bobdylan wrote:Mr Anurag, can you please explain a little more in detail. I'm not grasping the concept behind this question.
It would've helped if you had pointed out which steps you do not understand. Anyway I'm guessing that it is "√[(x-3)^2] = |x - 3|"
Every positive number x has two square roots: √x, which is positive, and -√x, which is negative. Together, these two roots are denoted ±√x. The positive square root, √x is known as the principal square root of x.
By definition √x is positive.
Note that the positive or negative sign comes before the √ sign. That's your clue.
Whenever the square root sign '√' is used it is used to mean the principal square root, i.e. the positive square root. Hence, square roots of 4 are ±√4, i.e. -2 and 2. But √4 is always equal to 2 NOT -2.
To remove ambiguities we use the modulus notation.
We write √(x²) = |x|, so that we always get the principal square root, i.e. the positive square root. Now, |x| is always positive. Hence,
For x > 0 --> √(x²) = |x| = x > 0
For x < 0 --> √(x²) = |x| = -x > 0
Thus we always get the positive square root.
Let's take two examples,
1. For x = 2 = √(x²) = √[(2)²] = |2| = 2
2. For x = -2 = √(x²) = √[(-2)²] = |-2| = 2
