Solving √(8x + 1) = X + 2 Finding Real Roots

In this article, we will delve into solving the equation √(8x + 1) = x + 2 for its real roots. This type of problem falls under the category of algebraic equations, specifically those involving square roots. Solving equations with radicals requires a methodical approach to ensure we arrive at the correct solutions while avoiding extraneous roots. Extraneous roots are solutions that emerge during the solving process but do not satisfy the original equation. This comprehensive guide aims to provide a step-by-step solution, complete with explanations and checks, to ensure clarity and accuracy. We'll start by understanding the initial equation and then proceed with the algebraic manipulations required to isolate x and find its possible values. Finally, we will verify each solution to confirm its validity, making this article an invaluable resource for students and math enthusiasts alike. Understanding how to solve radical equations is a fundamental skill in algebra, and mastering this technique will significantly enhance your problem-solving abilities. Let's embark on this mathematical journey together, ensuring every step is clear and easy to follow, so you can confidently tackle similar problems in the future.

Before we begin solving, let's take a moment to fully understand the equation √(8x + 1) = x + 2. This equation involves a square root, which means we are looking for values of x that, when substituted into the equation, make both sides equal. The expression inside the square root, (8x + 1), must be non-negative because the square root of a negative number is not a real number. This gives us our first constraint: 8x + 1 ≥ 0. Additionally, since the square root function always yields a non-negative result, the right side of the equation, (x + 2), must also be non-negative. This provides another constraint: x + 2 ≥ 0. These constraints are crucial because they help us identify potential extraneous solutions later in the process. Extraneous solutions are values of x that satisfy the transformed equations but not the original equation. Recognizing and addressing these constraints from the beginning ensures we only consider valid solutions. The presence of a square root requires careful algebraic manipulation to isolate x. Squaring both sides is a common technique, but it can also introduce extraneous solutions. Therefore, a thorough verification process is essential. By understanding the equation's structure and the implications of the square root, we can approach the problem methodically and accurately, ensuring we find all real roots and avoid any potential pitfalls. In the following sections, we will discuss the detailed steps for solving the equation, keeping these initial constraints in mind.

To solve the equation √(8x + 1) = x + 2, we will follow a series of algebraic steps. First, we need to eliminate the square root by squaring both sides of the equation. This gives us: (√(8x + 1))² = (x + 2)². Simplifying the left side, we get 8x + 1. Expanding the right side, we obtain x² + 4x + 4. Now, we have a quadratic equation: 8x + 1 = x² + 4x + 4. To solve the quadratic equation, we need to set it to zero. Subtracting 8x and 1 from both sides, we get: 0 = x² + 4x + 4 - 8x - 1, which simplifies to 0 = x² - 4x + 3. Next, we factor the quadratic equation. We are looking for two numbers that multiply to 3 and add to -4. These numbers are -1 and -3. Thus, we can factor the equation as: 0 = (x - 1)(x - 3). Setting each factor equal to zero gives us two potential solutions: x - 1 = 0 and x - 3 = 0. Solving these equations, we find x = 1 and x = 3. However, these are just potential solutions. We must verify them in the original equation to ensure they are not extraneous. The next step involves substituting each value back into the original equation to check its validity. This verification process is crucial to identify and eliminate any extraneous roots, ensuring that our final answers are accurate. In the subsequent sections, we will meticulously verify each potential solution to confirm whether it satisfies the original equation.

After finding the potential solutions x = 1 and x = 3, it is crucial to verify these solutions in the original equation √(8x + 1) = x + 2. This step ensures that we eliminate any extraneous solutions that may have arisen from squaring both sides of the equation. First, let's check x = 1. Substituting x = 1 into the original equation, we get: √(8(1) + 1) = 1 + 2. Simplifying the left side, we have √(8 + 1) = √9 = 3. Simplifying the right side, we have 1 + 2 = 3. Since both sides are equal, x = 1 is a valid solution. Next, we check x = 3. Substituting x = 3 into the original equation, we get: √(8(3) + 1) = 3 + 2. Simplifying the left side, we have √(24 + 1) = √25 = 5. Simplifying the right side, we have 3 + 2 = 5. Since both sides are equal, x = 3 is also a valid solution. Therefore, both x = 1 and x = 3 satisfy the original equation and are the real roots of the equation. This verification process highlights the importance of checking solutions, especially when dealing with radical equations. By confirming each solution, we can be confident in the accuracy of our results. In the conclusion, we will summarize our findings and restate the real roots of the equation.

In this comprehensive exploration, we successfully solved the equation √(8x + 1) = x + 2 for its real roots. We began by understanding the equation and the constraints imposed by the square root function, noting that the expressions inside and outside the square root must be non-negative. We then methodically worked through the algebraic steps, squaring both sides to eliminate the square root, resulting in the quadratic equation x² - 4x + 3 = 0. Factoring the quadratic equation, we found two potential solutions: x = 1 and x = 3. The crucial step of verification followed, where we substituted each potential solution back into the original equation. For x = 1, we found that √(8(1) + 1) = √9 = 3, which equals 1 + 2 = 3, thus confirming x = 1 as a valid solution. Similarly, for x = 3, we found that √(8(3) + 1) = √25 = 5, which equals 3 + 2 = 5, confirming x = 3 as a valid solution. Therefore, after careful analysis and verification, we conclude that the real roots of the equation √(8x + 1) = x + 2 are x = 1 and x = 3. This detailed process underscores the importance of each step in solving radical equations, from understanding initial constraints to verifying potential solutions. By following this systematic approach, you can confidently tackle similar problems and achieve accurate results. This exercise not only provides the solutions to this specific equation but also reinforces fundamental algebraic techniques that are essential in mathematical problem-solving. Thus, the real roots are 1 and 3, making option a. 3, 1 the correct answer.