The GMAT will NEVER require that you know Calculus/Derivatives to get to the correct answer to a Quant question.
IF the original poster had included the 5 answer choices, then we'd have those to use for reference (and we'd likely need to do LESS work to get to the solution). As it is, we can find the correct answer with a little bit of "brute force" and some Number Properties.
We're given X^2 - 4X - 5 and asked for the MINIMUM value that can be generated from this equation.
Here are the Number Properties worth noting:
1) X^2 will either be positive or zero.
2) If X is NEGATIVE, then -4X becomes positive (which INCREASES the value of the calculation and that is NOT what we want)
3) Since we're dealing with X^2, the graph of this equation will "curve" somehow (it won't be a straight line)
To minimize the outcome, X WILL NOT be negative....so let's look for a pattern in 0 and the positive possibilities:
If X=0, the result is 0 - 0 - 5 = -5
If X=1, the result is 1 - 4 - 5 = -8
If X=2 the result is 4 - 8 - 5 = -9
If X=3 the result is 9 - 12 - 5 = -8
If X=4 the result is 16 - 16 - 5 = -5
From this, the answer appears to be -9. The answer choices, if we had them, would help to confirm this (as we could eliminate answers that were "too big" or not possible).
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