Solving 2 - √(x - 2) = X Find Real Solutions

In the realm of mathematics, solving equations is a fundamental skill. This article delves into the process of finding real solutions for the equation 2 - √(x - 2) = x. We will explore the steps involved in isolating the variable, eliminating the square root, and verifying the solutions. It's crucial to remember that when dealing with square roots, we need to check for extraneous solutions, which are solutions obtained algebraically but do not satisfy the original equation.

Understanding the Equation

Before we begin, let's break down the equation 2 - √(x - 2) = x. This equation involves a square root, which means we need to be mindful of the domain. The expression inside the square root, x - 2, must be non-negative. This gives us the initial condition that x ≥ 2. This condition is crucial because it limits the possible values of x that can be considered as valid solutions. Ignoring this domain restriction can lead to extraneous solutions, which are values that emerge from the algebraic manipulation but do not actually satisfy the original equation. The equation presents a relationship between a linear term (x) and a square root term (√(x - 2)), requiring careful algebraic manipulation to isolate and solve for x. Our goal is to find the values of x that make this equation true while adhering to the domain constraint we've established. We will need to isolate the square root term, square both sides of the equation, and solve the resulting quadratic equation. Finally, we must check our solutions against the original equation and the domain restriction to ensure they are valid.

Steps to Solve the Equation

To solve the equation, we will follow a series of algebraic steps. First, we need to isolate the square root term. This involves moving the constant term to the other side of the equation. Then, we will square both sides of the equation to eliminate the square root. This step is critical, but it also introduces the possibility of extraneous solutions. After squaring, we will simplify the equation and rearrange it into a quadratic equation. Quadratic equations are equations of the form ax² + bx + c = 0, and they can be solved using various methods, such as factoring, completing the square, or using the quadratic formula. Once we have the solutions to the quadratic equation, we must check these solutions in the original equation. This is a vital step because squaring both sides can introduce extraneous solutions, which are values that satisfy the transformed equation but not the original equation. We also need to check if the solutions satisfy the initial condition x ≥ 2. Only the solutions that satisfy both the original equation and the domain restriction are considered valid real solutions. This methodical approach ensures that we find all genuine solutions and avoid including any extraneous ones.

1. Isolate the Square Root

The first step is to isolate the square root term in the equation 2 - √(x - 2) = x. To do this, we subtract 2 from both sides of the equation. This gives us -√(x - 2) = x - 2. Isolating the square root is a crucial step because it allows us to eliminate the square root in the next step by squaring both sides. Without isolating the square root, squaring both sides would result in a more complex expression that is difficult to simplify. The negative sign on the square root term requires careful attention. We can either multiply both sides by -1 at this stage or square the equation directly with the negative sign. Both approaches will lead to the same result, but multiplying by -1 first can sometimes simplify the subsequent steps. The resulting equation, √(x - 2) = 2 - x, sets the stage for eliminating the square root and transforming the equation into a more manageable form, such as a polynomial equation. This transformation is essential for solving for x, but it also necessitates the critical step of checking for extraneous solutions later in the process.

2. Square Both Sides

After isolating the square root, the next step is to square both sides of the equation -√(x - 2) = x - 2. Squaring both sides eliminates the square root on the left side. This gives us ( -√(x - 2) )² = (x - 2)², which simplifies to x - 2 = (x - 2)². Squaring both sides is a powerful technique for dealing with square roots in equations, but it is crucial to recognize that this step can introduce extraneous solutions. Extraneous solutions are values that satisfy the squared equation but do not satisfy the original equation. This is because squaring both sides can sometimes mask the sign differences that are present in the original equation. For instance, if we have an equation like √a = b, squaring both sides gives us a = b², but this does not necessarily imply that √a = b. It could also be the case that √a = -b. Therefore, it is absolutely essential to check all solutions obtained after squaring back in the original equation to ensure they are valid. The squared equation, x - 2 = (x - 2)², is a polynomial equation that can be solved using algebraic techniques. This step is a significant advancement in finding the solutions, but it comes with the responsibility of verifying those solutions later.

