Solving The Equation √x+4 - 3 = 1 Step-by-Step Guide
In this article, we will delve into the step-by-step process of solving the equation √x+4 - 3 = 1 for the variable x. This equation involves a square root, which adds a layer of complexity, but with a systematic approach, we can find the solution. Our discussion will cover the fundamental algebraic principles involved, common pitfalls to avoid, and a clear, easy-to-follow method to arrive at the correct answer. Understanding how to solve such equations is crucial for various mathematical applications, from basic algebra to more advanced calculus problems. This guide aims to provide not just the solution, but also a thorough understanding of the underlying concepts, making it a valuable resource for students and anyone looking to refresh their algebraic skills.
Understanding the Equation
Before we dive into the solution, let's first break down the equation √x+4 - 3 = 1. This equation is a radical equation, meaning it contains a variable within a radical, specifically a square root in this case. The presence of the square root necessitates a specific method of solving to ensure we obtain the correct solution and avoid extraneous roots. Extraneous roots are solutions that arise from the solving process but do not satisfy the original equation. These typically occur when we square both sides of an equation, as squaring can introduce solutions that weren't there initially. Therefore, it's crucial to verify our final solution by substituting it back into the original equation. The equation also involves basic arithmetic operations such as addition and subtraction, which we will use to isolate the square root term. Understanding the order of operations and how each operation affects the equation is paramount. Our primary goal is to isolate the square root term first, which will allow us to eliminate the radical by squaring both sides. This will transform the equation into a simpler algebraic form that we can easily solve for x. By meticulously following each step and understanding the reasoning behind it, we can confidently tackle radical equations of this type.
Step-by-Step Solution
To solve the equation √x+4 - 3 = 1, we'll proceed with a step-by-step approach to isolate x. Here's the breakdown:
1. Isolate the Square Root
The first crucial step is to isolate the square root term. To do this, we add 3 to both sides of the equation:
√x+4 - 3 + 3 = 1 + 3
√x+4 = 4
This step simplifies the equation and brings us closer to eliminating the square root. Isolating the radical is essential because it allows us to apply the inverse operation (squaring) effectively. By adding 3 to both sides, we maintain the equality of the equation while moving the constant term to the right side. This prepares the equation for the next step, where we will square both sides to remove the square root. This initial step is a fundamental technique in solving radical equations and sets the stage for the subsequent steps.
2. Eliminate the Square Root
Now that we have isolated the square root, we can eliminate it by squaring both sides of the equation:
(√x+4)² = 4²
x + 4 = 16
Squaring both sides removes the square root, transforming the equation into a simple linear equation. Squaring both sides is a valid operation because if two quantities are equal, their squares are also equal. However, it's important to remember that this step can introduce extraneous solutions, which we'll need to check later. The resulting equation, x + 4 = 16, is much easier to solve than the original radical equation. It's a straightforward linear equation that we can solve by isolating x. This step demonstrates the power of inverse operations in solving equations and how strategically applying them can simplify complex expressions.
3. Solve for x
To solve for x, we subtract 4 from both sides of the equation:
x + 4 - 4 = 16 - 4
x = 12
This gives us a potential solution for x. Solving for x involves isolating the variable on one side of the equation. In this case, by subtracting 4 from both sides, we achieve this goal. The result, x = 12, is our candidate solution. However, before we declare this as the final answer, we must verify it in the original equation to ensure it's not an extraneous solution. This verification step is crucial in solving radical equations because, as mentioned earlier, squaring both sides can sometimes introduce solutions that don't actually satisfy the original equation.
4. Verify the Solution
To verify the solution, we substitute x = 12 back into the original equation:
√12+4 - 3 = 1
√16 - 3 = 1
4 - 3 = 1
1 = 1
The solution x = 12 satisfies the original equation. Verifying the solution is the final and crucial step in solving radical equations. By substituting the candidate solution back into the original equation, we confirm whether it's a valid solution or an extraneous one. In this case, when we substitute x = 12 into the original equation √x+4 - 3 = 1, we find that it holds true. This confirms that x = 12 is indeed a valid solution. If the substitution had resulted in an inequality, it would have indicated that the solution was extraneous, and we would have needed to discard it. This step highlights the importance of rigor in mathematical problem-solving and the necessity of checking our work to ensure accuracy.
Common Mistakes to Avoid
When solving equations like √x+4 - 3 = 1, several common mistakes can lead to incorrect answers. Being aware of these pitfalls can help you avoid them and ensure you arrive at the correct solution. One of the most frequent errors is forgetting to verify the solution. As we discussed earlier, squaring both sides of an equation can introduce extraneous solutions. Therefore, always substitute your solution back into the original equation to check its validity. Another mistake is incorrectly applying the order of operations. For example, some students might try to subtract 3 from 4 inside the square root before isolating the radical. This is incorrect; you must isolate the square root term first before performing any operations within it. Additionally, errors can occur when squaring both sides of the equation if not done carefully. Ensure you square the entire side, not just individual terms. For instance, if you had an equation like √(x+4) = x, squaring both sides should result in x+4 = x², not x+4 = x. Finally, mistakes can arise from simple arithmetic errors, such as incorrect addition or subtraction. To minimize these errors, take your time, write down each step clearly, and double-check your work. By being mindful of these common mistakes and taking precautions to avoid them, you can increase your accuracy in solving radical equations.
Alternative Methods
While the step-by-step method we discussed is the most common and straightforward approach for solving the equation √x+4 - 3 = 1, exploring alternative methods can provide a deeper understanding of the underlying principles and potentially offer different perspectives on problem-solving. One alternative approach involves using graphical methods. By graphing the functions y = √x+4 - 3 and y = 1 on the same coordinate plane, the solution to the equation corresponds to the x-coordinate of the point where the two graphs intersect. This method provides a visual representation of the solution and can be particularly helpful for understanding the nature of solutions in more complex equations. Another alternative method could involve using numerical techniques, such as iterative methods or approximations, especially if an exact algebraic solution is difficult or impossible to obtain. However, for this particular equation, the algebraic method is the most efficient and accurate. Exploring these alternative methods not only enhances your problem-solving skills but also broadens your mathematical toolkit, allowing you to approach different types of equations with greater flexibility and confidence. In the case of this equation, while alternative methods exist, the algebraic method remains the most practical and reliable way to find the solution.
Conclusion
In conclusion, solving the equation √x+4 - 3 = 1 involves a systematic approach that includes isolating the square root, squaring both sides, solving for x, and, most importantly, verifying the solution. We found that x = 12 is the solution to the equation. This process highlights the importance of understanding algebraic principles and the need to be meticulous in each step. By following these steps and being aware of common mistakes, you can confidently solve radical equations. Furthermore, exploring alternative methods provides a broader perspective on problem-solving. Remember, the key to mastering algebra is practice and a thorough understanding of the underlying concepts. This detailed guide should serve as a valuable resource for anyone seeking to improve their equation-solving skills. Whether you are a student learning algebra for the first time or someone looking to refresh your knowledge, the principles discussed here will be invaluable in tackling a wide range of mathematical problems.
