Question

How do you solve $2{n^2} = - 144?$

Answer 1

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.