Solving The Equation √2x - 1 = 3 A Step-by-Step Guide

Solving radical equations like √2x - 1 = 3 might seem daunting at first, but with a systematic approach, it becomes a manageable task. This article delves deep into the step-by-step process of solving this equation, ensuring a clear understanding of each stage. We'll not only arrive at the solution but also explore the underlying mathematical principles that make it work. Understanding these principles is crucial because they empower you to tackle a variety of similar problems with confidence. This detailed exploration will cover isolating the radical, squaring both sides, and verifying solutions, which are the core techniques for solving radical equations. So, let's embark on this mathematical journey and master the art of solving equations involving square roots.

The first key step in solving any equation containing a radical is isolating the radical term. In the equation √2x - 1 = 3, the radical term is √2x. To isolate this term, we need to get rid of the '-1' that's being subtracted from it. This is achieved by performing the inverse operation, which in this case is addition. We add 1 to both sides of the equation, ensuring that we maintain the balance of the equation. This fundamental principle of equality dictates that any operation performed on one side must also be performed on the other to preserve the equation's validity. By adding 1 to both sides, we transform the equation from √2x - 1 = 3 to √2x = 4. This step is vital because it sets the stage for the next crucial operation – squaring both sides – which will eliminate the square root and allow us to solve for x directly. Without properly isolating the radical, we wouldn't be able to effectively apply the squaring operation and move forward in the solving process. This isolation process simplifies the equation, making it easier to manipulate and ultimately find the solution.

With the radical term isolated, the next crucial step is to eliminate the square root. To achieve this, we employ the operation that's the inverse of taking a square root: squaring. Squaring both sides of the equation is a fundamental technique in solving radical equations. This is based on the principle that if two expressions are equal, then their squares are also equal. When we square √2x, we effectively undo the square root, leaving us with 2x. Similarly, squaring the right side, which is 4, gives us 16. This transforms the equation from √2x = 4 into 2x = 16. It's important to understand that squaring both sides can sometimes introduce extraneous solutions – solutions that satisfy the transformed equation but not the original equation. Therefore, it's always essential to verify the solutions we obtain in the original equation. However, this step of squaring both sides is critical because it removes the radical, making the equation significantly easier to solve for the variable. By squaring both sides, we transition from an equation with a square root to a simple linear equation, which can be solved using basic algebraic techniques.

Now that we've eliminated the square root and simplified the equation to 2x = 16, the next step is to isolate x and solve for its value. To achieve this, we need to undo the multiplication by 2 that's currently acting on x. The inverse operation of multiplication is division, so we divide both sides of the equation by 2. This operation is another application of the principle of equality, ensuring that we maintain the balance of the equation. Dividing both sides by 2 transforms the equation from 2x = 16 to x = 8. This step directly reveals the potential solution for x, which is 8. However, as we mentioned earlier, it's crucial to verify this solution in the original equation to ensure that it's not an extraneous solution. This verification process is a vital safeguard against errors and ensures that we arrive at the correct answer. The division step is the final algebraic manipulation required to isolate x and determine its value, bringing us closer to the complete solution of the equation.

After obtaining a potential solution, the final and arguably most important step is verification. This step involves substituting the obtained value of x back into the original equation to ensure that it satisfies the equation. In our case, we found that x = 8. So, we substitute 8 for x in the original equation, which is √2x - 1 = 3. This gives us √2(8) - 1 = 3. Simplifying the expression under the square root, we get √16 - 1 = 3. The square root of 16 is 4, so the equation becomes 4 - 1 = 3. Finally, 4 - 1 indeed equals 3, which confirms that the left-hand side of the equation equals the right-hand side. This verification process is essential because, as mentioned earlier, squaring both sides of an equation can sometimes introduce extraneous solutions. These are values that satisfy the transformed equation but not the original equation. By verifying the solution, we ensure that it's a valid solution and not an extraneous one. In this case, since x = 8 satisfies the original equation, it is indeed the correct solution. This step adds a layer of rigor to the solving process and guarantees the accuracy of the final answer. Verification is not just a check; it's a fundamental part of solving radical equations.

In conclusion, by systematically applying the principles of algebra and following a step-by-step approach, we have successfully solved the equation √2x - 1 = 3. We began by isolating the radical term, then squared both sides to eliminate the square root, and finally solved for x. The potential solution we obtained was x = 8. To ensure the validity of this solution, we performed a crucial verification step, substituting x = 8 back into the original equation. This verification confirmed that x = 8 is indeed the correct solution. The process we've outlined highlights the importance of not only knowing the steps to solve an equation but also understanding the underlying mathematical principles that make those steps valid. This understanding empowers you to solve a wider range of equations with confidence. Remember, solving radical equations involves a combination of algebraic manipulation and careful verification. With practice and a firm grasp of these concepts, you can confidently tackle any radical equation that comes your way. This exercise provides a solid foundation for more advanced mathematical concepts and problem-solving skills. The key takeaways from this process are the techniques of isolating the radical, squaring both sides, solving the resulting equation, and, most importantly, verifying the solution. These steps are the building blocks for solving a variety of equations involving radicals and are fundamental to mathematical problem-solving.

The correct answer is C) x = 8.