# Solution

Everything was fine till (4 - 9/2)^{2} = (5 - 9/2)^{2}.

But, then, my friend took the square roots and concluded that (4 - 9/2) = (5 - 9/2). That was absolutely wrong!

Square root of (4 - 9/2)^{2} can be either +(4 - 9/2) or -(4 - 9/2). Similarly, square root of (5 - 9/2)^{2}
can be either +(5 - 9/2) or -(5 - 9/2).

For instance, square root of 4 can be either +2 or -2. So, although 2^{2} = (-2)^{2} = 4, 2 is not equal to -2.

So, we cannot simply remove the squares; we will have to work out four equalities in this case to see which is true:

1. +(4 - 9/2) = +(5 - 9/2) ...... this, obviously, does not hold true.

2. +(4 - 9/2) = -(5 - 9/2) ..... this holds.

3. -(4 - 9/2) = +(5 - 9/2) ..... this holds, and

4. -(4 - 9/2) = -(5 - 9/2) ..... this does not hold true.

Instead of considering all the four alternatives, my friend conveniently went along with the first alternative which, of course, does not hold true.

Let's consider the step (4 - 9/2)^{2} = (5 - 9/2)^{2}. Now, 4 - 9/2 = -1/2 and 5 - 9/2 = 1/2; so, what we have is;
(-1/2)^{2} = (1/2)^{2}. This is true because 1/4 = 1/4, but -1/2 is not equal to 1/2. But -(-1/2) = 1/2 is true, and
so also -1/2 = -(1/2).