let's embark on this journey of algebraic解明, starting with the vital first step.
The Significance of Isolating a Radical Term
The key to efficiently tackling equations like √(5 + x) - √x = 3 lies in the strategic isolation of radical terms. Isolating a radical means getting a single radical expression by itself on one side of the equation. Why is this so important? The answer lies in the process of eliminating the radical. When we square both sides of an equation, our goal is to eliminate the square root. However, if we have multiple radical terms on the same side, squaring the entire side can lead to a more complex expression due to the cross-terms that arise from the binomial expansion (think (a - b)² = a² - 2ab + b²). These cross-terms often reintroduce radicals, defeating the purpose of squaring in the first place. Imagine squaring the original equation √(5 + x) - √x = 3 directly. We would get (√(5 + x) - √x)² = 3². Expanding the left side would give us (5 + x) - 2√(5 + x)√x + x = 9, which still contains a radical term (-2√(5 + x)√x). This illustrates why isolating a radical term first is so crucial. By isolating one of the square roots, we set ourselves up for a cleaner squaring process. When we square a single square root, the square and the square root effectively cancel each other out, simplifying the equation. This simplification makes the subsequent steps of solving for x much easier and less prone to errors. In the context of our equation, isolating one of the radicals before squaring significantly streamlines the solution process, making it the most logical and efficient first step. It's a testament to the power of strategic algebraic manipulation.
Option 3: Transposing for Isolation - The Optimal First Step
Among the options presented, transposing one of the radical expressions to the other side of the equation stands out as the most effective initial move. Transposing, in this context, means adding or subtracting a term from both sides of the equation to move it to the opposite side. For our equation, √(5 + x) - √x = 3, transposing the term -√x to the right side results in √(5 + x) = 3 + √x. Notice what this accomplishes: we have successfully isolated the radical term √(5 + x) on the left side of the equation. This isolation is a crucial prerequisite for the next step, which will involve squaring both sides. By isolating the radical, we've paved the way for a much cleaner and more manageable squaring operation. Squaring both sides of √(5 + x) = 3 + √x will eliminate the square root on the left side, giving us 5 + x. On the right side, we'll have to expand (3 + √x)², but this expansion is far simpler than squaring the original expression with two radicals. In contrast, consider the consequences of not transposing first. If we were to square the original equation √(5 + x) - √x = 3 directly, we'd encounter the complications of squaring a binomial containing two radical terms, as discussed earlier. This would lead to a more complex equation with a radical term still present, making it harder to solve. Therefore, transposing one of the radical expressions to isolate the other is the most strategic and efficient first step in solving the equation √(5 + x) - √x = 3. It sets the stage for a smoother solution process and minimizes the chances of algebraic errors.
Why Other Options Fall Short
While option 3, transposing a radical, is the optimal first step, it's crucial to understand why the other options are less effective or even incorrect at this stage. Let's examine why options 1 and 2 are not the ideal starting points. Option 1, squaring both sides of the equation immediately, might seem like a direct approach to eliminate the square roots. However, as we've discussed, squaring √(5 + x) - √x = 3 directly leads to a more complicated expression due to the binomial expansion. The cross-term -2√(5 + x)√x reintroduces a radical, making the equation more difficult to solve. This approach bypasses the crucial step of isolating a radical, which is essential for efficient simplification. Therefore, while squaring both sides is a necessary step in solving radical equations, doing it prematurely, before isolating a radical, is counterproductive. Option 2, getting the square root of both sides, is fundamentally incorrect in this context. We are trying to eliminate square roots, not introduce more. Taking the square root of both sides would further complicate the equation, making it even harder to isolate x. This option demonstrates a misunderstanding of the strategy for solving radical equations. The correct approach involves using the inverse operation of the square root, which is squaring, to eliminate the radical. In summary, options 1 and 2 fail to address the core strategy of isolating a radical term before attempting to eliminate it. Option 3, transposing a radical, directly addresses this strategy and sets the stage for a much more manageable solution process.
Subsequent Steps in Solving the Equation
Having established that transposing one of the radical expressions is the correct first step, let's briefly outline the subsequent steps involved in solving the equation √(5 + x) - √x = 3. After transposing -√x to the right side, we have √(5 + x) = 3 + √x. The next step is to square both sides of the equation. This eliminates the square root on the left side and expands the right side: (√(5 + x))² = (3 + √x)² which simplifies to 5 + x = 9 + 6√x + x. Notice that while we've eliminated one square root, another remains on the right side. This is a common occurrence in radical equations, and it simply means we need to repeat the process of isolating and squaring. In this case, we can simplify the equation 5 + x = 9 + 6√x + x by subtracting x from both sides and subtracting 9 from both sides, resulting in -4 = 6√x. Now, we isolate the remaining radical term by dividing both sides by 6: -4/6 = √x, which simplifies to -2/3 = √x. Finally, we square both sides again to eliminate the last square root: (-2/3)² = (√x)² which gives us 4/9 = x. It's crucial to remember to check our solution in the original equation. Substituting x = 4/9 into √(5 + x) - √x = 3, we get √(5 + 4/9) - √(4/9) = √(49/9) - 2/3 = 7/3 - 2/3 = 5/3. Since 5/3 ≠ 3, our solution x = 4/9 is an extraneous solution, meaning it doesn't satisfy the original equation. This highlights the importance of checking solutions in radical equations, as the squaring process can sometimes introduce extraneous solutions. In this case, the equation has no real solutions.
Conclusion: Mastering the First Step for Success
In conclusion, the first and most crucial step in solving the equation √(5 + x) - √x = 3 is to transpose one of the radical expressions to the other side of the equation. This strategic move, option 3, sets the stage for a much smoother and more efficient solution process. By isolating a radical term, we avoid the complexities of squaring a binomial containing multiple radicals, which can lead to unnecessary complications and errors. Understanding the importance of this initial step is key to mastering the art of solving radical equations. While squaring both sides is a necessary step, doing it prematurely, before isolating a radical, can be counterproductive. The subsequent steps involve squaring both sides, isolating the remaining radical (if any), squaring again, and finally, checking for extraneous solutions. The example we worked through illustrates the importance of checking solutions, as the squaring process can sometimes introduce solutions that don't actually satisfy the original equation. By mastering the first step of transposing and understanding the subsequent steps, you'll be well-equipped to tackle a wide range of radical equations with confidence and accuracy. Remember, a strategic approach, starting with the right first step, is the key to解明 mathematical challenges.
