Determine The Number Of Valid Solutions For The Equation \(\sqrt{x-1} - 5 = X - 8\).

Determining the number of valid solutions for a given equation is a fundamental concept in mathematics. In this article, we will delve into the process of solving a radical equation and identifying the valid solutions. We'll specifically focus on the equation ${\sqrt{x-1} - 5 = x - 8}$, providing a step-by-step approach to isolate the variable x and verify the solutions obtained. Through this exploration, we aim to enhance your understanding of radical equations and solution validation techniques.

Understanding Radical Equations

Radical equations involve variables within radical expressions, such as square roots, cube roots, and so on. Solving radical equations requires careful manipulation to eliminate the radical and isolate the variable. One common technique is to square both sides of the equation, but this can sometimes introduce extraneous solutions, which are solutions that satisfy the transformed equation but not the original one. Therefore, it's crucial to check all solutions in the original equation to ensure their validity. The initial step in solving ${\sqrt{x-1} - 5 = x - 8}$ is to isolate the radical term on one side of the equation. This involves adding 5 to both sides, which gives us ${\sqrt{x-1} = x - 3}$. By isolating the radical, we prepare the equation for the next step: squaring both sides to eliminate the square root. However, it's important to keep in mind the potential for extraneous solutions, which we will address later in the verification process. Before proceeding, let's briefly discuss why squaring both sides can introduce extraneous solutions. When we square both sides of an equation, we are essentially applying a function to both sides. This function, in this case squaring, is not necessarily one-to-one, meaning that different inputs can produce the same output. This property can lead to solutions that work in the squared equation but not in the original equation. For example, if we have the equation ${a = b}$, squaring both sides gives ${a^2 = b^2}$. While it's true that if ${a = b}$ then ${a^2 = b^2}$, the converse is not necessarily true. If ${a^2 = b^2}$, it could be that ${a = b}$ or ${a = -b}$. This difference is the root cause of extraneous solutions in radical equations. To avoid mistakenly accepting extraneous solutions, it is imperative to check all potential solutions in the original equation. This verification step is a cornerstone of solving radical equations accurately. In the context of our equation, after squaring both sides, we will obtain a quadratic equation. Solving this quadratic equation will give us two potential solutions, both of which must be checked in the original equation ${\sqrt{x-1} - 5 = x - 8}$. This check involves substituting each potential solution for x in the original equation and determining whether the equation holds true. Only solutions that satisfy the original equation are considered valid solutions.

Solving the Equation ${\sqrt{x-1} - 5 = x - 8}$

To effectively solve the given radical equation, a systematic approach is required. Isolating the radical term is the first critical step, followed by the elimination of the square root through squaring. However, this process can potentially introduce extraneous solutions, making it essential to verify all obtained solutions in the original equation. The process begins by adding 5 to both sides of the equation ${\sqrt{x-1} - 5 = x - 8}$, which results in ${\sqrt{x-1} = x - 3}$. This step isolates the radical term on one side of the equation, preparing it for the next operation. With the radical term isolated, the next step involves squaring both sides of the equation to eliminate the square root. Squaring both sides of ${\sqrt{x-1} = x - 3}$ yields ${(\sqrt{x-1})^2 = (x - 3)^2}$, which simplifies to ${x - 1 = x^2 - 6x + 9}$. This transformation removes the radical, but it also introduces the potential for extraneous solutions, as mentioned earlier. After squaring both sides, we are left with a quadratic equation. To solve this quadratic equation, we need to rearrange the terms to bring it into the standard form ${ax^2 + bx + c = 0}$. Subtracting x and adding 1 to both sides of ${x - 1 = x^2 - 6x + 9}$ gives us ${0 = x^2 - 7x + 10}$. This quadratic equation can now be solved using various methods, such as factoring, completing the square, or using the quadratic formula. Factoring is often the most straightforward method when the quadratic equation can be factored easily. In this case, the quadratic equation ${x^2 - 7x + 10 = 0}$ can be factored as ${(x - 2)(x - 5) = 0}$. This factorization leads to two potential solutions: ${x = 2}$ and ${x = 5}$. These are the potential solutions that need to be checked in the original equation to determine their validity. It is crucial to remember that squaring both sides of an equation can introduce solutions that do not satisfy the original equation. Therefore, the next essential step is to substitute each of these potential solutions back into the original equation, ${\sqrt{x-1} - 5 = x - 8}$, to verify their validity. This verification process will help us identify any extraneous solutions and ensure that we only accept the true solutions of the equation.

