Solving -1/2 N^2 + 18 = 0 A Step-by-Step Guide

This article provides a detailed solution to the quadratic equation $-\frac{1}{2}n^2 + 18 = 0$. Solving quadratic equations is a fundamental skill in algebra, with applications spanning various fields, including physics, engineering, and computer science. Our focus will be on using algebraic manipulation to isolate the variable n and find its possible values. This step-by-step guide will help you understand the process, ensuring you grasp the core concepts and can apply them to similar problems. Understanding how to solve quadratic equations not only helps in academic settings but also equips you with problem-solving skills applicable in real-world scenarios. We will explore each step meticulously, making sure the reasoning behind every action is clear and understandable. Whether you are a student tackling homework or someone brushing up on your algebra, this guide is designed to make the solution process straightforward and effective. By the end of this article, you will be confident in your ability to tackle quadratic equations of this type and apply the underlying principles to more complex problems. Mastering these techniques is crucial for further studies in mathematics and related disciplines. The goal is to provide a comprehensive understanding, ensuring you not only get the correct answer but also appreciate the mathematical principles at play. We will also touch upon the importance of checking your solutions to ensure accuracy, a vital step in mathematical problem-solving. Let's dive in and unravel the solution together!

Step-by-Step Solution

To solve the quadratic equation $-\frac{1}{2}n^2 + 18 = 0$, we need to isolate n. The first step involves moving the constant term to the other side of the equation. This is a fundamental algebraic manipulation that helps us simplify the equation and bring it closer to a form where we can directly solve for the variable. By isolating the term containing n, we set the stage for further steps that will eventually lead us to the solution. This initial move is crucial as it separates the variable term from the constant, making the subsequent operations more straightforward. Understanding this principle is key to solving not just this equation, but many others in algebra. The process of moving terms across the equals sign is based on the properties of equality, ensuring that the equation remains balanced and the solution remains valid. This concept is the bedrock of algebraic problem-solving, and mastering it will greatly enhance your ability to tackle various mathematical challenges. We will now proceed with the next step, building upon this foundation to further isolate n. Remember, each step is carefully chosen to systematically bring us closer to the solution, and a solid grasp of these basic manipulations is essential for success in algebra. Keeping the equation balanced is paramount, and every operation must be performed on both sides to maintain the integrity of the solution process. This attention to detail is what separates a correct solution from an incorrect one. Let's continue our journey to unraveling the value of n.

1. Isolate the Term with $n^2$

Subtract 18 from both sides of the equation:

$-\frac{1}{2}n^2 + 18 - 18 = 0 - 18$

This simplifies to:

$-\frac{1}{2}n^2 = -18$

Now that we have isolated the term containing $n^2$, the next step is to eliminate the coefficient $-\frac{1}{2}$. Removing this coefficient will further simplify the equation, bringing us closer to isolating n. This process involves multiplying both sides of the equation by the reciprocal of the coefficient, which is a common technique in algebra. By doing so, we maintain the balance of the equation while making progress towards our goal of finding the value of n. This step is crucial because it transforms the equation into a simpler form where we can apply more direct methods to solve for the variable. The choice of multiplying by the reciprocal is strategic, as it effectively cancels out the existing coefficient, leaving us with a single $n^2$ term. Understanding the concept of reciprocals and their use in solving equations is fundamental to algebraic manipulation. It's a technique that you'll find useful in a wide range of mathematical problems. This step demonstrates the power of using inverse operations to isolate variables, a cornerstone of algebraic problem-solving. Let's proceed to the next step where we will perform this multiplication and further simplify the equation.

2. Eliminate the Coefficient of $n^2$

Multiply both sides by -2:

$(-2) imes (-\frac{1}{2}n^2) = (-2) imes (-18)$

This simplifies to:

$n^2 = 36$

Now that we have $n^2$ isolated on one side of the equation, the next step involves finding the values of n that satisfy this condition. Finding the square root of both sides is the key to solving for n. This is because the square root operation is the inverse of squaring, allowing us to undo the exponent and isolate the variable. When taking the square root, it's crucial to remember that there are two possible solutions: a positive and a negative value. This is because both the positive and negative roots, when squared, will result in the same positive number. Understanding this duality is essential for correctly solving quadratic equations. For example, both 6 and -6, when squared, equal 36. Therefore, we must consider both possibilities to find all solutions to the equation. This step highlights the importance of being thorough in mathematical problem-solving and not overlooking potential solutions. By considering both positive and negative roots, we ensure that we have a complete and accurate answer. Let's proceed to the next step where we will explicitly calculate these square roots and state the solutions for n. The careful consideration of both positive and negative roots is a hallmark of precise mathematical reasoning.