3. Simplify and Rearrange

Now, let's simplify and rearrange the equation x - 2 = (x - 2)². Expanding the right side, we get x - 2 = x² - 4x + 4. To solve this, we need to set the equation to zero. Subtracting x and adding 2 to both sides gives us 0 = x² - 5x + 6. This is a quadratic equation in the standard form ax² + bx + c = 0, where a = 1, b = -5, and c = 6. Simplifying and rearranging the equation into a standard quadratic form is essential because it allows us to apply well-established methods for solving quadratic equations, such as factoring, completing the square, or using the quadratic formula. The process of expanding and rearranging terms ensures that all terms are on one side of the equation, leaving zero on the other side. This is a necessary condition for applying most solution techniques for quadratic equations. The resulting quadratic equation, 0 = x² - 5x + 6, can be solved by factoring, which is often the quickest method if the quadratic expression can be easily factored. If factoring is not straightforward, the quadratic formula provides a guaranteed method for finding the solutions. The solutions to this quadratic equation will be potential solutions to the original equation, but they must be checked for validity.

4. Solve the Quadratic Equation

The quadratic equation we have is 0 = x² - 5x + 6. We can solve this equation by factoring. We are looking for two numbers that multiply to 6 and add up to -5. These numbers are -2 and -3. Therefore, we can factor the quadratic as (x - 2)(x - 3) = 0. Setting each factor equal to zero gives us two potential solutions: x - 2 = 0 and x - 3 = 0. Solving these equations gives us x = 2 and x = 3. Factoring is a highly efficient method for solving quadratic equations when the quadratic expression can be easily factored. It involves rewriting the quadratic expression as a product of two linear factors. The solutions are then found by setting each factor equal to zero and solving for x. In this case, the quadratic factors neatly into (x - 2)(x - 3), making factoring a straightforward approach. The solutions obtained, x = 2 and x = 3, are potential solutions to the original equation. However, it is crucial to remember that squaring both sides of an equation can introduce extraneous solutions. Therefore, the next step is absolutely essential: we must check these potential solutions in the original equation to ensure they are valid. Only those solutions that satisfy the original equation are considered true solutions.

5. Check for Extraneous Solutions

The most crucial step in solving equations involving square roots is to check the solutions in the original equation 2 - √(x - 2) = x. We have two potential solutions: x = 2 and x = 3. Let's check x = 2 first:

Substituting x = 2 into the original equation, we get 2 - √(2 - 2) = 2, which simplifies to 2 - √0 = 2, and further simplifies to 2 - 0 = 2, which is 2 = 2. This is true, so x = 2 is a valid solution. Now, let's check x = 3:

Substituting x = 3 into the original equation, we get 2 - √(3 - 2) = 3, which simplifies to 2 - √1 = 3, and further simplifies to 2 - 1 = 3, which is 1 = 3. This is false, so x = 3 is an extraneous solution. Checking for extraneous solutions is a critical step because squaring both sides of an equation can introduce solutions that do not satisfy the original equation. These extraneous solutions arise due to the loss of information about the sign when squaring. By substituting the potential solutions back into the original equation, we can verify whether they make the equation true. In this case, x = 2 satisfies the original equation, while x = 3 does not. Therefore, x = 3 is an extraneous solution and must be discarded. The process of checking solutions ensures that we only accept valid solutions and avoid including any extraneous ones in our final answer. This meticulous verification is a hallmark of rigorous mathematical problem-solving.

Final Answer

After checking for extraneous solutions, we find that only x = 2 is a real solution to the equation 2 - √(x - 2) = x. The solution x = 3 was an extraneous solution, resulting from squaring both sides of the equation during the solution process. It is imperative to always check potential solutions back in the original equation when dealing with radicals to avoid such extraneous results. Therefore, the final and only real solution to the given equation is x = 2. This solution satisfies both the algebraic manipulations performed and the initial domain restriction x ≥ 2, making it a valid and complete answer to the problem.

Final Answer: The final answer is 2.