Solving -(1/2)n^2 + 18 = 0 A Step By Step Guide

Jul 13, 2025 by ADMIN 48 views

Solving Quadratic Equations: A Step-by-Step Guide to Finding the Solution of -1/2 n^2 + 18 = 0

Introduction: Understanding Quadratic Equations

In the realm of mathematics, quadratic equations play a pivotal role, appearing in diverse fields ranging from physics to engineering. These equations, characterized by a variable raised to the power of two, often present a unique set of challenges and solutions. Understanding how to solve quadratic equations is a fundamental skill for anyone delving into advanced mathematical concepts.

In this comprehensive guide, we will explore the intricacies of solving a specific quadratic equation: -1/2 n^2 + 18 = 0. This equation serves as an excellent example to illustrate the various techniques and approaches used to find solutions. By breaking down the problem into manageable steps, we will empower you to tackle similar equations with confidence.

Before we dive into the specifics, let's first establish a solid foundation by understanding the general form of a quadratic equation. A quadratic equation is typically expressed as:

ax^2 + bx + c = 0

where:

a, b, and c are constants, with a not equal to 0.
x is the variable we aim to solve for.

In our case, the equation -1/2 n^2 + 18 = 0 can be mapped to this general form, where:

a = -1/2
b = 0 (since there is no term with just 'n')
c = 18

Now that we have a clear understanding of the equation's structure, let's embark on the journey of finding its solutions. We'll explore a step-by-step approach, highlighting the underlying principles and techniques involved.

Step 1: Isolating the Squared Term

The first crucial step in solving the equation -1/2 n^2 + 18 = 0 is to isolate the term containing the squared variable, which in this case is n^2. To achieve this, we need to eliminate the constant term (+18) from the left side of the equation. We can accomplish this by subtracting 18 from both sides of the equation. This maintains the equality and moves us closer to isolating the n^2 term.

Let's perform this operation:

-1/2 n^2 + 18 - 18 = 0 - 18

This simplifies to:

-1/2 n^2 = -18

Now we have successfully isolated the term with the squared variable on one side of the equation. The next step involves dealing with the coefficient of n^2, which is -1/2. To get n^2 by itself, we need to eliminate this coefficient.

Step 2: Eliminating the Coefficient

To eliminate the coefficient of n^2, which is -1/2, we need to perform an operation that will result in n^2 having a coefficient of 1. This can be achieved by multiplying both sides of the equation by the reciprocal of -1/2, which is -2. Remember, multiplying both sides of an equation by the same non-zero value maintains the equality.

Let's perform this multiplication:

(-2) * (-1/2 n^2) = (-2) * (-18)

This simplifies to:

n^2 = 36

Now we have successfully isolated n^2 on one side of the equation. We are one step closer to finding the solution for n. The next step involves undoing the square to find the value(s) of n.

Step 3: Taking the Square Root

Now that we have the equation in the form n^2 = 36, we need to find the value(s) of n that, when squared, equal 36. This is where the concept of square roots comes into play. The square root of a number is a value that, when multiplied by itself, gives the original number. However, it's crucial to remember that both positive and negative numbers can have the same square. For example, both 6 and -6, when squared, result in 36.

To find the solutions for n, we need to take the square root of both sides of the equation:

√(n^2) = ±√36

This gives us:

n = ±6

The symbol ± indicates that there are two possible solutions: a positive value and a negative value. Therefore, the solutions for n are 6 and -6.

Step 4: Expressing the Solution

We have successfully found the two solutions for the equation -1/2 n^2 + 18 = 0. The solutions are n = 6 and n = -6. It's important to express the solution in a clear and concise manner. In this case, we can express the solution as:

n = ±6

This notation efficiently represents both solutions. In the context of the original problem, where the answer box is represented as n = ± □, we would fill the box with the numeral 6.

Conclusion: Mastering Quadratic Equations

In this comprehensive guide, we have successfully navigated the process of solving the quadratic equation -1/2 n^2 + 18 = 0. By following a step-by-step approach, we have demonstrated how to isolate the squared term, eliminate coefficients, and utilize square roots to find the solutions. This process not only solves the specific equation at hand but also provides a framework for tackling a wide range of quadratic equations.

Key takeaways from this guide:

Understanding the general form of a quadratic equation (ax^2 + bx + c = 0) is crucial for identifying the coefficients and applying appropriate solving techniques.
Isolating the squared term is a fundamental step in simplifying the equation and preparing it for further manipulation.
Eliminating the coefficient of the squared term allows us to work with a simpler equation, making it easier to find the solutions.
Taking the square root introduces the concept of both positive and negative solutions, as both values, when squared, can result in the same positive number.
Expressing the solution clearly and concisely is essential for communicating the results effectively.

By mastering these techniques, you will be well-equipped to solve various quadratic equations and apply these skills to more complex mathematical problems. Remember, practice is key to solidifying your understanding. Work through additional examples and challenge yourself with different types of quadratic equations. With dedication and persistence, you will unlock the power of quadratic equations and their applications in the world around us.

This step-by-step approach to solving quadratic equations can be applied to many scenarios. Understanding the underlying principles is key to successfully tackling these types of problems. Remember to always check your work and ensure that your solutions make sense in the context of the original equation. This process will enhance your problem-solving skills and boost your confidence in mathematics.

Common Mistakes to Avoid When Solving Quadratic Equations

When tackling quadratic equations, it's easy to fall into common traps that can lead to incorrect solutions. Recognizing these potential pitfalls and understanding how to avoid them is crucial for achieving accurate results. Here are some of the most frequent mistakes students make when solving quadratic equations, along with guidance on how to prevent them:

1. Forgetting the Negative Root

One of the most common errors is neglecting the negative square root when solving for a variable. As we discussed earlier, both the positive and negative square roots of a number, when squared, yield the same positive result. For instance, both 6 and -6, when squared, equal 36. Therefore, when taking the square root of both sides of an equation, it's essential to remember to include both the positive and negative possibilities.

Example:

If we have the equation n^2 = 25, the correct solution is n = ±5, not just n = 5. Forgetting the negative root would lead to an incomplete and incorrect answer.