What is value of x - y?

You semplify those terms(x+y) : the condition #0 is not necessary here.
Yes, from A we can deduce the answer.

1. (x^2 - y^2) / (x+y) = (x+y)(x-y)/(x+y)
If (x+y) is 0, (x+y)/(x+y) will be 0/0 which is undefined.
In that case (x^2 - y^2) / (x+y) cannot be equal to 6.
That means, we can safely say (x+y) <> 0

So, (x^2 - y^2) / (x+y) = x-y = 6
Sufficient

2. (x^2 -2*x*y + y^2) = 36
--> (x-y)^2 = 6^2
--> x - y = +6 or x - y = -6
Not sufficient

Answer is A.

I want to make something extremely explicit that is implicit in the fine responses of Zarrolou and doe007 above. Everything those two people say is 100% correct, and well explained, but I can see that it might not make perfect sense to someone who is missing some very basic pieces of algebra.

There are three fundamental algebra formulas, three fundamental patterns, that everyone taking the GMAT needs to know cold.

1) The Difference of Squares
\(A^2 - B^2 = (A + B)(A - B)\)

2) The Square of a Sum
\((A + B)^2 = A^2 + 2AB + B^2\)

3) The Square of a Difference
\((A - B)^2 = A^2 - 2AB + B^2\)

Here are two blogs that explains the processes you would use to derive or verify these:
https://magoosh.com/gmat/2012/foil-on-th ... expanding/
https://magoosh.com/gmat/2012/algebra-on ... to-factor/

Here's another blog on the first:
https://magoosh.com/gmat/2012/gmat-quant ... o-squares/

It's not just that you need to know them. You need to know them better than you know you own phone number. You need to recognize these patterns in whatever way they appear on the GMAT. The difference between recognizing these patterns and being fluent in them vs. not knowing them at all, is the difference between solving the DS problem at the topic in 30 seconds vs. staring at it in utter frustration, not knowing how to begin. Furthermore, this DS question is not unique. You are guaranteed to see some of these formulas sprinkled throughout the 37 question you see on any GMAT Quant section. GMAT Quant questions are specifically designed to test your knowledge of and fluency with these three patterns, and if you don't know them, any so designed question will be impenetrable to you. Knowing these is absolutely essential for GMAT Quant success.

Without going through a full solution (it's already been done quite handily above), I'll just mention --- in this DS question:
Statement #1 simplifies immensely with the use of the Difference of Squares formula
Statement #2 simplifies immensely with the use of the Square of a Sum formula

BTW, Sujit2k7, it's completely unnecessary to specify in the first that (x + y) is unequal to zero. You see, (x + y) is the denominator of a fraction that equals 6. The only way a fraction can equal any well-defined number is if it's denominator is unequal to zero. If a fraction has a denominator of zero, then any statement involving that fraction and equal sign would be a mathematical obscenity, which never would appear on the GMAT.

Please let me know if anyone reading this has any further questions.

Mike

Mike, well said. All the required algebraic and geometric formulas need to be well absorbed so that those formulas can become handy to respond to the questions quickly and correctly during GMAT exam where every second is precious.

In GMAT are not we supposed to consider +ve roots only, in this case (x-y)^2=36->x-y=6? Please clarify.

When the GMAT provides the square root sign for an even root, such as a square root, fourth root, etc. then the only accepted answer is the positive root. That is:

\(\sqrt{9} = 3\), NOT +3 or -3;
\(\sqrt[4]{16} = 2\), NOT +2 or -2;

Notice that in contrast, the equation \(x^2 = 9\) has TWO solutions, +3 and -3. Because \(x^2 = 9\) means that \(x =-\sqrt{9}=-3\) or \(x=\sqrt{9}=3\).