Solving √[7x-4] = 2√[x] A Step-by-Step Guide
In this article, we will delve into the process of solving the equation √[7x - 4] = 2√[x]. This is a common type of algebraic problem that involves dealing with square roots and requires a systematic approach to find the correct solution. Such equations often appear in mathematics courses, standardized tests, and various real-world applications. The ability to solve equations involving radicals is a fundamental skill in algebra and is essential for more advanced mathematical topics. We will explore each step of the solution in detail, providing explanations and justifications to ensure a clear understanding of the method. By the end of this article, you will not only know the answer but also the reasoning behind each step, empowering you to tackle similar problems with confidence. This detailed exploration aims to enhance your problem-solving skills and deepen your understanding of algebraic manipulations involving radicals. Understanding the nuances of such equations is crucial for success in mathematics and related fields.
The equation √[7x - 4] = 2√[x] presents a classic problem in algebra where we need to find the value(s) of x that satisfy the equation. The key to solving equations involving square roots is to eliminate the radicals by squaring both sides. However, it's important to remember that squaring both sides can sometimes introduce extraneous solutions, which are solutions that satisfy the transformed equation but not the original equation. Therefore, it is crucial to check all potential solutions in the original equation to ensure they are valid. In this article, we will meticulously go through each step, explaining the rationale behind it, and highlighting the importance of checking for extraneous solutions. This methodical approach will provide a comprehensive understanding of how to solve such equations and avoid common pitfalls. The goal is not just to find the answer but to grasp the underlying mathematical principles and techniques involved.
This article will not only provide a step-by-step solution to the equation √[7x - 4] = 2√[x] but also explain the underlying concepts and potential pitfalls. We will begin by squaring both sides of the equation to eliminate the square roots. This will lead to a quadratic equation, which we will then solve using standard algebraic techniques. After finding the potential solutions, we will carefully check each one in the original equation to ensure that they are valid. This step is crucial because squaring both sides of an equation can sometimes introduce extraneous solutions. By emphasizing this verification process, we aim to reinforce the importance of rigor and accuracy in mathematical problem-solving. The detailed explanation and step-by-step approach will make the solution accessible to students and anyone interested in refreshing their algebra skills. Ultimately, this article aims to empower readers with the knowledge and confidence to solve similar equations independently.
1. Square Both Sides
The initial step in solving the equation √[7x - 4] = 2√[x] is to eliminate the square roots. We accomplish this by squaring both sides of the equation. Squaring both sides is a valid algebraic manipulation because if two quantities are equal, then their squares are also equal. However, it is crucial to remember that this operation can sometimes introduce extraneous solutions, which are solutions that satisfy the squared equation but not the original equation. This is why it is essential to check all potential solutions in the original equation at the end of the solving process. Squaring both sides allows us to transform the equation into a simpler form that we can solve using standard algebraic techniques. The squared equation will not contain any square roots, making it easier to manipulate and solve for x. This step is the foundation for the rest of the solution process, and understanding its implications is vital for solving radical equations correctly. By carefully squaring both sides, we set the stage for isolating x and finding its value(s).
Squaring both sides of √[7x - 4] = 2√[x], we get:
(√[7x - 4])² = (2√[x])²
This simplifies to:
7x - 4 = 4x
Now, we have a linear equation that is much easier to solve than the original equation with square roots. The next step is to isolate x by performing algebraic manipulations on the equation. The process of squaring both sides eliminates the radicals, but it's a double-edged sword. While it simplifies the equation, it also opens the door to extraneous solutions. This is because the squaring operation can make a false statement true. For example, if we have -2 = 2 (which is false), squaring both sides gives us 4 = 4 (which is true). This illustrates why checking the solutions in the original equation is so important. The simplification we achieve by squaring both sides is crucial for making progress towards the solution, but we must remain vigilant about the potential for extraneous solutions.
2. Isolate x
After squaring both sides and obtaining the equation 7x - 4 = 4x, the next step is to isolate x. This involves rearranging the terms so that all terms containing x are on one side of the equation and the constant terms are on the other side. This is a fundamental technique in algebra used to solve for an unknown variable. By isolating x, we are essentially getting closer to determining its value. The goal is to perform algebraic operations that maintain the equality of the equation while moving terms around strategically. This step often involves adding or subtracting terms from both sides of the equation to cancel out terms and consolidate like terms. The process of isolating x is a critical skill in algebra and is used extensively in solving various types of equations. Understanding how to isolate a variable is essential for success in more advanced mathematical topics.
