Solving Radical Equations √x+12 - √x-15 = 3 A Step-by-Step Guide
Radical equations, which involve variables inside radical symbols like square roots, cube roots, and so on, can appear daunting at first. However, with a systematic approach, these equations can be solved effectively. This article delves into the process of solving the radical equation √x+12 - √x-15 = 3, providing a detailed, step-by-step solution while emphasizing the importance of checking proposed solutions to avoid extraneous roots. Radical equations, which involve variables inside radical symbols like square roots, cube roots, and so on, can appear daunting at first. However, with a systematic approach, these equations can be solved effectively. This article delves into the process of solving the radical equation √x+12 - √x-15 = 3, providing a detailed, step-by-step solution while emphasizing the importance of checking proposed solutions to avoid extraneous roots. This comprehensive guide will break down each step, ensuring a clear understanding of the techniques involved. Whether you're a student grappling with algebra or someone looking to refresh their math skills, this guide will provide you with the tools and knowledge necessary to tackle radical equations confidently.
The journey through solving radical equations involves isolating the radical terms, squaring both sides (or raising to the appropriate power), simplifying, and ultimately solving for the variable. However, the process doesn't end there; verifying the solutions is crucial to eliminate extraneous roots that may arise from the algebraic manipulations. Extraneous roots are solutions that satisfy the transformed equation but not the original radical equation. Therefore, every solution obtained must be substituted back into the original equation to ensure its validity. This article will not only provide the steps to solve the equation but also emphasize the critical importance of this verification step. Let's embark on this mathematical journey and unravel the solution to √x+12 - √x-15 = 3.
Understanding Radical Equations
Radical equations are equations where the variable appears inside a radical symbol, most commonly a square root. To effectively solve radical equations, it's essential to understand the properties of radicals and how they interact with algebraic operations. The core strategy involves isolating the radical term and then raising both sides of the equation to a power that cancels the radical. For instance, if you have a square root, you would square both sides; if it's a cube root, you would cube both sides, and so on. This process is based on the fundamental principle that if two expressions are equal, then raising both to the same power maintains the equality. However, it's crucial to remember that this operation can sometimes introduce extraneous solutions, which are solutions that satisfy the transformed equation but not the original equation. This is why checking the solutions in the original equation is a non-negotiable step.
When dealing with radical equations, it's also important to consider the domain of the variables. The expressions inside the radical must be non-negative, especially for even-indexed radicals (like square roots). This constraint can help in narrowing down the possible solutions or identifying extraneous ones early in the process. Understanding the domain of the variables is a crucial aspect of solving radical equations, as it provides a framework for the solutions to be valid. For example, in the equation √x+12 - √x-15 = 3, the expressions inside the square roots, x+12 and x-15, must both be greater than or equal to zero. This leads to the conditions x ≥ -12 and x ≥ 15, respectively. Combining these conditions, we find that x must be greater than or equal to 15 for the solutions to be valid. This initial assessment of the domain can save time and effort by preventing the pursuit of solutions that are inherently invalid.
Step-by-Step Solution of √x+12 - √x-15 = 3
Now, let's dive into the step-by-step solution of the equation √x+12 - √x-15 = 3. We'll break down each step to ensure clarity and understanding. Following these steps meticulously will help you confidently tackle similar radical equations in the future. Remember, the key is to isolate the radical terms, eliminate them through appropriate algebraic manipulations, and always verify the solutions.
Step 1: Isolate One Radical Term
The first step in solving this radical equation is to isolate one of the radical terms. This means we want to get one of the square root expressions alone on one side of the equation. In our case, we can achieve this by adding √x-15 to both sides of the equation:
√x+12 = √x-15 + 3
This step simplifies the equation by separating the radicals, making it easier to eliminate them in the subsequent steps. Isolating a radical term is a critical first step because it sets the stage for squaring both sides of the equation. Without isolating a radical, squaring both sides can lead to a more complex expression that is harder to simplify. By isolating the radical, we can effectively eliminate one of the square roots in the next step, thus progressing towards solving for x. This strategic approach is fundamental to solving radical equations efficiently.
Step 2: Square Both Sides
To eliminate the isolated radical, we square both sides of the equation. Squaring both sides is a fundamental operation in solving radical equations, as it directly addresses the radical symbol. This is based on the principle that if two quantities are equal, their squares are also equal. However, it's crucial to remember that squaring both sides can introduce extraneous solutions, which is why we must check our solutions later.
(√x+12)² = (√x-15 + 3)²
This gives us:
x + 12 = (√x-15 + 3)(√x-15 + 3)
Now, we need to expand the right side of the equation. Expanding the right side of the equation involves applying the distributive property (also known as the FOIL method) to multiply the binomial (√x-15 + 3) by itself. This process is crucial for simplifying the equation and moving closer to isolating the variable x. Careful expansion and simplification are essential to avoid errors that can lead to incorrect solutions. The correct expansion will eliminate one of the square roots, making the equation easier to solve. This step is a pivotal moment in the solution process, and accuracy here is paramount.
