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Algebra
Absolute value
Problem solving
GMAT QUANTITATIVE REASONINGQ-51 Series
Question
If x and y are integers and |x - y| = 12, what is the minimum
possible value of xy?
A. -12
B. -18
C. -24
D. -36
E. -48
x and y are integers and |x - y| = 12
What is the minimum possible value of xy?
Square both sides of |x – y| = 12
· (x – y)2 = 144
x2 + y2 – 2xy = 144
x and y are integers and |x - y| = 12
What is the minimum possible value of xy?
Square both sides of |x – y| = 12
· (x – y)2 = 144
x2 + y2 – 2xy = 144
· Add 4xy to both sides of the equation
x2 + y2 – 2xy + 4xy = 144 + 4xy
x2 + y2 + 2xy = 144 + 4xy
(x + y)2 = 144 + 4xy
x and y are integers and |x - y| = 12
What is the minimum possible value of xy?
Square both sides of |x – y| = 12
· (x – y)2 = 144
x2 + y2 – 2xy = 144
· Add 4xy to both sides of the equation
x2 + y2 – 2xy + 4xy = 144 + 4xy
x2 + y2 + 2xy = 144 + 4xy
(x + y)2 = 144 + 4xy
(x + y)2 will NOT be negative for real x and y.
· i.e., (x + y)2 > 0
So, 144 + 4xy > 0
Or 4xy > -144
Hence, xy > -36
x and y are integers and |x - y| = 12
What is the minimum possible value of xy?
Square both sides of |x – y| = 12
· (x – y)2 = 144
x2 + y2 – 2xy = 144
· Add 4xy to both sides of the equation
x2 + y2 – 2xy + 4xy = 144 + 4xy
x2 + y2 + 2xy = 144 + 4xy
(x + y)2 = 144 + 4xy
(x + y)2 will NOT be negative for real x and y.
· i.e., (x + y)2 > 0
So, 144 + 4xy > 0
Or 4xy > -144
Hence, xy > -36
Minimum Possible value of xy = -36
Choice D.
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