How do you solve \[2{n^2} = - 144?\]

Hint: In order to solve this question, we have to follow certain steps and find the value of the variable n. The given equation contains a negative value on the right side of the equation and a square on the variable n therefore we use the definition of ‘i’ to solve this problem.

Complete step-by-step answer:
Given to us is an equation \[2{n^2} = - 144\]
We have to now find the value of the variable n.
In order to do this, let us first divide both the sides of the equation by $ 2 $
So now we can write this equation as $ \dfrac{{2{n^2}}}{2} = - \dfrac{{144}}{2} $
We can solve this equation to get $ {n^2} = - 72 $
In order to find the value of the variable n, let us now take square root on both sides of the equation.
The equation now becomes
 $ \sqrt {{n^2}} = \sqrt { - 72} $
The square and root on the left side of the equation get cancelled to give
 $ n = \sqrt { - 72} $
Here, we see that there is a negative value inside the square root. We know that a negative value inside a square root is an imaginary value. We can write this as follows.
 $ n = \sqrt { - 1 \times 72} $
This can also be written as
 $ n = \sqrt { - 1} \times \sqrt {72} $
Now, we know that $ \sqrt { - 1} = i $ , by substituting this in the above equation, we get
 $ n = i\sqrt {72} $
We can also write this as
 $ n = i\sqrt {36 \times 2} $
On solving this we get
 $ n = \pm 6i\sqrt 2 $
Hence we solve the given equation to get the final value as $ n = \pm 6i\sqrt 2 $
So, the correct answer is “ $ n = \pm 6i\sqrt 2 $ ”.

Note: It is to be noted that any negative value inside a square root does not exist so it is an imaginary value. This imaginary value is denoted as i where $ i = \sqrt { - 1} $ . This value denotes that the number is imaginary.