Verifying the Solutions

After obtaining potential solutions, the crucial step of verification cannot be overlooked. Substituting each potential solution back into the original equation allows us to identify and eliminate any extraneous solutions that may have been introduced during the solving process. For the equation ${\sqrt{x-1} - 5 = x - 8}$, we found two potential solutions: ${x = 2}$ and ${x = 5}$. We must now substitute each of these values into the original equation to determine if they hold true. First, let's verify ${x = 2}$. Substituting ${x = 2}$ into the original equation ${\sqrt{x-1} - 5 = x - 8}$ gives us ${\sqrt{2-1} - 5 = 2 - 8}$. This simplifies to ${\sqrt{1} - 5 = -6}$, which further simplifies to ${1 - 5 = -6}$. Since ${-4 = -6}$ is not a true statement, ${x = 2}$ is an extraneous solution and is not a valid solution to the original equation. This demonstrates the importance of the verification step in solving radical equations. Now, let's verify the second potential solution, ${x = 5}$. Substituting ${x = 5}$ into the original equation ${\sqrt{x-1} - 5 = x - 8}$ gives us ${\sqrt{5-1} - 5 = 5 - 8}$. This simplifies to ${\sqrt{4} - 5 = -3}$, which further simplifies to ${2 - 5 = -3}$. Since ${-3 = -3}$ is a true statement, ${x = 5}$ is a valid solution to the original equation. This confirms that ${x = 5}$ is indeed a solution that satisfies the given radical equation. By performing this verification process, we have successfully identified that ${x = 2}$ is an extraneous solution and ${x = 5}$ is the only valid solution. This underscores the importance of verifying potential solutions in radical equations to avoid including extraneous solutions in the final answer. In summary, the verification step is essential for ensuring the accuracy of solutions obtained when solving radical equations. It involves substituting each potential solution back into the original equation and checking for equality. Only solutions that satisfy the original equation are considered valid, while those that do not are classified as extraneous and discarded.

Conclusion: Valid Solutions for ${\sqrt{x-1} - 5 = x - 8}$

In summary, solving the radical equation ${\sqrt{x-1} - 5 = x - 8}$ involved a series of steps: isolating the radical, squaring both sides, solving the resulting quadratic equation, and, most importantly, verifying the solutions. The verification step is crucial in identifying and eliminating any extraneous solutions that may arise from squaring both sides of the equation. Through this process, we found two potential solutions, ${x = 2}$ and ${x = 5}$. However, upon verification, only ${x = 5}$ satisfied the original equation. The solution ${x = 2}$ turned out to be an extraneous solution, highlighting the necessity of checking all potential solutions in radical equations. Therefore, the equation ${\sqrt{x-1} - 5 = x - 8}$ has only one valid solution, which is ${x = 5}$. This methodical approach to solving radical equations ensures accuracy and a thorough understanding of the solution set. The techniques discussed in this article, including isolating the radical, squaring both sides, solving the resulting equation, and verifying solutions, are fundamental in solving a wide range of radical equations. These steps not only provide the correct solutions but also foster a deeper comprehension of the mathematical concepts involved. Understanding the potential for extraneous solutions and the importance of the verification step is critical in mastering radical equations. By applying these methods consistently, you can confidently tackle various radical equations and accurately determine their valid solutions. Furthermore, the concepts discussed here extend beyond radical equations. The idea of verifying solutions is a fundamental aspect of mathematical problem-solving, especially when dealing with transformations that might introduce extraneous solutions. This principle applies to other areas of mathematics, such as trigonometric equations and logarithmic equations. In conclusion, the equation ${\sqrt{x-1} - 5 = x - 8}$ serves as an excellent example for illustrating the process of solving radical equations and the significance of solution verification. The meticulous approach outlined in this article ensures that only valid solutions are accepted, leading to a comprehensive understanding of the equation's solution set. By mastering these techniques, you can confidently approach and solve similar mathematical problems with accuracy and precision.

The equation has:

• one valid solution