Solving Radical Equations A Comprehensive Guide To $\sqrt{x}+3=12$

In the realm of mathematics, equations serve as fundamental tools for modeling and solving real-world problems. Among these equations, radical equations, which involve variables within radical expressions (such as square roots, cube roots, etc.), often present a unique set of challenges. In this comprehensive guide, we will delve into the intricacies of solving radical equations, focusing on the specific equation $\sqrt{x}+3=12$. By the end of this exploration, you will gain a solid understanding of the underlying principles and techniques for tackling such equations with confidence.

This article aims to provide a detailed, step-by-step solution to the given radical equation, ensuring clarity and understanding for readers of all levels. Whether you're a student grappling with algebra or simply someone seeking to expand your mathematical horizons, this guide will equip you with the knowledge and skills to confidently solve radical equations.

Before we dive into the specifics of solving $\sqrt{x}+3=12$, let's first establish a firm grasp of what radical equations are and the key principles involved in solving them.

Radical equations are equations in which the variable appears inside a radical expression. The most common type of radical is the square root, denoted by the symbol $\sqrt{ }$, but radicals can also include cube roots, fourth roots, and so on. Solving radical equations involves isolating the radical term and then eliminating the radical by raising both sides of the equation to the appropriate power. For instance, to eliminate a square root, we square both sides; for a cube root, we cube both sides, and so forth. It's crucial to remember that raising both sides of an equation to an even power can sometimes introduce extraneous solutions, which are solutions that satisfy the transformed equation but not the original one. Therefore, it's always essential to check your solutions in the original equation to ensure their validity.

Radical equations are frequently encountered in various branches of mathematics and physics, including geometry, calculus, and mechanics. They arise naturally when dealing with distances, areas, volumes, and other quantities that involve square roots or other radicals. Mastering the techniques for solving radical equations is thus an invaluable skill for anyone pursuing studies or careers in these fields. The equation $\sqrt{x}+3=12$ serves as an excellent starting point for understanding these concepts, as it embodies the fundamental principles of radical equation solving in a clear and concise manner. By working through this example, you will gain a solid foundation for tackling more complex radical equations in the future.

Now, let's tackle the equation $\sqrt{x}+3=12$ step by step. Our goal is to isolate the variable x and determine its value. We will proceed by isolating the radical term, eliminating the radical, and finally solving for x. Each step is crucial and must be performed with careful attention to detail to avoid errors.

Step 1: Isolate the Radical Term

The first step in solving the equation is to isolate the radical term, which in this case is $\sqrt{x}$. To do this, we need to get rid of the '+3' on the left side of the equation. We can achieve this by subtracting 3 from both sides of the equation. This maintains the equality and moves us closer to isolating the square root. Subtracting 3 from both sides gives us:

$\sqrt{x}+3-3=12-3$

This simplifies to:

$\sqrt{x}=9$

Now we have successfully isolated the radical term on one side of the equation. This is a critical step because it sets the stage for eliminating the radical in the next step. Without isolating the radical, it would be much more difficult to proceed with solving for x. This initial isolation is a common strategy in solving various types of equations, including those involving logarithms, exponents, and trigonometric functions. By focusing on isolating the term containing the variable, we can often simplify the equation and make it more manageable.

Step 2: Eliminate the Radical

With the radical term isolated, our next step is to eliminate the square root. To do this, we square both sides of the equation. Remember, whatever operation we perform on one side of the equation, we must also perform on the other side to maintain the balance. Squaring both sides of $\sqrt{x}=9$ gives us:

$(\sqrt{x})^2=9^2$

This simplifies to:

$x=81$

By squaring both sides, we have successfully eliminated the square root and obtained a simple algebraic equation. This is a standard technique for dealing with radical equations and is based on the property that the square of a square root is the original number. However, as we mentioned earlier, squaring both sides can sometimes introduce extraneous solutions. Therefore, it is essential to check our solution in the original equation to ensure its validity. This step highlights the importance of understanding the properties of mathematical operations and their potential consequences. In the next step, we will verify whether the solution we obtained is indeed a valid solution for the original equation.

Step 3: Check the Solution

Now that we have found a potential solution, x = 81, it's crucial to check whether this solution satisfies the original equation. This step is essential because, as we discussed earlier, squaring both sides of an equation can sometimes introduce extraneous solutions. Extraneous solutions are values that satisfy the transformed equation but not the original equation. To check our solution, we substitute x = 81 back into the original equation $\sqrt{x}+3=12$:

$\sqrt{81}+3=12$

Simplifying the square root, we get:

$9+3=12$

Which further simplifies to:

$12=12$

Since this statement is true, our solution x = 81 satisfies the original equation. Therefore, it is a valid solution. This verification step provides us with confidence in our answer and ensures that we have correctly solved the radical equation. In this case, we found that x = 81 is indeed a solution. However, in other situations, you might encounter extraneous solutions, and it's crucial to identify and discard them. The process of checking solutions is a fundamental aspect of solving any equation, not just radical equations, and it helps to ensure the accuracy of your results.

In conclusion, the solution to the equation $\sqrt{x}+3=12$ is x = 81. We arrived at this solution by carefully following the steps of isolating the radical term, eliminating the radical by squaring both sides, and then verifying the solution in the original equation. This process highlights the importance of each step in solving radical equations and the need to be vigilant about potential extraneous solutions.

Solving radical equations requires a solid understanding of algebraic principles and careful execution of each step. By mastering these techniques, you will be well-equipped to tackle a wide range of mathematical problems involving radicals. Remember, the key to success lies in understanding the underlying concepts and practicing regularly. The equation $\sqrt{x}+3=12$ serves as a valuable example for grasping the fundamentals of solving radical equations, and the skills you've gained here can be applied to more complex problems in the future. Keep practicing, and you'll become proficient in solving radical equations and other algebraic challenges.

To further enhance your problem-solving skills with radical equations, consider the following tips:

• Always isolate the radical term first: Before you eliminate the radical, make sure it is isolated on one side of the equation. This simplifies the process and reduces the chances of errors.
• Check for extraneous solutions: As emphasized earlier, checking your solutions in the original equation is crucial. This step will help you identify and discard any extraneous solutions that may arise from squaring or raising both sides to an even power.
• Be mindful of the domain: When dealing with square roots, remember that the expression inside the square root must be non-negative. This can help you identify potential restrictions on the solutions.
• Practice regularly: The more you practice, the more comfortable you will become with solving radical equations. Work through various examples and try different types of problems to broaden your understanding.

By incorporating these tips into your problem-solving approach, you will be well-prepared to tackle radical equations with confidence and accuracy. Remember, mathematics is a skill that improves with practice, so keep exploring and challenging yourself.

Therefore, the correct answer to the equation $\sqrt{x}+3=12$ is:

B. 81