Problem Solving

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

Cheers,
Brent

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

Cheers,
Brent