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If (x - y)^2 = x^2 - y^2, what is the value of the non-zero integer xy

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Bunuel
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If (x - y)^2 = x^2 - y^2, what is the value of the non-zero integer xy [#permalink] New post  14 Jan 2020, 01:46
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If \((x - y)^2 = x^2 - y^2\), what is the value of the non-zero integer \(xy\)?

(1) \(x = 5\)

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

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Re: If (x - y)^2 = x^2 - y^2, what is the value of the non-zero integer xy [#permalink] New post  14 Jan 2020, 02:18
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If \((x - y)^2 = x^2 - y^2\), what is the value of the non-zero integer \(xy\)?

(1) \(x = 5\)
\((x - y)^2 = x^2 - y^2\) ... opening the \((x - y)^2\)
\(x^2 + y^2 - 2xy = x^2 - y^2 \) .. cancelling out \(x^2\) , we get
\(y^2 - 2xy = - y^2 \) .. re-writing
\(2y^2 = 2xy \) .. cancelling \(2y\) both side, we get
\(y = x\) .. sufficient ( xy = 25)

A D / B C E

(2) \(x-y = 0\)
It says \(x = y\) ... if we put into the main equation, we can get Zero value which we do not want
Neither x nor y value is known apart from 'Zero'. They could be anything. So, Insufficient.

'A' is the winner
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Re: If (x - y)^2 = x^2 - y^2, what is the value of the non-zero integer xy [#permalink] New post  14 Jan 2020, 03:31
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Constraint: (x-y)^2 = x^2-y^2
Question: Value of non-zero Integer xy?

Note! xy = non-zero Integer, so neither x nor y is zero.
Guys! let’s breakdown the constraint
x^2+y^2 -2xy = x^2 - y^2
2y^2 = 2xy ——> y^2= xy
.: y=x (we can divide by y since it ain’t zero)
Now constraint: y=x

(1) x= 5
We know y=x ,and x= 5
.: xy = (5)(5)=25
(Sufficient)

(2) x=y
Just reiterating the constraint
Say x=5 ,xy =25 , again x=-2 ,xy=4
(Ain’t Sufficient)

Hit that A ;)

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Re: If (x - y)^2 = x^2 - y^2, what is the value of the non-zero integer xy [#permalink] New post  14 Jan 2020, 06:27
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(x-y)^2 = x^2 - y^2 --> y^2 = xy.

1) If x=5, then y=0 or y=5. As a result, xy=0 or xy=25. xy is non-zero integer, thus value of xy is invariably 25.
SUFFICIENT

2)x=y --> x^2=y^2 and and xy can be 1,4,9,16,...
NOT SUFFICIENT

FINAL ANSWER IS (A)

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Re: If (x - y)^2 = x^2 - y^2, what is the value of the non-zero integer xy [#permalink] New post  14 Jan 2020, 06:42
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If \(((x - y)^2 = x^2 - y^2)\), what is the value of the non-zero integer \((xy)\)?
xy ≠ 0 means x ≠ 0 or y ≠ 0
\((x^2 + y^2 - 2xy = x^2 - y^2)\)
\((y^2 - 2xy = - y^2)\)
\((2y^2 - 2xy = 0)\)
\((y(y - x) = 0)\)
y = x

So either x = ? or y = ?

(1) \((x = 5)\)

xy = 5 * 5 = 25

SUFFICIENT.

(2) \((x-y = 0)\)
x = y MANY Possibilities

INSUFFICIENT.

IMO Answer A.
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If (x - y)^2 = x^2 - y^2, what is the value of the non-zero integer xy [#permalink] New post  Updated on: 15 Jan 2020, 00:56
from given info
we can say ; ( x-y)^2 = ( x+y) ( x-y)
simplify the given expression we get
2y^2= 2xy
or y=x
#1
x=5
so y=5 ; xy= 25
sufficient
#2
x-y=0
so x+y=0
value of x & y are same but opposite sign insufficient as x*y will vary as it can be zero and non zero both of many possible values
insufficient
IMO A

If (x−y)2=x2−y2, what is the value of the non-zero integer xyxy?

