Solving Radical Equations A Step By Step Guide To √x-9 + 9 = X

In the realm of mathematics, solving equations is a fundamental skill. Among the various types of equations, radical equations often pose a unique challenge due to the presence of radicals (square roots, cube roots, etc.). These equations require careful manipulation to isolate the variable and find the solution. In this article, we will delve into a detailed step-by-step solution of the equation √x-9 + 9 = x, providing clear explanations and insights along the way. Whether you're a student grappling with algebra or simply someone looking to sharpen your math skills, this guide will equip you with the knowledge and techniques to tackle radical equations with confidence. By the end of this comprehensive exploration, you will not only understand how to solve this specific equation but also gain a broader understanding of the principles and strategies involved in solving radical equations in general.

Understanding Radical Equations

Before diving into the solution, it's crucial to understand what radical equations are and the key principles involved in solving them. Radical equations are algebraic equations in which a variable appears inside a radical symbol, most commonly a square root (√). The goal in solving these equations is to isolate the variable by eliminating the radical. This typically involves raising both sides of the equation to a power that matches the index of the radical (e.g., squaring both sides for a square root). However, this process can sometimes introduce extraneous solutions, which are solutions that satisfy the transformed equation but not the original equation. Therefore, it's essential to check all potential solutions in the original equation to ensure their validity. When we talk about solving radical equations, we often use several key mathematical concepts. One of them is the principle of squaring (or raising to a power), which states that if a = b, then a² = b². This principle allows us to eliminate the radical. Another concept is the identification and elimination of extraneous solutions, which arise due to the non-reversible nature of the squaring operation. Extraneous solutions are particularly common in radical equations and can lead to incorrect answers if not properly addressed. Thus, a thorough understanding of these principles is vital in solving radical equations effectively and accurately.

The Given Equation: √x-9 + 9 = x

Our focus is on solving the radical equation √x-9 + 9 = x. This equation presents a typical radical equation structure, where the variable x is present both inside and outside the square root. The equation's complexity arises from the interplay between the square root and the linear term, making it a compelling example for illustrating the techniques required to solve such equations. Solving equations like √x-9 + 9 = x requires careful algebraic manipulation to isolate the variable and eliminate the radical. The presence of the square root necessitates the use of specific strategies, such as squaring both sides of the equation, which can potentially introduce extraneous solutions. Therefore, a meticulous approach is crucial to ensure the correctness of the solution. By working through this equation step-by-step, we will gain a deeper understanding of how to handle radical equations and the importance of verifying solutions. This particular equation serves as an excellent example due to its moderate level of difficulty and the clear illustration of the steps involved in solving radical equations. It provides an opportunity to apply the principles of algebra and the properties of radicals in a practical context, making it a valuable learning exercise for anyone studying algebra or related mathematical topics.

Step-by-Step Solution

To solve the equation √x-9 + 9 = x, we will follow these steps:

Step 1: Isolate the Radical Term

Isolate the radical term by subtracting 9 from both sides of the equation: √x-9 = x - 9.

This initial step is crucial because it sets the stage for eliminating the square root. Isolating the radical term allows us to square both sides of the equation in the next step without introducing unnecessary complexity. By subtracting 9 from both sides, we effectively separate the square root term from the other terms, making it easier to manipulate. This is a standard technique in solving radical equations and is essential for a clean and efficient solution process. The goal here is to get the square root term by itself on one side of the equation, which is a prerequisite for the subsequent steps.

Step 2: Square Both Sides

Square both sides of the equation to eliminate the square root: (√x-9)² = (x - 9)². This simplifies to x - 9 = (x - 9)². Squaring both sides is a fundamental step in solving radical equations. It eliminates the square root, allowing us to work with a more manageable algebraic expression. However, it's important to recognize that this operation can introduce extraneous solutions, which we'll need to check later. The expression (x - 9)² expands to x² - 18x + 81, transforming our equation into a quadratic form. This step essentially converts the radical equation into a more familiar algebraic form, which we can then solve using standard techniques.

Step 3: Expand and Simplify

Expand the right side and simplify the equation: x - 9 = x² - 18x + 81. Rearrange the equation to form a quadratic equation: 0 = x² - 19x + 90.

Expanding the squared term and rearranging the equation into a quadratic form is a crucial step in solving for x. The quadratic equation x² - 19x + 90 = 0 is now in a standard form that we can solve using factoring, completing the square, or the quadratic formula. This step bridges the gap between the radical equation and the more familiar territory of quadratic equations. By setting the equation to zero, we prepare it for factoring or the application of the quadratic formula, which are standard methods for finding the roots of a quadratic equation. This transformation is a key technique in solving many types of equations, not just radical ones.

Step 4: Factor the Quadratic Equation

Factor the quadratic equation: 0 = (x - 10)(x - 9). This gives us two potential solutions: x = 10 and x = 9. Factoring the quadratic equation is an efficient method to find the potential solutions for x. By factoring x² - 19x + 90 into (x - 10)(x - 9), we identify the values of x that make the equation equal to zero. These values, x = 10 and x = 9, are our candidate solutions. However, it's crucial to remember that these are only potential solutions, and we must verify them in the original equation to ensure they are not extraneous.

Step 5: Check for Extraneous Solutions

Check the solutions in the original equation √x-9 + 9 = x:

For x = 10: √10-9 + 9 = √1 + 9 = 1 + 9 = 10. This solution is valid.

For x = 9: √9-9 + 9 = √0 + 9 = 0 + 9 = 9. This solution is also valid.

Checking for extraneous solutions is a critical step in solving radical equations. Squaring both sides of an equation can sometimes introduce solutions that do not satisfy the original equation. By substituting each potential solution back into the original equation, we can verify its validity. In this case, both x = 10 and x = 9 satisfy the original equation √x-9 + 9 = x. For x = 10, the equation holds true: √10-9 + 9 = 10. Similarly, for x = 9, the equation also holds true: √9-9 + 9 = 9. This confirmation step ensures that our solutions are accurate and not artifacts of the algebraic manipulation.

Final Answer

The solutions to the equation √x-9 + 9 = x are x = 9 and x = 10.

Having verified both potential solutions, we can confidently state that the solutions to the given equation are x = 9 and x = 10. This conclusion is the result of a methodical approach, starting with isolating the radical term, squaring both sides, solving the resulting quadratic equation, and, most importantly, checking for extraneous solutions. The solutions x = 9 and x = 10 are the values that satisfy the original radical equation, making them the valid solutions. This comprehensive process demonstrates the importance of each step in solving radical equations and highlights the significance of verifying solutions to avoid errors.

In this comprehensive guide, we've meticulously walked through the process of solving the radical equation √x-9 + 9 = x. From the initial step of isolating the radical term to the crucial final step of checking for extraneous solutions, we've covered each aspect in detail. Solving radical equations like this one requires a systematic approach, combining algebraic manipulation with careful verification. The key takeaways from this exercise include the importance of isolating the radical, the potential for introducing extraneous solutions when squaring both sides, and the necessity of checking all potential solutions in the original equation. This step-by-step solution not only provides the answer to the specific equation but also offers a broader understanding of the techniques involved in solving radical equations in general. By mastering these techniques, you'll be well-equipped to tackle a variety of algebraic challenges and build a stronger foundation in mathematics. The ability to solve radical equations is a valuable skill in algebra and beyond, applicable in various fields of science and engineering. Therefore, understanding the underlying principles and the step-by-step process is essential for anyone looking to excel in mathematics and its applications.