Step 3: Expand and Simplify
Expanding the right side, we get:
x + 12 = (x - 15) + 6√x-15 + 9
Simplifying, we have:
x + 12 = x - 6 + 6√x-15
Now, we can simplify the equation further by canceling the x terms and isolating the remaining radical term. This is a crucial step in solving radical equations because it helps to isolate the remaining radical, making it easier to eliminate. By canceling the x terms, we reduce the complexity of the equation and bring it closer to a form where we can solve for the variable. This simplification process is a testament to the power of algebraic manipulation in solving complex equations. Each step brings us closer to the solution, highlighting the importance of methodical and precise algebraic techniques.
Step 4: Isolate the Remaining Radical
Subtract x from both sides and add 6 to both sides:
18 = 6**√x-15**
Now, divide both sides by 6:
3 = √x-15
At this stage, we have successfully isolated the remaining radical term. This is a pivotal step in solving radical equations, as it sets the stage for eliminating the radical through squaring or raising to the appropriate power. Isolating the radical term is a strategic move that simplifies the equation and makes it easier to solve for the variable. It demonstrates a clear understanding of the properties of radicals and algebraic manipulation techniques. With the radical term isolated, we can now proceed to the final steps of eliminating the radical and solving for x.
Step 5: Square Both Sides Again
To eliminate the square root, square both sides of the equation again:
3² = (√x-15)²
This gives us:
9 = x - 15
Squaring both sides again is a necessary step to eliminate the remaining square root and solve for x. This operation is based on the same principle as before: if two quantities are equal, their squares are also equal. By squaring both sides, we transform the radical equation into a simple linear equation, which is much easier to solve. This step highlights the importance of strategic algebraic manipulation in simplifying complex equations. However, it's crucial to remember that each time we square both sides, we increase the possibility of introducing extraneous solutions, which underscores the necessity of checking our solutions in the original equation.
Step 6: Solve for x
Add 15 to both sides:
x = 24
Now we have a proposed solution: x = 24. Solving for x involves isolating the variable on one side of the equation. In this case, we achieved this by adding 15 to both sides of the equation, resulting in x = 24. This step represents the culmination of our algebraic manipulations, where we have successfully isolated x and obtained a potential solution. However, it's crucial to remember that this solution is not yet confirmed. We must now proceed to the critical step of checking the solution in the original equation to ensure it is valid and not an extraneous solution. The proposed solution x = 24 is a significant milestone, but the journey is not complete until we verify its correctness.
Checking the Solution
Checking the solution is a crucial step in solving radical equations. This is because squaring both sides of an equation can sometimes introduce extraneous solutions, which are solutions that satisfy the transformed equation but not the original equation. To check our proposed solution, x = 24, we substitute it back into the original equation:
√24+12 - √24-15 = 3
Simplifying, we get:
√36 - √9 = 3
6 - 3 = 3
3 = 3
The equation holds true, so x = 24 is a valid solution. Checking the solution is not merely a formality; it is an essential safeguard against extraneous solutions. By substituting the proposed solution back into the original equation, we ensure that it satisfies the initial conditions of the problem. This verification step confirms that our algebraic manipulations have led us to a valid solution and not a false one. In the case of x = 24, the check confirms that it is indeed a solution to the radical equation √x+12 - √x-15 = 3. This thorough approach is the hallmark of a meticulous problem-solver, ensuring accuracy and validity in mathematical solutions.
Conclusion
In conclusion, solving the radical equation √x+12 - √x-15 = 3 involves a series of steps: isolating a radical term, squaring both sides, simplifying, isolating the remaining radical (if any), squaring again, solving for the variable, and, most importantly, checking the solution. We found that x = 24 is the valid solution to the equation. This process highlights the importance of careful algebraic manipulation and the necessity of verifying solutions in the original equation to avoid extraneous roots. Solving radical equations is a valuable skill in algebra, requiring a combination of algebraic techniques and a keen eye for detail. The systematic approach outlined in this guide provides a framework for tackling similar problems with confidence. Remember, the key is to isolate the radical terms, eliminate them through appropriate algebraic operations, and always verify the solutions to ensure their validity.
By mastering these steps, you can confidently tackle a wide range of radical equations. The ability to solve radical equations is not only a valuable mathematical skill but also a testament to your problem-solving abilities. Each step in the process, from isolating radicals to squaring both sides and checking solutions, reinforces fundamental algebraic concepts. As you continue to practice and apply these techniques, you'll develop a deeper understanding of radical equations and their solutions. The journey of solving √x+12 - √x-15 = 3 serves as a microcosm of the broader mathematical landscape, where precision, verification, and systematic approaches are the cornerstones of success.
