Is x > y ?
(1) x + 2√(xy) + y = 0
(2) x² – y² = 0


A. Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by itself is not.
B. Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by itself is not.
C. Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, even though NEITHER statement BY ITSELF is sufficient.
D. Either statement BY ITSELF is sufficient to answer the question.
E. Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question, meaning that further information would be needed to answer the question.

(A) Statement (1) has the term √(xy). So xy must be a non-negative number. It’s easy to see that if at least one of them is 0, then the other one must be 0 as well. If the both variables are positive, then the left side will be positive and it will NOT equal 0. If the both variables are negative then the equation transforms into
x + 2 × √(-x) × √(-y) + y = 0. Let’s multiply it by -1.
(-x) – 2√(-x)√(-y) + (-y) = 0. Note, that (-x) and (-y) are positive numbers. Let’s apply the formula of the sum of squares, where (-x) = (√(-x))² and (-y) = (√(-y))². This formula transforms the equation into
(√(-x) – √(-y))² = 0. This yields √(-x) = √(-y).
Therefore x = y, where x and y are NOT positive, is the solution of statement (1). Therefore we have the definite answer to the main question. Is x > y? NO. Statement (1) by itself is sufficient to answer the question.

Statement (2) transforms into x² = y². The solutions are
x = y
x = -y

As we already know, the first solution yields a definite answer NO to the main question. While the second solution can yield a positive answer to the main question. If y is negative (y = -1), then x is positive (x = +1) and so x > y (1 > -1). Statement (2) by itself is NOT sufficient to answer the question.

Statement (1) by itself is sufficient to answer the question, but statement (2) by itself is not. The correct answer is A.
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Hi,

Can you please explain this in any alternative way?
I was not able to follow the explanation properly.

Thanks.

Statement (2) is quite simple, so let's deal with it first.
x² – y² = 0
x² = y²
|x| = |y|
This yields the two possible variants: x = y and x = -y .

Plug in a positive value for y into x = -y.
If y = 1, then x = -1 and "x > y" becomes -1 > 1 ,which is NOT true.

Plug in a negative value for y into x = -y.
If y = -1, then x = 1 and "x > y" becomes 1 > -1 ,which IS true.

Thus we have no definite answer for the question "Is x > y ?". Statement (2) by itself is NOT sufficient.


Statement (1) is much harder.
x + 2√(xy) + y = 0

If we had x + 2(√x)(√y) + y = 0 then we could easily turn it into
a² + 2ab + b², where a = √x and b =√y.
Think about it. Feel the difference with Statement (1). It is the key moment here.

But we have √(xy). The case x = 0 and y = 0 is obvious so I skip it. xy must be positive. So either the two numbers are both positive or both negative.
In other words √(xy) turns into √(x)√(y) if x and y are positive, and √(-x)√(-y) if x and y are negative.

If x and y are both positive then we have
positive + positive + positive = 0, which is impossible.

The only case is x and y are negative
x + 2(√-x)(√-y) + y = 0
Multiply by (-1) and it turns into
a² – 2ab + b² = 0, where a = √-x and b = √-y.
(a – b)² = 0
a = b
√-x = √-y
-x = -y
x = y

And they both are non-positive numbers, but it is not important here. We know that x = y, so the definite answer to the question "Is x > y ?" is "NO".

(√x + √y)² = 0 ONLY if x = y = 0 . But (√x + √y)² = 0 is NOT statement (1). Statement (1) is
x + 2√(xy) + y = 0

while the proposed expression, (√x + √y)² = 0, is
x + 2(√x)(√y) + y = 0

Statement (1) implies that xy must be non-negative, while the proposed statement implies that x and y must be non-negative. Try to feel the difference.