What Is The Solution Of This Equation?$\[ -\frac{1}{2} N^2 + 18 = 0 \\]$\[ n = \pm \square \\] Type The Correct Answer In The Box. Use Numerals Instead Of Words.

Introduction

In mathematics, solving equations is a fundamental concept that helps us find the value of unknown variables. When we encounter an equation like $-\frac{1}{2} n^2 + 18 = 0$, our goal is to isolate the variable $n$ and determine its value. In this article, we will explore the solution to this quadratic equation and provide a step-by-step guide on how to solve it.

Understanding the Equation

The given equation is a quadratic equation in the form of $ax^2 + bx + c = 0$, where $a = -\frac{1}{2}$, $b = 0$, and $c = 18$. To solve this equation, we need to find the value of $n$ that satisfies the equation.

Step 1: Rearrange the Equation

The first step in solving the equation is to rearrange it to get all the terms on one side of the equation. We can do this by subtracting $18$ from both sides of the equation:

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

Step 2: Multiply Both Sides by -2

To eliminate the fraction, we can multiply both sides of the equation by $-2$:

$$n^2 = 36$$

Step 3: Take the Square Root

Now that we have the equation in the form of $n^2 = 36$, we can take the square root of both sides to find the value of $n$. Since the square root of a number can be positive or negative, we need to consider both possibilities:

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

Step 4: Simplify the Square Root

The square root of $36$ is $6$, so we can simplify the equation to:

$$n = \pm 6$$

Conclusion

In conclusion, the solution to the equation $-\frac{1}{2} n^2 + 18 = 0$ is $n = \pm 6$. This means that the value of $n$ can be either $6$ or $-6$, and both values satisfy the equation.

Final Answer

The final answer is: $\boxed{6}$

Introduction

In our previous article, we explored the solution to the equation $-\frac{1}{2} n^2 + 18 = 0$. We provided a step-by-step guide on how to solve the equation and found that the solution is $n = \pm 6$. In this article, we will address some frequently asked questions (FAQs) about solving this equation.

Q: What is the difference between a quadratic equation and a linear equation?

A: A quadratic equation is a polynomial equation of degree two, which means it has a squared variable. In contrast, a linear equation is a polynomial equation of degree one, which means it has a variable but no squared variable. The equation $-\frac{1}{2} n^2 + 18 = 0$ is a quadratic equation because it has a squared variable $n^2$.

Q: How do I know if an equation is quadratic or linear?

A: To determine if an equation is quadratic or linear, look for the highest power of the variable. If the highest power is two, the equation is quadratic. If the highest power is one, the equation is linear.

Q: What is the significance of the coefficient $-\frac{1}{2}$ in the equation $-\frac{1}{2} n^2 + 18 = 0$?

A: The coefficient $-\frac{1}{2}$ affects the shape of the parabola that represents the equation. In this case, the coefficient is negative, which means the parabola opens downward. If the coefficient were positive, the parabola would open upward.

Q: Can I use the quadratic formula to solve the equation $-\frac{1}{2} n^2 + 18 = 0$?

A: Yes, you can use the quadratic formula to solve the equation $-\frac{1}{2} n^2 + 18 = 0$. The quadratic formula is:

$$n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

In this case, $a = -\frac{1}{2}$, $b = 0$, and $c = 18$. Plugging these values into the quadratic formula, we get:

$$n = \frac{0 \pm \sqrt{0^2 - 4(-\frac{1}{2})(18)}}{2(-\frac{1}{2})}$$

Simplifying the expression, we get:

$$n = \frac{0 \pm \sqrt{36}}{-1}$$

$$n = \pm 6$$

Q: What is the difference between the solutions $n = 6$ and $n = -6$?

A: The solutions $n = 6$ and $n = -6$ are both valid solutions to the equation $-\frac{1}{2} n^2 + 18 = 0$. However, they represent different values of $n$. The solution $n = 6$ means that $n$ is equal to $6$, while the solution $n = -6$ means that $n$ is equal to $-6$.

Q: Can I use the equation $-\frac{1}{2} n^2 + 18 = 0$ to model real-world problems?

A: Yes, you can use the equation $-\frac{1}{2} n^2 + 18 = 0$ to model real-world problems. For example, you could use this equation to model the height of a projectile as a function of time, or the amount of money in a savings account as a function of time.

Conclusion

In conclusion, solving the equation $-\frac{1}{2} n^2 + 18 = 0$ requires a step-by-step approach. We can use the quadratic formula to solve the equation, and the solutions $n = 6$ and $n = -6$ represent different values of $n$. We can also use this equation to model real-world problems, such as the height of a projectile or the amount of money in a savings account.

Final Answer

The final answer is: $\boxed{6}$