What Is The Solution Of Sqrt X 12 X

Solving the Equation: √(x) + 12 = x

The equation √(x) + 12 = x presents a fascinating challenge in algebra. It's not immediately obvious how to solve it, but with a methodical approach, we can uncover its solution(s) and delve into the mathematical concepts involved. This article will guide you through the process, exploring different methods and discussing the validity of our solutions. We'll also examine the underlying principles of solving radical equations and how to check our work for accuracy.

Understanding the Problem

Our objective is to find the value(s) of 'x' that satisfy the equation √(x) + 12 = x. This is a radical equation, meaning it contains a variable inside a radical (square root, in this case). Solving radical equations often requires specific techniques to eliminate the radical and obtain a polynomial equation that is easier to manage.

Method 1: Isolating the Radical and Squaring

The most common approach to solving radical equations involves isolating the radical term on one side of the equation and then squaring both sides. This process eliminates the square root, but it's crucial to remember that squaring both sides can introduce extraneous solutions – solutions that satisfy the squared equation but not the original equation. Therefore, we must always check our solutions in the original equation.

1. Isolate the radical: Subtract 12 from both sides of the equation: √(x) = x - 12

2. Square both sides: (√(x))² = (x - 12)² This simplifies to: x = x² - 24x + 144

3. Rearrange into a quadratic equation: Subtract 'x' from both sides to obtain a standard quadratic equation: 0 = x² - 25x + 144

4. Solve the quadratic equation: We can solve this quadratic equation using various methods, such as factoring, the quadratic formula, or completing the square. Factoring is the simplest method in this case: 0 = (x - 9)(x - 16)

This gives us two potential solutions: x = 9 and x = 16.

5. Check for extraneous solutions: We must substitute each potential solution back into the original equation, √(x) + 12 = x, to verify its validity.

• Checking x = 9: √(9) + 12 = 3 + 12 = 15 ≠ 9. Therefore, x = 9 is an extraneous solution.

• Checking x = 16: √(16) + 12 = 4 + 12 = 16. This satisfies the original equation.

Therefore, the only valid solution to the equation √(x) + 12 = x is x = 16.

Method 2: Graphical Solution

Another effective method for solving this equation is to use a graphical approach. We can represent both sides of the equation as separate functions and find their intersection point(s).

Let's define two functions:

• f(x) = √(x) + 12
• g(x) = x

By plotting these functions on a graph, we can visually identify the point(s) where they intersect. The x-coordinate of the intersection point represents the solution to the equation. Graphing calculators or software like Desmos can easily perform this task. You'll observe that the two functions intersect at only one point, where x = 16. This confirms our previous result.

Understanding Extraneous Solutions

The appearance of extraneous solutions is a common characteristic of radical equations. The act of squaring both sides can introduce solutions that don't satisfy the original equation. This is because squaring changes the nature of the equation, potentially creating new solutions that are not part of the original problem's solution set. Always verify your solutions by substituting them back into the original equation to eliminate any extraneous results. This crucial step ensures the accuracy and validity of your findings.

Further Exploration: Domain and Range

It's important to consider the domain and range of the functions involved. The square root function, √(x), is only defined for non-negative values of x (x ≥ 0). Therefore, the domain of the function √(x) + 12 is [0, ∞). Furthermore, the function is monotonically increasing within its domain.

The linear function, x, has a domain of all real numbers (-∞, ∞). The intersection point must lie within the domain of both functions, reinforcing the significance of checking the solution within the context of the original equation.

Applications and Extensions

Radical equations like this appear in various areas of mathematics and its applications. They can arise in problems involving geometry, physics, and engineering. Understanding how to solve them is essential for addressing problems involving distances, areas, volumes, and other quantitative relationships.

Conclusion: A Systematic Approach

Solving the equation √(x) + 12 = x requires a systematic approach. Isolating the radical, squaring both sides, and checking for extraneous solutions are crucial steps. Graphical methods offer a visual confirmation of the solution. By understanding these techniques and the potential pitfalls (such as extraneous solutions), you can confidently solve a wide range of radical equations and gain a deeper appreciation for the nuances of algebraic manipulation. Remember, always verify your solutions by substituting them back into the original equation. This diligence ensures the accuracy and completeness of your mathematical work. The solution x=16 is not just a numerical answer; it's the result of a process that showcases the importance of careful algebraic techniques and solution verification.