3. Solve for $n$

Take the square root of both sides:

$\sqrt{n^2} = \pm\sqrt{36}$

This gives us two solutions:

$n = \pm 6$

Therefore, the solutions are $n = 6$ and $n = -6$. It's always a good practice to verify your solutions by plugging them back into the original equation. This step ensures that the answers obtained are correct and satisfy the initial equation. Verification is a crucial part of the problem-solving process, as it helps to catch any potential errors made during the algebraic manipulations. By substituting the solutions back into the original equation, we can confirm whether they make the equation true. This process not only validates the answers but also reinforces the understanding of the equation and its properties. Checking solutions is a habit that should be cultivated in mathematical problem-solving, as it adds an extra layer of confidence and accuracy. It's a simple yet powerful technique that can prevent mistakes and solidify your understanding. Now, let's go ahead and verify our solutions to ensure they are indeed correct. This final step is a testament to the thoroughness and precision required in mathematics. The satisfaction of confirming a correct solution is a rewarding experience that encourages continued diligence in problem-solving. We will substitute each value of n back into the original equation and check if the equality holds true.

Verification

Let's verify the solutions by substituting them back into the original equation:

For $n = 6$:

$-\frac{1}{2}(6)^2 + 18 = -\frac{1}{2}(36) + 18 = -18 + 18 = 0$

For $n = -6$:

$-\frac{1}{2}(-6)^2 + 18 = -\frac{1}{2}(36) + 18 = -18 + 18 = 0$

Both solutions satisfy the original equation. Now that we have verified the solutions, we can confidently state the final answer. The solutions to the equation $-\frac{1}{2}n^2 + 18 = 0$ are $n = 6$ and $n = -6$. Throughout this article, we have walked through each step of the solution process, emphasizing the importance of algebraic manipulation, understanding inverse operations, and the necessity of considering both positive and negative roots. We also highlighted the crucial role of verification in ensuring the accuracy of the solutions. Mastering these techniques is fundamental to success in algebra and other mathematical disciplines. The ability to solve quadratic equations is a valuable skill with wide-ranging applications, and a thorough understanding of the underlying principles is key to tackling more complex problems. This comprehensive guide has aimed to provide not only the solution but also the reasoning behind each step, fostering a deeper understanding of the mathematical concepts involved. We encourage you to practice solving similar equations to further solidify your understanding and build confidence in your problem-solving abilities. The journey through algebra is one of continuous learning and improvement, and with consistent effort, you will undoubtedly excel in this field.

Final Answer

$n = \pm 6$

In conclusion, we have successfully solved the quadratic equation $-\frac{1}{2}n^2 + 18 = 0$, finding the solutions to be $n = 6$ and $n = -6$. This process involved several key steps, including isolating the term with $n^2$, eliminating its coefficient, and taking the square root of both sides. Each step was carefully explained to ensure a clear understanding of the algebraic manipulations involved. We also emphasized the importance of considering both positive and negative roots when solving for the square root, a crucial aspect of solving quadratic equations. Furthermore, we highlighted the necessity of verifying the solutions by substituting them back into the original equation, a practice that guarantees the accuracy of the answers. Verification is a cornerstone of mathematical problem-solving, and it should be a standard part of your approach. The skills and techniques discussed in this article are fundamental to algebra and have broad applications in various fields. Mastering these concepts will not only help you in academic settings but also equip you with valuable problem-solving skills applicable to real-world scenarios. We encourage you to continue practicing and exploring different types of quadratic equations to further enhance your understanding and proficiency. Remember, consistent practice is the key to success in mathematics. The journey through algebra is a rewarding one, and with dedication and effort, you can achieve mastery. We hope this guide has been helpful and informative, providing you with the tools and knowledge to confidently tackle similar problems in the future.