Square root problem!

Interesting question. First, before looking at the statements, let's look at the question itself. Actually, let's start with a simpler question:

What is the sqrooot(y^2) equal to? By sqroot, I mean the 'non-negative sqrooot' (I can't write the root symbol here unfortunately!).

If y is positive, sqrooot(y^2) is just y.
But, if y is negative, sqrooot(y^2) is not y; the sqroot can't be negative. Instead, it's equal to |y|, which is equal to -y.

Now let's look at the question:

Is sqroot((x-5)^2) = 5 - x ?

Well, just as with y above:
sqroot((x-5)^2) will be equal to x-5 if x-5 is positive.
sqroot((x-5)^2) will be equal to -(x-5) = 5-x if x-5 is negative.

So the question is just asking: is x-5 negative? That is, is x<5? Statement 2) tells us that exactly, so it's sufficient. Statement 1) requires a bit more work:

-x|x| > 0

We can divide both sides by |x| here since we know |x| is positive (no need to worry about whether you'd need to reverse the inequality), so

-x > 0
x < 0

So again, x is certainly less than 5, and Statement 1 is also sufficient on its own.

D.

(technically, I should have also included the possibility that x-5 = 0 in the above, but it doesn't affect the solution in this case, and I've left it out for clarity).

anshul265 wrote: I am still stuck at one thing. Please find a problem with the following.

Sqroot of a^2 is normally written as +/- a. As both +a and -a on squaring will become a^2.

There is a technicality here. It is definitely true that a^2 has two square roots, a and -a, as you point out (unless, of course, a = 0). But the square root symbol, which I'll try to draw:
_____
\/

*means* the non-negative square root *only*. If you see this:

______
\/ a^2

this has only one solution, either a or -a, whichever of these is positive (or zero). It does not have two solutions, and you cannot assume it to be equal to a unless you know whether a is positive. So it is incorrect to say that

________
\/ (x-5)^2 = x-5

This is only true if x - 5 >= 0. And that's where your answer differs from the GMAT's (and mine). Hope that clears things up.

anshul265 wrote:I am getting what you are trying to explain, but in GMAT I have gotten in trouble a lot of times for not considering the negative values. Anyway, thanks a lot.

Yes, and you will often get in trouble for not considering negatives. If you see the equation x^2 = 1, for example, there are two solutions for x, 1 and -1. The difference in the question you posted above is that it contains the square root symbol. That symbol means 'the non-negative square root', so can never produce a negative result. If you see the number 1 underneath the square root symbol, the value is 1, not -1.