Bunuel wrote:
If \(a#b = a^2 - ab\), for values of a and b, which of the following expressions cannot be negative?
I. |x|#|y|
II. |x-y|#-2
III. (-x- 2)#0
A. I only
B. II only
C. I and II only
D. II and III only
E. I, II, and III
Quick method:
First test (I):
Take a to be 2 and b to be 3.
\(2^2\) - \(2^3\) results in -4. This rules out A, C, and E answer choices.
Looking at B and D, you see that both include (II) so you do not have to test this case. Move on to testing case (III).
\((-x-2)^2\) - \((-x-2)^0\)
Factorize the right side and the left side = 1.
results in \(x^2\) + 4x + 3.
Resulting in (x+1)(x+3). Which means x = -1, -3. Plugging both back into the original equation, it is proven that both will be positive. Thus, both (II) and (III) will work and the correct answer is D.