Solving Radical Equation √[3x-4] = 2√[x-5] A Step-by-Step Guide

In this article, we will delve into the process of solving the equation √[3x-4] = 2√[x-5]. This type of equation, involving square roots, falls under the category of radical equations. Solving radical equations often requires a series of algebraic manipulations to isolate the variable and find its value. However, it's crucial to verify the solutions obtained, as the process may introduce extraneous solutions. We aim to provide a comprehensive, step-by-step guide to solve the given equation, ensuring clarity and understanding for readers of all levels. Understanding how to solve these equations is a fundamental skill in algebra, which has numerous applications in various fields of science and engineering. The ability to manipulate and solve equations is crucial for problem-solving, and mastering this skill enhances mathematical proficiency. Let's embark on this journey to unravel the solution to this interesting equation.

Before we tackle the specific equation at hand, it's essential to grasp the basics of radical equations. Radical equations are algebraic equations where the variable appears inside a radical symbol, most commonly a square root. The primary strategy for solving these equations involves isolating the radical term on one side of the equation and then eliminating the radical by raising both sides of the equation to the appropriate power. In the case of a square root, we square both sides. However, this process can sometimes lead to extraneous solutions, which are solutions that satisfy the transformed equation but not the original equation. Therefore, it is imperative to check all solutions in the original equation.

Radical equations appear frequently in various mathematical contexts, including geometry, physics, and engineering. They arise in scenarios where relationships involve square roots, such as distance formulas, velocity calculations, and modeling certain physical phenomena. A solid understanding of how to solve these equations is crucial for success in higher-level mathematics and its applications. Furthermore, mastering the techniques for solving radical equations builds problem-solving skills and enhances algebraic manipulation abilities. Therefore, let’s proceed with the solution, keeping in mind these fundamental principles and the importance of verifying solutions.

Let's systematically solve the equation √[3x-4] = 2√[x-5]. We will go through each step in detail, ensuring clarity and understanding. This step-by-step approach will make it easier to follow the logic and reasoning behind each transformation.

Step 1: Square Both Sides

The first step in solving this radical equation is to eliminate the square roots. To do this, we square both sides of the equation. This operation will remove the radical symbols and allow us to work with a more standard algebraic equation. Squaring both sides gives us:

(√[3x-4])² = (2√[x-5])²

This simplifies to:

3x - 4 = 4(x - 5)

It's crucial to perform the squaring operation correctly, ensuring that the entire side of the equation is squared. In this case, the right side involves squaring both the constant 2 and the term √[x-5]. This step is fundamental in transforming the radical equation into a more manageable form.

Step 2: Expand and Simplify

Next, we need to expand and simplify the equation obtained in the previous step. This involves distributing any constants and combining like terms to bring the equation into a simpler form. Starting from:

3x - 4 = 4(x - 5)

We distribute the 4 on the right side:

3x - 4 = 4x - 20

Now, the equation is in a linear form, which is easier to solve. This step is essential for isolating the variable x and finding its value. Simplifying the equation helps in reducing the complexity and making the subsequent steps more straightforward.

Step 3: Isolate the Variable

To solve for x, we need to isolate the variable on one side of the equation. This involves rearranging terms so that all terms containing x are on one side and constant terms are on the other side. From our simplified equation:

3x - 4 = 4x - 20

We can subtract 3x from both sides:

-4 = x - 20

Then, add 20 to both sides:

16 = x

So, we have found a potential solution: x = 16. Isolating the variable is a key step in solving any algebraic equation, and it requires careful manipulation of terms to maintain equality.

Step 4: Check for Extraneous Solutions

As mentioned earlier, when solving radical equations, it's essential to check for extraneous solutions. These are solutions that satisfy the transformed equation but not the original equation. To check if x = 16 is a valid solution, we substitute it back into the original equation:

√[3x-4] = 2√[x-5]

Substitute x = 16:

√[3(16)-4] = 2√[16-5]

Simplify:

√[48-4] = 2√[11]

√[44] = 2√[11]

2√[11] = 2√[11]

Since the equation holds true, x = 16 is indeed a valid solution. This step is crucial to ensure the accuracy of the solution and to avoid incorrect answers. Checking for extraneous solutions is a hallmark of solving radical equations.

Conclusion

Therefore, the solution to the equation √[3x-4] = 2√[x-5] is x = 16. We have followed a step-by-step process, ensuring each step is clear and understandable. This process included squaring both sides, simplifying the equation, isolating the variable, and, most importantly, checking for extraneous solutions. This comprehensive approach ensures that we have arrived at the correct solution. Solving radical equations requires careful attention to detail, and the techniques used here can be applied to a wide range of similar problems. By understanding these methods, you can confidently tackle various algebraic challenges. Now you should have a clear understanding of how to solve this type of equation effectively.

While we have detailed one method for solving the equation √[3x-4] = 2√[x-5], it's beneficial to explore alternative approaches to broaden our problem-solving skills. Different methods can offer unique insights and may be more efficient for certain types of equations. Understanding multiple techniques allows for a more flexible and adaptable approach to problem-solving. Let's discuss an alternative method that could be used in this case, even though the primary method we used is quite efficient.

