Hi Abhinav,
This is an important concept. And when one first comes around it, it seems a little difficult to digest.
Let's try to understand this.
First of all: By definition the square root of any Quantity is a positive number. Always!
If x > 0,
x^2 > 0.
square root(x^2) = x (a positive number)
But if x < 0,
x^2 > 0
While taking the square root of x^2 now we have to keep in mind that square root(x^2) should be a positive number.
So, if square root(x^2) = x, then it's negative because x is negative.
To respect the definition of square root, a '-1' is multiplied to x to retain the value but change the sign.
now,
square root(x^2) = -x (which is a positive number. Isn't it?)
Thus
square root(x^2) = x when x > 0
square root(x^2) = -x when x < 0.
If you notice, this is same as the definition of modulus, which states that
|x| = x when x > 0
|x| = -x when x < 0.
To put the definition of square root(x^2) concisely
square root(x^2) = |x|
