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For all real numbers x and y, let x# y = (xy)^2 − x + y^2

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by BTGmoderatorDC » Tue Aug 14, 2018 10:48 pm

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Answers

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D

E

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Difficulty—

For all real numbers x and y, let x# y = (xy)^2 − x + y^2 . What is the value of y that makes x # y equal to -x for all values of x ?

(A) 0
(B) 2
(C) 5
(D) 7
(E) 10

OA A

Source: Princeton Review
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by Brent@GMATPrepNow » Wed Aug 15, 2018 5:13 am
BTGmoderatorDC wrote:For all real numbers x and y, let x# y = (xy)² − x + y² . What is the value of y that makes x # y equal to -x for all values of x ?

(A) 0
(B) 2
(C) 5
(D) 7
(E) 10
APPROACH #1
We want the following equation to hold true: x # y = -x
Replace x # y with its equivalent to get: (xy)² − x + y² = -x
Add x to both sides to get: (xy)² + y² = 0
Simplify (xy)² to get: x²y² + y² = 0
Factor out the y² to get: y²(x² + 1) = 0
So, the equation will hold true when EITHER y² = 0 OR x² + 1 = 0
If y² = 0, then y =0

Answer: A

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by Brent@GMATPrepNow » Wed Aug 15, 2018 5:16 am
BTGmoderatorDC wrote:For all real numbers x and y, let x#y = (xy)^2 − x + y^2 . What is the value of y that makes x # y equal to -x for all values of x ?

(A) 0
(B) 2
(C) 5
(D) 7
(E) 10
APPROACH #2 - Test the answer choices

A) 0
Take x#y = (xy)² − x + y², and replace y with 0
We get: [(x)(0)]² − x + 0² = [0]² − x + 0
= -x
So, when y = 0, x#y = -x
PERFECT!

Answer: A

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by Scott@TargetTestPrep » Fri Apr 19, 2019 9:15 am
BTGmoderatorDC wrote:For all real numbers x and y, let x# y = (xy)^2 − x + y^2 . What is the value of y that makes x # y equal to -x for all values of x ?

(A) 0
(B) 2
(C) 5
(D) 7
(E) 10

OA A

Source: Princeton Review
If y = 0, then x # 0 is:

0 - x + 0 = -x

Alternate solution:

We want x # y = -x, i.e.,

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

x^2y^2 + y^2 = 0

y^2(x^2 + 1) = 0

Since x^2 + 1 can't be 0, y^2 must be 0, and hence y must be 0.


Answer: A

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