Solving Radical Equations A Step-by-Step Guide To X = √(2x + 3)

Radical equations, which involve variables inside a radical symbol (like a square root), can appear daunting at first. However, with a systematic approach, they can be solved effectively. This article will guide you through the process of solving the radical equation x = √(2x + 3). We will break down each step, provide clear explanations, and emphasize the crucial step of checking the solution to ensure its validity. Understanding how to solve radical equations is a fundamental skill in algebra and is essential for more advanced mathematical concepts. We aim to make this process as clear and straightforward as possible, so you can confidently tackle similar problems in the future. Our focus will be on showing every step involved, explaining the reasoning behind each step, and highlighting common pitfalls to avoid. This comprehensive guide will ensure that you not only get the correct solution but also understand the underlying principles of solving radical equations. So, let's dive in and master the art of solving these equations.

Understanding Radical Equations

Before diving into the specific equation, let's understand what radical equations are and the key principles involved in solving them. A radical equation is an equation that contains a variable within a radical expression, most commonly a square root. The main challenge in solving these equations is dealing with the radical, which requires a specific approach to isolate the variable. The primary strategy is to eliminate the radical by raising both sides of the equation to the power that corresponds to the index of the radical. For example, if you have a square root (index 2), you would square both sides. If it's a cube root, you would cube both sides, and so on. However, this process can sometimes introduce extraneous solutions, which are solutions that satisfy the transformed equation but not the original radical equation. This is why checking your solutions is an absolutely crucial step in solving radical equations. Understanding the concept of extraneous solutions and the importance of checking solutions is essential for avoiding errors and ensuring the correctness of your results. Remember, mathematical precision is key, and verifying your solutions is a critical part of that precision. With a solid understanding of these principles, you are well-equipped to tackle a variety of radical equations.

The Given Equation: x = √(2x + 3)

Our task is to solve the radical equation x = √(2x + 3). This equation presents a typical example of a radical equation, where the variable x appears both inside and outside the square root. To solve this, we will follow a standard procedure that involves isolating the radical term and then eliminating it by squaring both sides of the equation. This process will transform the equation into a more familiar form, typically a quadratic equation, which we can then solve using standard algebraic techniques. However, as we proceed, we must be mindful of the potential for extraneous solutions. These can arise because squaring both sides of an equation can introduce solutions that don't actually satisfy the original equation. Therefore, we will need to check any solutions we obtain by substituting them back into the original equation. This step is not just a formality; it's a critical part of the solution process that ensures we only accept valid answers. The methodical approach we will take will not only help us find the correct solutions but also reinforce the importance of verification in solving mathematical problems. Now, let's begin the step-by-step solution of the equation.

Step-by-Step Solution

Step 1: Isolate the Radical Term

The first step in solving a radical equation is to isolate the radical term on one side of the equation. In our case, the radical term is √(2x + 3). Fortunately, in the equation x = √(2x + 3), the radical term is already isolated on the right side. This simplifies our initial setup and allows us to move directly to the next step, which involves eliminating the radical. Isolating the radical is a crucial first step because it sets the stage for the next operation, which is raising both sides of the equation to a power that will cancel out the radical. If the radical term were not isolated, we would need to perform algebraic manipulations to get it by itself before proceeding. This might involve adding, subtracting, multiplying, or dividing terms to move them to the other side of the equation. However, since our radical term is already isolated, we can proceed more efficiently. This highlights the importance of careful observation and strategic planning when solving equations. Recognizing that the radical is already isolated saves us time and effort, allowing us to focus on the next phase of the solution.

Step 2: Eliminate the Radical

To eliminate the square root in the equation x = √(2x + 3), we need to square both sides of the equation. Squaring both sides is the inverse operation of taking the square root, and it effectively cancels out the radical. This transforms our equation into a more manageable form that we can solve using standard algebraic techniques. When we square the left side (x), we get x². When we square the right side (√(2x + 3)), the square root symbol is removed, leaving us with 2x + 3. Thus, our equation becomes x² = 2x + 3. This step is crucial because it converts the radical equation into a polynomial equation, specifically a quadratic equation, which we know how to solve. However, it is important to remember that squaring both sides can introduce extraneous solutions, which is why we will need to check our answers later. The transformation from a radical equation to a polynomial equation is a common strategy in solving these types of problems, and mastering this step is essential for success. Now that we have a quadratic equation, we can move on to solving it.

