Gmatasap wrote:Which of the following numbers CANNOT have a negative value?
A. |a+b|-|a-b|
B. |a+b|-|a|
C. |2a+b|-|a+b|
D. a^2+b^2-2|ab|
E. a^3+b^3-a-b
Always look for the following common quadratic identities:
a² + b² + 2ab = (a+b)².
a² + b² - 2ab = (a-b)².
a² - b² = (a+b)(a-b).
Answer choice
D resembles the identity in red.
If we add absolute values to (a-b)², we get:
(|a| - |b|)² = |a|² + |b|² - 2|a||b| = a² + b² - 2|ab|.
Thus:
a² + b² - 2|ab| = (|a| - |b|)².
Since (|a| - |b|)² cannot be negative, answer choice
D cannot be negative.
The correct answer is
D.
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