Solving √[3x + 7] = √[x + 9] A Step-by-Step Guide
Introduction
In this article, we will delve into the step-by-step process of solving the algebraic equation √[3x + 7] = √[x + 9]. This equation involves square roots and requires a systematic approach to isolate the variable x and find its value. Understanding how to solve such equations is a fundamental skill in algebra and is crucial for various mathematical applications. We will break down each step, providing clear explanations and justifications to ensure a comprehensive understanding of the solution. Furthermore, we'll explore the underlying principles and concepts that make this solution valid and accurate. Whether you're a student learning algebra or someone looking to refresh your math skills, this guide will provide you with the necessary tools and insights to solve similar equations confidently.
The journey to solving algebraic equations often begins with recognizing the structure of the equation. In this case, we have a specific type of equation—one that involves square roots. Square root equations require a particular approach, as squaring both sides is a common technique to eliminate the radical signs. However, this method can sometimes introduce extraneous solutions, which are values that satisfy the transformed equation but not the original one. Therefore, a crucial part of the solution process is to check the solutions obtained against the original equation. This step ensures that we only accept the solutions that truly make the original equation true. Through this detailed exploration, we aim to equip you with a robust understanding of how to tackle square root equations effectively and accurately.
Moreover, the techniques used to solve this equation are broadly applicable to a wide range of algebraic problems. The ability to manipulate equations, isolate variables, and verify solutions is a cornerstone of mathematical problem-solving. By mastering these skills, you'll be better equipped to handle more complex mathematical challenges in the future. Our focus will not only be on finding the correct answer but also on fostering a deep understanding of the why behind each step. This understanding will allow you to adapt these techniques to various scenarios and build a strong foundation in algebra. So, let's embark on this mathematical journey together, carefully dissecting each step and uncovering the insights needed to solve the equation √[3x + 7] = √[x + 9].
Step 1: Squaring Both Sides
To begin, the primary goal is to eliminate the square root signs from the equation. The most direct way to achieve this is by squaring both sides of the equation. This technique is based on the fundamental algebraic principle that if two quantities are equal, their squares are also equal. By squaring both sides, we transform the equation √[3x + 7] = √[x + 9] into a more manageable form that allows us to work with the terms inside the square roots directly. This step is crucial because it simplifies the equation, making it easier to isolate the variable x and ultimately solve for its value. Squaring both sides is a common and effective method for dealing with square root equations, but it's important to remember that this operation can sometimes introduce extraneous solutions, which we'll address later in the process.
When we square both sides of the equation √[3x + 7] = √[x + 9], we apply the square function to each side. This results in (√[3x + 7])² = (√[x + 9])². The square function effectively cancels out the square root, leaving us with the expressions inside the square roots. Thus, the equation transforms into 3x + 7 = x + 9. This new equation is linear and much simpler to solve than the original equation with square roots. The simplification achieved in this step is a testament to the power of algebraic manipulation in transforming complex problems into more manageable ones. It’s a crucial step in our journey towards finding the value of x.
However, it's essential to proceed with caution. While squaring both sides simplifies the equation, it can also introduce extraneous solutions. Extraneous solutions are values of x that satisfy the transformed equation (in this case, 3x + 7 = x + 9) but do not satisfy the original equation (√[3x + 7] = √[x + 9]). This phenomenon occurs because the squaring operation can sometimes make unequal quantities appear equal. Therefore, after solving the simplified equation, we must check our solutions against the original equation to ensure they are valid. This verification step is a critical component of solving square root equations, preventing us from accepting solutions that are not truly solutions of the original problem. Keeping this in mind, we move forward to the next step, where we solve the simplified linear equation.
Step 2: Isolating the Variable x
After squaring both sides, we've arrived at the equation 3x + 7 = x + 9. Now, the next logical step is to isolate the variable x. Isolating the variable means manipulating the equation algebraically to get x by itself on one side. This is a fundamental technique in algebra, and it's essential for solving a wide range of equations. Isolating x involves a series of operations that maintain the equality of the equation while moving all terms involving x to one side and all constant terms to the other. By doing so, we progressively simplify the equation until we have x expressed in terms of constants, which gives us the solution.