(1) x=5x=5

(2) x−y=0

Originally posted by Archit3110 on 14 Jan 2020, 07:43.
Last edited by Archit3110 on 15 Jan 2020, 00:56, edited 1 time in total.
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Re: If (x - y)^2 = x^2 - y^2, what is the value of the non-zero integer xy [#permalink] New post  14 Jan 2020, 09:42
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Given, \((x−y)^2=x^2−y^2\) & non-zero integer xy. So, \(x, y ≠ 0\)
--> \(x^2 + y^2 - 2xy - x^2 + y^2\)
--> \(2y^2 - 2xy = 0\)
--> \(2y(y - x) = 0\)
Since, \(y ≠ 0, y - x = 0\)
--> \(y = x\)

(1) \(x = 5\)
--> \(y = x = 5\)
--> \(xy = 5*5 = 25\) --> Sufficient

(2) \(x - y = 0\)
--> \(x = y\)
So, \(xy = x*x = x^2\) --> No definite value! --> Insufficient

Option A
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Re: If (x - y)^2 = x^2 - y^2, what is the value of the non-zero integer xy [#permalink] New post  14 Jan 2020, 22:03
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Re: If (x - y)^2 = x^2 - y^2, what is the value of the non-zero integer xy

(x-y)2= x2+y2-2xy

x^2+y^2-2xy=x^2 - y^2

x^2 cancels out

y^2-2xy=-y^2

2y^2-2xy=0

y^2-xy=0

Therefor y=0 or y-x=0

Statement 1:

X=5. if we substitute this in the previous equation we get the values x and y as 5

hence statement 1 is sufficient

Statement 2:
x-y=0

This is non conclusive as the values can be same or can also be 0

Hence IMO A
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Re: If (x - y)^2 = x^2 - y^2, what is the value of the non-zero integer xy [#permalink] New post  14 Jan 2020, 22:23
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Ans: A

(x−y)2=x2−y2
x2+y2-2xy=x2-y2
y(y-x)=0

so, y=0 or y=x

now y cannot be 0 as xy is the non-zero integer as mentioned

so y=x only possible

a states x=5

so, xy=25
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Re: If (x - y)^2 = x^2 - y^2, what is the value of the non-zero integer xy [#permalink] New post  14 Jan 2020, 23:15
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Given that (x-y)^2 = x^2 - y^2, we are to find the value of the non-zero integer xy.
It is worth noting that (x-y)^2 can only equal x^2 - y^2 when x=y or x-y=0.
So, to determine xy, we only need to know the value of x or y as xy=x^2=y^2

Statement 1: x=5
This is sufficient as x=y=5 and xy=5*5=25.

Statement 2: x−y=0
Insufficient. we already know from the question stem that x-y=0 or x=y. We need an actual value of x or y in order to determine xy. Since that information is not provided, there are infinite possibilities.

The answer is A.
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Re: If (x - y)^2 = x^2 - y^2, what is the value of the non-zero integer xy [#permalink] New post  15 Jan 2020, 00:44
Remember (x+y)^2 =x^2+y^2+2xy
(x-y)^2= x^2+y^2-2xy
(x^2-y^2) = (x-y)(x+y)
Think that’s what you missed :)
Archit3110 wrote:
from given info
we can say ; ( x-y)^2 = ( x+y) ( x-y)
(x-y)=(x+y)
#1
x=5
insufficient as y is not know ; and y has to be 0 but its against condition
#2
x-y=0
so x+y=0
value of x & y are same but opposite sign insufficient as x*y will vary
from 1 & 2
x=5 so y = -5
x*y = -25
IMO C sufficient

If (x−y)2=x2−y2, what is the value of the non-zero integer xyxy?

(1) x=5x=5

(2) x−y=0


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Re: If (x - y)^2 = x^2 - y^2, what is the value of the non-zero integer xy [#permalink] New post  03 Dec 2020, 17:20
Bunuel wrote:

Competition Mode Question



If \((x - y)^2 = x^2 - y^2\), what is the value of the non-zero integer \(xy\)?

(1) \(x = 5\)

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

Are You Up For the Challenge: 700 Level Questions


\(x^2 - 2xy + y^2 = x^2 - y^2\)

\(2y^2 - 2xy = 0\)

\(y^2 - xy = 0\)

\(y(y-x) = 0\)

\(y = x\)

(1) \(x = 5\); therefore \(xy = 25\).

SUFFICIENT.

(2) Clearly insufficient.

Answer is A.
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Re: If (x - y)^2 = x^2 - y^2, what is the value of the non-zero integer xy [#permalink]
03 Dec 2020, 17:20
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