"The seemingly innocuous equation x squared = - 1".......i read that in one of the math books in a DC library.......

Imaginary Numbers

An Imaginary Number, when squared, gives a negative result.

Try 

Let's try squaring some numbers to see if we can get a negative result:

2 × 2 = 4
(−2) × (−2) = 4 (because a negative times a negative gives a positive)
0 × 0 = 0
0.1 × 0.1 = 0.01

No luck! Always positive, or zero. 

It seems like we cannot multiply a number by itself to get a negative answer ...

... but imagine that there is such a number (call it i for imaginary) that could do this: i × i = −1  Would it be useful, and what could we do with it?

Well, by taking the square root of both sides we get this:

Which means that i is the answer to the square root of −1.

Which is actually very useful because ... 

... by simply accepting that i exists we can solve things
that need the square root of a negative number.

Let us have a go:

Example: What is the square root of −9 ?

Answer: = √(9 × −1) = √(9) × √(−1) = 3 × √(−1) = 3i

(see how to simplify square roots)

Hey! that was interesting! The square root of −9 is simply the square root of +9, times i.

In general:

√(−x) = i√x

So long as we keep that little "i" there to remind us that we still
need to multiply by √−1 we are safe to