Solving 5 + √x = 16 A Step-by-Step Guide

In the realm of mathematics, solving equations is a fundamental skill. This article delves into the step-by-step process of solving the equation 5 + √x = 16. Understanding the equation is the first key step. This equation combines basic arithmetic operations with a square root, making it a classic example of an algebraic problem. The main goal is to isolate the variable 'x' to find its value. This process involves using inverse operations to undo the operations performed on 'x'. For instance, the square root operation requires squaring as its inverse, and addition requires subtraction. Before diving into the solution, it's important to grasp the underlying principles of algebraic manipulation, such as maintaining equality by performing the same operation on both sides of the equation. This ensures that the equation remains balanced and the solution remains valid. Additionally, understanding the properties of square roots and how they interact with other mathematical operations is crucial for solving this type of equation effectively. The process involves several steps, each building upon the previous one, to gradually isolate 'x' and determine its value. This method not only helps in solving this specific equation but also provides a framework for tackling other similar algebraic problems. Throughout this guide, each step will be explained in detail to ensure a clear understanding of the solution process. By the end of this article, you will have a solid grasp of how to solve the equation 5 + √x = 16 and similar algebraic challenges. This foundational knowledge is essential for further studies in mathematics and related fields.

Step-by-Step Solution: Isolating the Square Root

The initial step in solving the equation 5 + √x = 16 involves isolating the square root term. To achieve this, we need to eliminate the constant term added to the square root. In this case, the constant term is 5. The operation connecting 5 and √x is addition, so we apply the inverse operation, which is subtraction. Subtracting 5 from both sides of the equation maintains the balance and ensures the equation remains valid. This step transforms the original equation into a simpler form, making it easier to work with. The process can be written mathematically as follows:

5 + √x - 5 = 16 - 5

Simplifying both sides of the equation, we get:

√x = 11

Now, the square root term is isolated on one side of the equation, which is a crucial milestone in solving for 'x'. This isolation allows us to address the square root operation directly. The next step will involve undoing the square root to finally determine the value of 'x'. Isolating the square root is a common strategy in solving equations involving radicals, and it's a technique that can be applied to a variety of similar problems. This step not only simplifies the equation but also sets the stage for the final step of finding the value of the unknown variable. By meticulously following this process, we ensure that each step is mathematically sound and contributes to the accurate solution of the equation. This foundational understanding of algebraic manipulation is essential for tackling more complex mathematical problems in the future.

Eliminating the Square Root: Squaring Both Sides

With the square root term isolated, the next step is to eliminate the square root itself. The inverse operation of taking a square root is squaring. Therefore, to eliminate the square root, we square both sides of the equation. This ensures that we maintain the equality and do not alter the solution. Squaring both sides of the equation √x = 11 means raising each side to the power of 2. Mathematically, this is represented as:

(√x)² = (11)²

When you square a square root, you essentially undo the operation, leaving you with the original value under the radical. In this case, (√x)² simplifies to x. On the other side of the equation, (11)² means 11 multiplied by itself, which equals 121. Therefore, the equation becomes:

x = 121

This step is critical because it directly reveals the value of 'x'. By squaring both sides, we have effectively removed the square root and found the solution. It's important to remember that squaring both sides is a valid operation as long as it's applied to the entire side of the equation. This technique is widely used in solving equations involving radicals and is a fundamental concept in algebra. Once we have found the value of 'x', it's always a good practice to verify the solution by substituting it back into the original equation to ensure its correctness. This confirmation step helps to avoid errors and reinforces the understanding of the solution process.

Verification: Confirming the Solution

After finding a potential solution, verification is a crucial step to ensure accuracy. To verify the solution x = 121, we substitute this value back into the original equation: 5 + √x = 16. This process involves replacing 'x' with 121 and checking if the equation holds true. Substituting x = 121 into the original equation gives us:

5 + √121 = 16

The next step is to evaluate the square root of 121. The square root of 121 is 11 because 11 multiplied by itself equals 121. So, the equation becomes:

5 + 11 = 16

Now, we perform the addition on the left side of the equation:

16 = 16

Since both sides of the equation are equal, this confirms that our solution x = 121 is correct. The verification process is essential because it helps identify any potential errors made during the solution process. It also reinforces the understanding of how the solution satisfies the original equation. Verifying the solution is a standard practice in mathematics and is particularly important when dealing with equations involving radicals, as squaring both sides can sometimes introduce extraneous solutions. By taking the time to verify, we can be confident in the accuracy of our answer and the validity of our solution process. This step not only confirms the solution but also enhances the overall understanding of the problem and the techniques used to solve it.

Final Answer: The Value of x

After successfully isolating the variable and verifying the solution, we arrive at the final answer. Based on the steps taken, the solution to the equation 5 + √x = 16 is x = 121. This result is a whole number, as requested in the problem statement. The final answer, x = 121, represents the value that satisfies the original equation. It is the unique value that, when substituted into the equation, makes the equation true. The process of solving this equation involved several key steps, including isolating the square root term, squaring both sides, and verifying the solution. Each step was performed with the goal of isolating 'x' and determining its value. The solution x = 121 is not only the answer to this specific problem but also a demonstration of the application of fundamental algebraic principles. These principles can be applied to solve a wide range of equations and mathematical problems. Understanding the process of solving equations like this is crucial for building a strong foundation in mathematics. The ability to manipulate equations and isolate variables is a skill that is essential for further studies in algebra, calculus, and other related fields. Therefore, the final answer, x = 121, is not just a numerical result; it is also a testament to the successful application of mathematical techniques and problem-solving skills.

In conclusion, solving the equation 5 + √x = 16 provides a valuable exercise in mastering algebraic equations. The process involved several key steps: isolating the square root, squaring both sides, and verifying the solution. Each step is a critical component of the overall solution and demonstrates the importance of understanding inverse operations and maintaining equality in equations. The solution, x = 121, not only answers the specific problem but also illustrates the broader principles of algebraic manipulation. These principles are applicable to a wide range of mathematical problems and are essential for building a strong foundation in mathematics. Mastering algebraic equations requires a combination of understanding the underlying concepts and practicing problem-solving techniques. This article has provided a detailed guide to solving this particular equation, but the methods and principles discussed can be applied to other similar problems. The ability to solve algebraic equations is a fundamental skill in mathematics and is crucial for further studies in various fields, including science, engineering, and economics. By understanding the steps involved and practicing regularly, one can develop confidence and proficiency in solving algebraic equations. This skill is not only valuable in academic settings but also in real-world applications where mathematical problem-solving is required. Therefore, the journey of mastering algebraic equations is a continuous process of learning, practicing, and applying these skills to new and challenging problems. The equation 5 + √x = 16 serves as a stepping stone in this journey, providing a solid foundation for tackling more complex mathematical challenges.