Solving √x-4 + 5 = 2 A Step-by-Step Mathematical Guide
Introduction
In mathematics, solving equations is a fundamental skill. Solving equations involves finding the value(s) of the variable(s) that make the equation true. This often requires using algebraic manipulations to isolate the variable on one side of the equation. One common type of equation involves radicals, such as square roots. These equations require special care to solve correctly. In this article, we will thoroughly explore the equation √x-4 + 5 = 2, providing a step-by-step guide to its solution and discussing the underlying principles of solving radical equations. This comprehensive guide aims to provide you with a clear understanding of the process, ensuring you can confidently tackle similar problems in the future. Mastering the techniques for solving radical equations is essential for success in algebra and beyond, as these equations frequently appear in various mathematical contexts and applications. We will delve into the nuances of each step, clarifying why certain operations are performed and highlighting potential pitfalls to avoid. By the end of this guide, you will not only know the solution to this specific equation but also possess a deeper understanding of how to approach and solve radical equations in general. This knowledge will be invaluable as you progress in your mathematical studies and encounter more complex problems. The ability to solve equations involving radicals is a crucial skill that extends beyond the classroom, finding applications in fields such as physics, engineering, and computer science. Therefore, a thorough understanding of the concepts and techniques involved is highly beneficial.
Understanding Radical Equations
Before diving into the solution, let's understand what radical equations are. Radical equations are equations that contain a variable within a radical, most commonly a square root. The presence of the radical requires a specific approach to isolate the variable and find its value. Solving these equations involves careful algebraic manipulation and a critical step of checking for extraneous solutions. Extraneous solutions are values obtained through the solving process that do not satisfy the original equation. These can arise when squaring both sides of an equation, as this operation can introduce solutions that were not present initially. Understanding the nature of radical equations is crucial because they require a systematic method to ensure accurate solutions. The goal is to eliminate the radical by raising both sides of the equation to the appropriate power, which is the index of the radical. For example, to remove a square root (index 2), we square both sides; to remove a cube root (index 3), we cube both sides, and so on. However, this step is where extraneous solutions can be introduced. Therefore, verifying the solutions obtained by substituting them back into the original equation is an essential step in solving radical equations. This verification process ensures that the solutions are valid and that no extraneous roots have been included. Moreover, understanding the domain of the variable within the radical is also important. For square roots, the expression inside the radical must be non-negative, as the square root of a negative number is not a real number. This constraint can help to identify potential solutions that are not valid in the context of the real number system.
Step-by-Step Solution of √x-4 + 5 = 2
Now, let's solve the given equation step by step. Our equation is √x-4 + 5 = 2. The primary goal is to isolate the radical term, which in this case is √x-4. This is achieved by performing algebraic operations that maintain the equality of the equation. We will start by subtracting 5 from both sides of the equation. This will move the constant term to the right side and bring us closer to isolating the square root. This step is crucial because it simplifies the equation and prepares it for the next operation, which involves eliminating the radical. Isolating the radical is a standard first step in solving radical equations, as it allows us to address the radical directly without interference from other terms. After subtracting 5 from both sides, the equation becomes √x-4 = -3. Now, we have the square root term isolated on one side of the equation. The next step is to eliminate the square root by squaring both sides of the equation. Squaring both sides is a valid operation because if two quantities are equal, then their squares are also equal. However, this is also the step where extraneous solutions might be introduced, so we need to be mindful of this. By squaring both sides, we get (√x-4)^2 = (-3)^2, which simplifies to x-4 = 9. Now the equation is free of radicals and we can proceed to solve for x using basic algebraic techniques. Adding 4 to both sides gives us x = 13. This appears to be our solution, but it is crucial to verify whether it is a valid solution or an extraneous one.
Step 1: Isolate the Radical
The first step in solving the equation √x-4 + 5 = 2 is to isolate the radical term. This means getting the term with the square root by itself on one side of the equation. To do this, we subtract 5 from both sides of the equation:
√x-4 + 5 - 5 = 2 - 5
This simplifies to:
√x-4 = -3
Now, the radical term, √x-4, is isolated on the left side of the equation. This step is crucial because it sets the stage for eliminating the square root, which is the next key step in solving the equation. Isolating the radical is a common technique used in solving various types of equations, not just those involving radicals. It helps to simplify the equation and focus on the specific term that needs to be addressed. In this case, by isolating the square root, we can now apply the inverse operation, which is squaring, to eliminate the radical. This process is similar to isolating variables in linear or quadratic equations, where we perform operations to get the variable by itself on one side of the equation. By following this systematic approach, we can effectively solve more complex equations involving radicals and other types of functions. This first step is often the most straightforward, but it is essential for setting up the subsequent steps and ultimately finding the correct solution.