Step 3: Solve the Resulting Equation

After squaring both sides, our equation is now x² = 2x + 3. This is a quadratic equation, which we can solve by rearranging it into the standard form (ax² + bx + c = 0) and then factoring, using the quadratic formula, or completing the square. In this case, let's rearrange the equation by subtracting 2x and 3 from both sides, which gives us x² - 2x - 3 = 0. Now, we can try to factor this quadratic. We are looking for two numbers that multiply to -3 and add to -2. These numbers are -3 and 1. Therefore, we can factor the quadratic as (x - 3)(x + 1) = 0. To find the solutions for x, we set each factor equal to zero: x - 3 = 0 and x + 1 = 0. Solving these linear equations gives us x = 3 and x = -1. These are our potential solutions, but we must remember the crucial step of checking for extraneous solutions. Factoring is a powerful technique for solving quadratic equations, and it's often the quickest method when the quadratic can be factored easily. However, if factoring is difficult, the quadratic formula or completing the square can always be used. Now, let's proceed to the crucial step of verifying our potential solutions.

Step 4: Check Your Solution

As we mentioned earlier, checking for extraneous solutions is a vital step in solving radical equations. Extraneous solutions are potential solutions that satisfy the transformed equation (in our case, the quadratic equation) but do not satisfy the original radical equation. To check our solutions, we will substitute each potential solution (x = 3 and x = -1) back into the original equation, x = √(2x + 3), and see if the equation holds true.

• Checking x = 3:

  Substituting x = 3 into the original equation, we get 3 = √(2(3) + 3). Simplifying the right side, we have 3 = √(6 + 3), which further simplifies to 3 = √9. Since √9 is indeed 3, the equation 3 = 3 holds true. Therefore, x = 3 is a valid solution.

• Checking x = -1:

  Substituting x = -1 into the original equation, we get -1 = √(2(-1) + 3). Simplifying the right side, we have -1 = √(-2 + 3), which simplifies to -1 = √1. Since √1 is 1, the equation becomes -1 = 1, which is false. Therefore, x = -1 is an extraneous solution and not a valid solution to the original equation.

This step clearly demonstrates the importance of checking solutions. We found that while x = 3 is a valid solution, x = -1 is not. If we had skipped this step, we would have incorrectly included x = -1 in our solution set. By performing this verification, we ensure that our solution is accurate and complete.

Final Answer

After solving the equation x = √(2x + 3) and checking for extraneous solutions, we have determined that the only valid solution is x = 3. The potential solution x = -1 was found to be extraneous, meaning it does not satisfy the original radical equation.

Therefore, the final answer is:

B. 3

This solution represents the value of x that makes the original equation true. It is crucial to remember that in solving radical equations, the process of checking for extraneous solutions is as important as the algebraic manipulations used to find the potential solutions. This practice ensures that we arrive at the correct and complete answer.

Conclusion

In this article, we have thoroughly walked through the process of solving the radical equation x = √(2x + 3). We began by understanding the nature of radical equations and the importance of isolating the radical term. We then eliminated the radical by squaring both sides of the equation, which transformed it into a quadratic equation. After solving the quadratic equation, we obtained two potential solutions: x = 3 and x = -1. Crucially, we emphasized the necessity of checking these solutions in the original equation to identify any extraneous solutions.

By substituting x = 3 and x = -1 back into the original equation, we found that x = 3 is a valid solution, while x = -1 is an extraneous solution. This step highlighted the critical role of verification in ensuring the accuracy of our results. The final answer, therefore, is x = 3.

This exercise not only demonstrates the specific steps involved in solving radical equations but also reinforces the broader mathematical principles of careful manipulation, attention to detail, and the importance of checking one's work. Mastering these skills is essential for success in algebra and beyond. Remember, solving mathematical problems is not just about finding the answer; it's about understanding the process and developing a methodical approach that can be applied to a wide range of problems.