Graphical Method

One alternative approach is the graphical method. This involves graphing both sides of the equation as separate functions and finding the points of intersection. The x-coordinates of these intersection points represent the solutions to the equation. For the equation √[3x-4] = 2√[x-5], we can graph y = √[3x-4] and y = 2√[x-5] on the same coordinate plane. The point where the two graphs intersect will give us the solution for x. This method is particularly useful for visualizing the solutions and understanding the behavior of the functions involved.

The graphical method can be a powerful tool, especially when dealing with more complex equations that are difficult to solve algebraically. It provides a visual representation of the solutions and can help in identifying the number of real solutions. While this method might not provide an exact solution in all cases, it can give a good approximation and serve as a check for algebraic solutions. However, it is important to use accurate graphing tools or software to ensure the reliability of the graphical solution. Additionally, the graphical method highlights the importance of understanding the domain and range of the functions involved, which is crucial in interpreting the solutions correctly.

Solving radical equations can be tricky, and there are several common mistakes that students often make. Recognizing these pitfalls can help in avoiding errors and ensuring accurate solutions. Let's discuss some of the most frequent mistakes and how to prevent them. By being aware of these potential issues, you can improve your problem-solving skills and achieve better results. Attention to detail and careful execution are crucial when dealing with radical equations.

Forgetting to Check for Extraneous Solutions

One of the most common mistakes is forgetting to check for extraneous solutions. As mentioned earlier, squaring both sides of an equation can introduce solutions that do not satisfy the original equation. Always substitute the obtained solutions back into the original equation to verify their validity. This step is non-negotiable when solving radical equations. Failing to check for extraneous solutions can lead to incorrect answers and a misunderstanding of the problem.

Incorrectly Squaring Both Sides

Another frequent mistake is incorrectly squaring both sides of the equation. It's essential to square the entire side, not just individual terms. For example, when squaring (2√[x-5]), it's necessary to square both the 2 and the √[x-5]. This requires a solid understanding of algebraic operations and careful execution. A misstep in squaring can lead to a completely different equation and an incorrect solution. Therefore, pay close attention to detail when performing this operation.

Not Isolating the Radical First

Failing to isolate the radical before squaring is another common error. The radical term should be isolated on one side of the equation before squaring both sides. This simplifies the process and reduces the chances of making mistakes. If the radical is not isolated, the squaring operation can become more complex and may lead to unnecessary difficulties. Isolating the radical is a fundamental step in solving radical equations, and it should always be the first step.

Radical equations are not just abstract mathematical concepts; they have numerous real-world applications across various fields. Understanding these applications can provide a deeper appreciation for the practical relevance of this mathematical skill. From physics to engineering, radical equations play a crucial role in solving real-world problems. Let's explore some specific examples where radical equations are used.

Physics

In physics, radical equations are used to model phenomena involving square roots, such as calculating the period of a pendulum or determining the speed of an object in free fall. For instance, the period (T) of a simple pendulum is given by the equation T = 2π√[L/g], where L is the length of the pendulum and g is the acceleration due to gravity. Solving for L or g in this equation involves manipulating a radical equation. Understanding and solving these equations is essential for physicists to make accurate predictions and analyze physical systems. The principles of mechanics often rely on radical equations to describe relationships between different physical quantities.

Engineering

Engineering also utilizes radical equations extensively. For example, in civil engineering, they are used in calculations related to the design of bridges and structures, particularly in determining the stress and strain on materials. In electrical engineering, radical equations can be used to calculate impedance in AC circuits. These applications highlight the importance of a solid understanding of radical equations for engineers. The ability to solve these equations accurately is crucial for designing safe and efficient systems and structures. Engineering relies on precise mathematical models, and radical equations are often a necessary component of these models.

Geometry

Geometry provides another context where radical equations are frequently encountered. The distance formula, which calculates the distance between two points in a coordinate plane, involves a square root. The Pythagorean theorem, a fundamental concept in geometry, also involves square roots and can lead to radical equations when solving for unknown side lengths in right triangles. These geometric applications demonstrate the versatility of radical equations and their significance in understanding spatial relationships. Geometry provides a visual and intuitive setting for applying algebraic concepts, and radical equations are no exception.

In conclusion, solving the equation √[3x-4] = 2√[x-5] demonstrates the process of handling radical equations, which involve isolating the radical, squaring both sides, and, crucially, checking for extraneous solutions. The solution x = 16 was obtained through a step-by-step algebraic manipulation and confirmed by substitution into the original equation. This article not only provided a detailed solution but also explored alternative methods, common mistakes to avoid, and real-world applications of radical equations.

Understanding radical equations is a fundamental skill in algebra with applications in various fields such as physics, engineering, and geometry. By mastering the techniques discussed in this article, you can confidently tackle similar problems and appreciate the practical relevance of mathematics in real-world scenarios. Remember to always check for extraneous solutions and be mindful of common mistakes to ensure accurate results. With a solid grasp of these concepts, you are well-equipped to handle a wide range of algebraic challenges. Continued practice and application of these skills will further enhance your mathematical proficiency.