To isolate x, we subtract 4x from both sides:
7x - 4 - 4x = 4x - 4x
This simplifies to:
3x - 4 = 0
Next, we add 4 to both sides:
3x - 4 + 4 = 0 + 4
Which gives us:
3x = 4
Now, to completely isolate x, we divide both sides by 3:
3x / 3 = 4 / 3
Thus, we find a potential solution:
x = 4/3
This value of x is a potential solution to the original equation, but it is crucial to remember the earlier discussion about extraneous solutions. Squaring both sides of an equation can introduce solutions that do not satisfy the original equation. Therefore, the next step is to check this potential solution in the original equation to ensure that it is a valid solution. This verification step is not merely a formality; it is a necessary step to ensure the correctness of the solution. By substituting x = 4/3 back into the original equation, we can confirm whether it truly satisfies the equation or if it is an extraneous solution.
3. Check for Extraneous Solutions
As emphasized earlier, checking for extraneous solutions is a critical step when solving equations involving square roots. This is because squaring both sides of an equation, while a valid algebraic manipulation, can introduce solutions that do not satisfy the original equation. These extraneous solutions arise because the squaring operation can make a false statement true, as illustrated in the previous example. Therefore, to ensure the validity of our solution, we must substitute the potential solution we found (x = 4/3) back into the original equation (√[7x - 4] = 2√[x]) and verify that it holds true. This verification process is not optional; it is a mandatory step to guarantee the correctness of the solution. By carefully checking for extraneous solutions, we can avoid the pitfall of accepting an incorrect answer and reinforce the importance of rigor in mathematical problem-solving.
Substitute x = 4/3 into the original equation:
√[7(4/3) - 4] = 2√(4/3)
Simplify the left side:
√[(28/3) - 4] = √[(28/3) - (12/3)] = √(16/3)
Simplify the right side:
2√(4/3) = 2(2/√3) = 4/√3
Now, we need to see if √(16/3) is equal to 4/√3. We can rewrite √(16/3) as:
√(16/3) = √16 / √3 = 4/√3
Since both sides are equal, x = 4/3 is a valid solution.
The process of checking for extraneous solutions involves careful substitution and simplification. Each step must be performed accurately to avoid errors. By verifying the solution in the original equation, we are essentially reversing the steps we took to solve the equation and ensuring that the solution is consistent with the initial conditions. This step not only confirms the correctness of the solution but also deepens our understanding of the equation and the algebraic manipulations involved. The successful verification of x = 4/3 reinforces the importance of this step and highlights the value of thoroughness in mathematical problem-solving. It also gives us confidence that we have found the correct solution to the equation.
In conclusion, the solution to the equation √[7x - 4] = 2√[x] is x = 4/3. We arrived at this solution by systematically following the steps of squaring both sides, isolating x, and, most importantly, checking for extraneous solutions. The process of squaring both sides allowed us to eliminate the square roots and transform the equation into a more manageable form. However, this step also introduced the possibility of extraneous solutions, which necessitated a careful verification process. By substituting the potential solution back into the original equation, we confirmed its validity and ensured that it was not an extraneous solution. This comprehensive approach highlights the importance of a methodical and rigorous problem-solving strategy in algebra. The ability to solve equations involving radicals is a fundamental skill, and mastering this skill requires a deep understanding of algebraic manipulations and the potential pitfalls involved. The solution presented here serves as a model for solving similar equations and reinforces the importance of checking for extraneous solutions.
The step-by-step solution provided in this article not only answers the specific question but also aims to enhance your understanding of algebraic techniques and problem-solving strategies. By carefully explaining each step and emphasizing the importance of checking for extraneous solutions, we hope to empower you to tackle similar problems with confidence. The process of solving equations involving radicals requires a combination of algebraic manipulation, attention to detail, and a thorough verification process. The solution presented here demonstrates the importance of each of these elements. By mastering these techniques, you will be well-equipped to solve a wide range of algebraic problems and deepen your understanding of mathematics. The emphasis on checking for extraneous solutions is a crucial aspect of this approach, as it ensures the accuracy and validity of the solution.
Therefore, the correct answer is B. 4/3. This solution was obtained through a careful and methodical approach, which included squaring both sides of the equation, isolating x, and verifying the solution in the original equation. The emphasis on checking for extraneous solutions is a key takeaway from this problem, as it highlights the importance of rigor and accuracy in mathematical problem-solving. By following these steps, you can confidently solve similar equations involving radicals and avoid common pitfalls. The solution presented here serves as a comprehensive guide to solving such equations and reinforces the importance of a systematic approach.
