Verified Answer :

  • The equation \(- \frac{1}{2} n^2 + 18 = 0\) is a quadratic equation.
  • Isolate the variable term by adding \(\frac{1}{2} n^2\) to both sides, then multiplying by \(2\) to get \(n^2 = 36\).
  • Calculate the roots by taking the square root of both sides, resulting in \(n = \pm 6\).
  • Final answer: \(n = \pm \boxed{6}\).

Explanation

  1. Analyzing the Equation

The given equation is \(-\frac{1}{2} n^2 + 18 = 0\). This is a quadratic equation where the variable \(n\) appears as a squared term. To solve for \(n\), we need to isolate \(n^2\) and then find the square root.

  1. Isolating the Squared Variable

First, add \(\frac{1}{2} n^2\) to both sides of the equation to isolate the term containing \(n\):
\[18 = \frac{1}{2} n^2\]
Next, multiply both sides of the equation by 2 to solve for \(n^2\):
\[18 \times 2 = n^2\]
\[36 = n^2\]

  1. Solving for n

To find the value of \(n\), take the square root of both sides of the equation \(n^2 = 36\). Remember that the square root of a positive number yields both a positive and a negative solution:
\[n = \pm \sqrt{36}\]
\[n = \pm 6\]
Thus, the solutions to the equation are \(n = 6\) and \(n = -6\). The equation is satisfied when \(n = \pm 6\).

Examples

Consider a scenario where you are calculating the displacement of an object in physics. If the energy equation of an oscillating system is given by \(E = \frac{1}{2} k n^2\), where \(E\) represents the total energy and \(k\) is a constant, solving for \(n\) helps determine the position of the object at a specific energy level. For instance, if \(k = 1\) and \(E = 18\), the equation becomes \(\frac{1}{2} n^2 = 18\), which directly relates to finding the amplitude of motion in such systems.

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