Solving The Equation √x+14 = 3 A Step-by-Step Guide
When it comes to solving radical equations, the presence of a square root (or any other root) adds a layer of complexity compared to standard algebraic equations. Our main goal remains the same: to isolate the variable. However, we need to employ specific techniques to eliminate the radical. This article focuses on a detailed, step-by-step solution for the equation √x + 14 = 3. We'll break down each step, explaining the reasoning behind it, and ensure a clear understanding of how to tackle similar problems in the future. This approach is crucial not just for this specific equation, but for mastering the broader concept of solving radical equations, which frequently appears in various mathematical contexts. Understanding the process thoroughly empowers you to confidently approach more complex problems involving radicals and algebraic manipulations. Moreover, the ability to solve radical equations is foundational for advanced topics in algebra and calculus, highlighting the importance of grasping these core principles. The step-by-step methodology not only aids in arriving at the correct solution but also helps in building a solid foundation for problem-solving in mathematics, enhancing logical reasoning and analytical skills applicable across various disciplines.
Understanding the Equation
Before diving into the solution, it's important to understand the equation we're dealing with: √x + 14 = 3. This is a radical equation because it contains a variable, 'x', under a square root symbol. The square root acts as a grouping symbol, and we need to isolate 'x' to find its value. The constant term '14' is added to 'x' inside the square root, which affects how we approach the solution. The presence of this constant necessitates a careful sequence of operations to correctly isolate 'x'. Misinterpreting the equation or attempting to square both sides prematurely can lead to errors. Therefore, a clear understanding of the equation's structure is paramount. The initial step in solving this equation will be to isolate the square root term on one side of the equation. Once isolated, we can then apply the appropriate inverse operation to eliminate the radical. This careful and deliberate approach ensures the integrity of the equation throughout the solution process. Recognizing the type of equation and the specific operations required is a key aspect of problem-solving in mathematics, helping students develop a strategic approach rather than relying solely on rote memorization. This analytical approach not only helps in solving equations but also fosters a deeper understanding of mathematical concepts and their interrelationships.
Step-by-Step Solution
Now, let's walk through the step-by-step solution to the equation √x + 14 = 3. Our first task is to isolate the square root term. This means getting √x by itself on one side of the equation. To do this, we subtract 14 from both sides of the equation. This maintains the balance of the equation while moving the constant term away from the radical. The new equation becomes √x = 3 - 14, which simplifies to √x = -11. This step is crucial because squaring both sides before isolating the square root would lead to a more complicated equation. Now that we have the square root isolated, the next step is to eliminate the square root. We achieve this by squaring both sides of the equation. Squaring a square root effectively cancels out the radical, leaving us with the expression inside the square root. So, (√x)² = (-11)². This simplifies to x = 121. However, we need to perform a crucial verification step. Squaring both sides of an equation can sometimes introduce extraneous solutions, which are solutions that satisfy the transformed equation but not the original equation. Therefore, we must substitute x = 121 back into the original equation to check if it holds true. Substituting x = 121 into √x + 14 = 3, we get √121 + 14 = 3, which simplifies to 11 + 14 = 3, or 25 = 3. This is clearly false. Therefore, x = 121 is an extraneous solution. Since we found an extraneous solution and no other possible solutions, the equation has no real solution.
Step 1: Isolate the Square Root
The first step: isolate the square root is crucial when dealing with radical equations. To achieve this, our aim is to get the term containing the square root by itself on one side of the equation. In our equation, √x + 14 = 3, the square root term is √x + 14. We need to isolate √x. This involves performing the inverse operation to eliminate the constant term '+14'. The inverse operation of addition is subtraction. Therefore, we subtract 14 from both sides of the equation. This ensures that we maintain the balance of the equation, as any operation performed on one side must also be performed on the other side to preserve equality. Subtracting 14 from both sides, we get: √x + 14 - 14 = 3 - 14. This simplifies to √x = -11. Isolating the square root term is a critical step because it sets the stage for the next operation: squaring both sides. Squaring both sides before isolating the square root would lead to a more complex equation, potentially making the problem more difficult to solve. By isolating the square root, we create a direct pathway to eliminating the radical and solving for the variable. This step demonstrates a fundamental principle in solving equations: simplifying the equation as much as possible before applying more complex operations. A clear understanding of this principle is crucial for tackling a wide range of algebraic problems, not just radical equations.
Step 2: Square Both Sides
Once we step 2: square both sides, we move closer to solving for 'x'. In the previous step, we isolated the square root, resulting in the equation √x = -11. To eliminate the square root, we perform the inverse operation, which is squaring. We square both sides of the equation to maintain balance. Squaring the left side, (√x)², cancels out the square root, leaving us with just 'x'. Squaring the right side, (-11)², means multiplying -11 by itself, which equals 121. Therefore, the equation becomes x = 121. This step seems straightforward, but it's essential to remember a crucial caveat: squaring both sides of an equation can sometimes introduce extraneous solutions. Extraneous solutions are solutions that satisfy the squared equation but do not satisfy the original equation. This is because the squaring operation can mask the sign of a number. For example, both 11 and -11, when squared, result in 121. Therefore, it is imperative to verify any solution obtained after squaring both sides by substituting it back into the original equation. This verification step is not just a formality; it's a necessary safeguard against accepting incorrect solutions. Understanding the potential for extraneous solutions is a key aspect of solving radical equations and demonstrates a deeper understanding of algebraic manipulations.
Step 3: Check for Extraneous Solutions
After finding a potential solution, the step 3: check for extraneous solutions is a crucial and often overlooked step in solving radical equations. As mentioned earlier, squaring both sides of an equation can introduce extraneous solutions, which are values that satisfy the transformed equation but not the original one. In our case, we found x = 121 after squaring both sides. To check if this is a valid solution, we need to substitute it back into the original equation: √x + 14 = 3. Substituting x = 121, we get √121 + 14 = 3. Simplifying the square root, we have 11 + 14 = 3. This further simplifies to 25 = 3, which is clearly false. This demonstrates that x = 121 is an extraneous solution and not a valid solution to the original equation. The fact that we obtained a false statement highlights the importance of this verification step. If we had skipped this step, we would have incorrectly concluded that x = 121 is the solution. Since we found an extraneous solution and no other potential solutions, we conclude that the original equation, √x + 14 = 3, has no real solution. This outcome is not uncommon in radical equations, and recognizing the possibility of no solution is just as important as finding a solution. The process of checking for extraneous solutions reinforces the understanding of equation equivalence and the impact of algebraic manipulations on the solution set.
Conclusion
In conclusion, solving the equation √x + 14 = 3 involves isolating the square root, squaring both sides, and crucially, checking for extraneous solutions. We found that x = 121 is an extraneous solution, and therefore, the equation has no real solution. This exercise highlights the importance of each step in the process and emphasizes the need for careful verification when dealing with radical equations. The process of solving this equation provides valuable insights into handling radical expressions and underscores the significance of rigorous solution verification in algebra. By following a systematic approach and understanding the potential pitfalls, students can confidently tackle a wide range of radical equations. The key takeaways from this exercise are: first, isolate the radical term; second, eliminate the radical by squaring both sides; and third, always check for extraneous solutions by substituting the potential solution back into the original equation. Mastering these steps ensures accuracy and a thorough understanding of the underlying principles of algebraic manipulation. Furthermore, this approach not only applies to simple radical equations but also forms the foundation for solving more complex problems involving radicals and algebraic expressions, building a robust foundation for advanced mathematical studies.
