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Question

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

How do you solve 2n2=144?2{n^2} = - 144?

Explanation

Solution

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 2n2=1442{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 22
So now we can write this equation as 2n22=1442\dfrac{{2{n^2}}}{2} = - \dfrac{{144}}{2}
We can solve this equation to get n2=72{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
n2=72\sqrt {{n^2}} = \sqrt { - 72}
The square and root on the left side of the equation get cancelled to give
n=72n = \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=1×72n = \sqrt { - 1 \times 72}
This can also be written as
n=1×72n = \sqrt { - 1} \times \sqrt {72}
Now, we know that 1=i\sqrt { - 1} = i , by substituting this in the above equation, we get
n=i72n = i\sqrt {72}
We can also write this as
n=i36×2n = i\sqrt {36 \times 2}
On solving this we get
n=±6i2n = \pm 6i\sqrt 2
Hence we solve the given equation to get the final value as n=±6i2n = \pm 6i\sqrt 2
So, the correct answer is “ n=±6i2n = \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=1i = \sqrt { - 1} . This value denotes that the number is imaginary.