Step 2: Square Both Sides
After isolating the radical, the next step is to eliminate the square root. To do this, we square both sides of the equation. Squaring both sides is the inverse operation of taking the square root, and it will effectively remove the radical from the equation. This operation is based on the principle that if two quantities are equal, then their squares are also equal. However, it is important to note that squaring both sides can sometimes introduce extraneous solutions, which are solutions that satisfy the transformed equation but not the original equation. This is why we need to check our solutions later. Squaring both sides of the equation √x-4 = -3 gives us:
(√x-4)^2 = (-3)^2
This simplifies to:
x - 4 = 9
Now, the equation no longer contains a radical, and we have a simple linear equation that we can solve for x. This step is a significant advancement in solving the equation because it transforms the radical equation into a more manageable form. Squaring both sides is a common technique for solving equations involving square roots, and it is a direct application of the inverse relationship between squaring and taking the square root. However, it is crucial to remember the potential for introducing extraneous solutions and to check the solutions obtained in the original equation. The simplification achieved by this step allows us to apply basic algebraic techniques to isolate the variable and find its value. This step highlights the importance of understanding inverse operations in mathematics and how they can be used to solve equations.
Step 3: Solve for x
Now that we have eliminated the square root, we have a simple linear equation: x - 4 = 9. To solve for x, we need to isolate x on one side of the equation. We can do this by adding 4 to both sides:
x - 4 + 4 = 9 + 4
This simplifies to:
x = 13
So, we have found a potential solution: x = 13. However, it is crucial to remember that squaring both sides of an equation can introduce extraneous solutions. Therefore, we must check this solution in the original equation to make sure it is valid. Solving for x after eliminating the radical is a standard algebraic procedure. The goal is to use inverse operations to isolate the variable and determine its value. In this case, adding 4 to both sides is the inverse operation of subtracting 4, and it effectively isolates x on the left side of the equation. This step demonstrates the fundamental principles of solving equations by maintaining equality through inverse operations. Once we have a potential solution, it is tempting to assume that we have solved the problem. However, in the context of radical equations, the check for extraneous solutions is a critical step that cannot be overlooked. This step ensures the validity of the solution and prevents errors that can arise from the squaring operation. The value obtained for x represents a possible solution, but its true validity can only be determined by substituting it back into the original equation.
Step 4: Check for Extraneous Solutions
This is a crucial step when solving radical equations. We need to substitute the value we found for x back into the original equation to see if it holds true. If it does not, then it is an extraneous solution, and we must discard it. Our original equation was:
√x-4 + 5 = 2
Substitute x = 13 into the equation:
√13-4 + 5 = 2
√9 + 5 = 2
3 + 5 = 2
8 = 2
This statement is false. Since 8 does not equal 2, x = 13 is an extraneous solution. This means that there is no real solution to the original equation. Checking for extraneous solutions is a vital part of the process of solving radical equations. The reason why extraneous solutions can arise is that squaring both sides of an equation can introduce roots that were not present in the original equation. This is because the squaring operation can make a negative value positive, which can lead to solutions that do not satisfy the original equation with the radical. The process of checking involves substituting the potential solution back into the original equation and verifying that the equation holds true. If the equation does not hold true, then the solution is extraneous and must be rejected. In this case, the check clearly shows that x = 13 is not a valid solution, as it leads to a false statement. The conclusion is that the original equation has no real solution. This highlights the importance of the check step and demonstrates that obtaining a value for x does not automatically mean that a solution has been found.
Conclusion
In conclusion, the equation √x-4 + 5 = 2 has no real solution. We arrived at this conclusion by systematically isolating the radical, squaring both sides, solving for x, and then crucially, checking for extraneous solutions. The solution x = 13 that we obtained through the algebraic manipulations turned out to be extraneous, as it did not satisfy the original equation. This underscores the importance of the final verification step when dealing with radical equations. Solving radical equations requires careful attention to detail and a thorough understanding of the algebraic principles involved. The process involves several steps, each of which must be performed accurately to arrive at the correct solution. The first step is to isolate the radical term, which simplifies the equation and prepares it for the next step. The second step is to eliminate the radical by raising both sides of the equation to the appropriate power. This step is based on the inverse relationship between radicals and exponents. However, it is also the step that can introduce extraneous solutions, so caution is required. The third step is to solve the resulting equation for the variable. This usually involves basic algebraic techniques such as adding, subtracting, multiplying, and dividing. The final step, and perhaps the most crucial, is to check the solution(s) obtained in the original equation. This step ensures that the solutions are valid and that no extraneous roots have been included. By following this systematic approach, you can confidently solve radical equations and avoid common pitfalls. The example we discussed in this article clearly demonstrates the importance of the check step and highlights the need for a comprehensive understanding of the solution process.