The process of isolating x typically involves performing inverse operations on both sides of the equation. In this case, we start by subtracting x from both sides of the equation 3x + 7 = x + 9. This gives us 3x - x + 7 = x - x + 9, which simplifies to 2x + 7 = 9. Subtracting x from both sides helps to consolidate the terms involving x on the left side of the equation. The next step is to eliminate the constant term on the left side. We do this by subtracting 7 from both sides of the equation 2x + 7 = 9. This yields 2x + 7 - 7 = 9 - 7, which simplifies to 2x = 2. Now, we have a much simpler equation where the term involving x is isolated on the left side.
Finally, to completely isolate x, we need to eliminate the coefficient 2 that is multiplying x. We achieve this by dividing both sides of the equation 2x = 2 by 2. This gives us (2x) / 2 = 2 / 2, which simplifies to x = 1. We have now successfully isolated x and found a potential solution to the equation. However, remember that we squared both sides of the original equation, so we must verify this solution in the original equation to ensure it is not an extraneous solution. The ability to methodically isolate variables is a crucial skill in algebra, enabling us to solve equations and find the values that make them true. With a potential solution in hand, we proceed to the next step, which is the critical verification process.
Step 3: Verifying the Solution
After finding a potential solution for x, it's absolutely crucial to verify this solution in the original equation. This step is particularly important when dealing with equations involving square roots because, as previously mentioned, squaring both sides can introduce extraneous solutions. Verifying the solution involves substituting the value we found for x back into the original equation and checking if the equation holds true. This process ensures that the value we found not only satisfies the simplified equation but also the original equation with the square roots. If the solution does not satisfy the original equation, it is an extraneous solution and must be discarded.
In our case, we found that x = 1. To verify this solution, we substitute x = 1 into the original equation √[3x + 7] = √[x + 9]. This gives us √[3(1) + 7] = √[1 + 9]. Now, we simplify both sides of the equation separately. On the left side, we have √[3(1) + 7] = √[3 + 7] = √[10]. On the right side, we have √[1 + 9] = √[10]. Thus, our equation becomes √[10] = √[10].
Since √[10] is indeed equal to √[10], the equation holds true when x = 1. This confirms that x = 1 is a valid solution to the original equation. The verification process has ensured that our solution is not an extraneous one and that it truly satisfies the conditions of the original problem. This step is a testament to the importance of thoroughness in mathematical problem-solving. It's not enough to simply find a solution; we must also verify its validity to ensure accuracy. With the solution verified, we can confidently state that x = 1 is the correct solution to the equation √[3x + 7] = √[x + 9]. This final step completes our journey through the solution process, reinforcing the importance of each step along the way.
Final Answer
After carefully working through the steps of squaring both sides, isolating the variable x, and verifying the solution, we have arrived at the final answer. The solution to the equation √[3x + 7] = √[x + 9] is x = 1. This value satisfies the original equation, and we have confirmed its validity through the verification process. The final answer, x = 1, represents the value that, when substituted into the original equation, makes the equation a true statement. This result is the culmination of our algebraic efforts, and it represents a concrete solution to the problem.
The process of solving this equation has highlighted several important algebraic techniques, including squaring both sides to eliminate square roots, isolating variables, and verifying solutions. Each step plays a critical role in the overall solution, and understanding these steps is essential for tackling similar algebraic problems. The solution x = 1 is not just a number; it's the result of a logical and methodical process, demonstrating the power of algebraic manipulation in finding solutions to mathematical problems. Furthermore, the verification step underscores the importance of accuracy and thoroughness in mathematics. By verifying our solution, we ensure that it is not an extraneous one and that it truly satisfies the original equation.
In conclusion, solving equations like √[3x + 7] = √[x + 9] requires a systematic approach and a clear understanding of algebraic principles. The final answer, x = 1, is a testament to the effectiveness of these principles and the importance of a methodical approach to problem-solving. This solution not only provides the answer to the specific equation we addressed but also serves as a valuable learning experience, reinforcing essential algebraic skills and concepts. These skills are transferable to a wide range of mathematical problems, making the process of solving this equation a worthwhile endeavor in mathematical education.
