Is x y? (1) x + 2(xy)^(1/2) + y = 0 (2) x^2 - y^2 = 0

Question

Is x > y?

(1)  x + 2\sqrt{xy} + y = 0

(2) x^2 - y^2 = 0

Bunuel · Accepted Answer

Is x > y?

(1)  x + 2\sqrt{xy} + y = 0 ;

Re-arrange:  x + y = -2\sqrt{xy} ;

Square:  x^2 + 2xy + y^2 = 4xy ;

Re-arrange:  x^2 - 2xy + y^2 = 0 ;

(x-y)^2=0 ;

x=y .

x is not greater than y. Sufficient.

(2) x^2 - y^2 = 0 --> x^2 = y^2 --> |x| = |y|. x can be more than y (x=1, y=-1), less than y (x=-1, y=1) or equal to y (x=y=1). Not sufficient.

Answer: A.

mvictor · Answer

i arrived to the answer choice differently...kind of..

2 - started with this one, since it is easier - clearly insufficient: (x+y)(x-y)=0, so either x=y - which is a definite no, or x=-y. if y=positive, then x<y. if y=negative, then x>y. so 2 outcomes...B and D - out.

now 1.
x+y=-sqrt(4xy)
square everything:
x^2+y^2+2xy = 4xy
x^2+y^2=2xy
x^2+y^2-2xy=0
(x-y)(x-y)=0
we know for sure that x=y. so the answer to the main question is only NO.

A

Izzyjolly · Answer

Hi  Bunuel 
From my point of view, in statement 1 we know that xy >= 0, but we can't conclude that x>=0 and y>=0 . Thus, writing  \sqrt{x}  and  \sqrt{y}  seems not quite right.
Can you please check it?

Bunuel · Answer

Good catch. You are absolutely right. Edited the post. Thank you.

T1101 · Answer

Hi  Bunuel

I do not understand why we can not infer from the red part that x and y have to be 0? Why do we need to square the whole expression in statement 1 and can not use the square roots of x and y? Could you explain it in more detail?

Thank you!

Bunuel · Answer

____________________
Consider x = y = -1.

BelalHossain046 · Answer

Statement 1:
x+2 √ (xy)+ y=0
or, (√x)^2+2. √x.√y.+(√y)^2=0
or, (√x+√y)^2=0
or, (√x+√y)=0
or, √x=-√y
 Since any value inside the square root CAN NOT be negative , x and y are both ZERO or BOTH positive. 
 If x and y are both positive, the second segment of the equation will be negative and first portion will be positive , which is impossible in equation. 
So, x and y can NOT be both positive.  Thus, x and y are both ZERO.

BelalHossain046 · Answer

Statement 2: x^2 - y^2 = 0
or, (x+y) (x-y)=0, implies that 
(1) only (x+y) is zero, or 
(2) only(x-y) is zero or
(3) Both (x+y) and (x-y) is zero.

for example, if x+y=0 
x and y can be (1, -1 : YES to main question) or (-1 and 1, NO to main question)

BelalHossain046 · Answer

Number picking can also be a easy solution for statement 2.
Statement 2: x^2 - y^2 = 0
if x=1, y=1 (No to main question) =======> (1)^2-(1)^2=0
if x=1, y= -1  (YES to main question) =======> (1)^2-(-1)^2=0

Insufficient

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

Is x > y? (1) x + 2(xy)^(1/2) + y = 0 (2) x^2 - y^2 = 0

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Data Sufficiency (DS) Questions