himu wrote:If (x − y)² = x² − y², what is the value of nonzero integer xy ?

(1) x = 5
(2) x − y = 0

--> (x − y)² = x² − y²
--> (x - y)(x - y) = (x - y)(x + y)
--> (x - y)(x - y) - (x - y)(x + y) = 0
--> (x - y)[(x - y) - (x + y)] = 0
--> (x - y)(-2y) = 0
--> y(x - y) = 0

So, either x = y or y = 0
But it is mentioned that xy is non-zero.
Hence, y cannot be zero.
So, x = y ---> xy = x² = y²

Statement 1 x = 5
So, xy = 5² = 25

Sufficient

Statement 2 This is redundant information.

Not sufficient

The correct answer is A.

himu wrote:If (x−y)^2 = x^2 − y^2 what is the value of nonzero integer xy?

(1) x=5

(2) x−y=0

Target question: What is the value of nonzero integer xy?

Given: (x−y)^2 = x^2 − y^2
Expand left-side to get: x^2 - 2xy + y^2 = x^2 − y^2
Simplify: 2y^2 - 2xy = 0
Divide both sides by 2: y^2 - xy = 0
Factor: y(y - x) = 0

This means that y = 0 or x - y = 0

IMPORTANT: Since we're told that xy is a nonzero integer, we know that y cannot equal 0, which means x - y = 0

So, the given information is essentially telling us that x - y = 0, and our goal is to find the value of xy.

Statement 1: x = 5
Since we also know that x - y = 0, we can conclude that y = 5 as well.
So, we can be certain that xy = (5)(5) = 25
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: x - y = 0
We already knew that x - y = 0 from the given information.
As such statement 2 does not provide any new information.
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Answer = A

Cheers,
Brent

hutch27 wrote:"Expand left-side to get: x^2 - 2xy + y^2 = x^2 - y^2
Simplify: 2y^2 - 2xy = 0
Divide both sides by 2: y^2 - xy = 0
Factor: y(y - x) = 0 "


How does one simplify 2y^2 -2xy? In particular, how to get rid of the 2 all together. Thank you.

One approach.
Start with 2y^2 - 2xy = 0
Factor: 2(y^2 - xy) = 0
Divide both sides by 2 to get: [2(y^2 - xy)]/2 = 0/2
Simplify: y^2 - xy = 0


Different approach.
Start with 2y^2 - 2xy = 0
Factor: 2(y^2 - xy) = 0
We have 2(something) = 0
So, it must be true that something = 0
This means tbat y^2 - xy = 0

Cheers,
Brent