Bunuel wrote:
What is the minimum possible value of \(x^2 – 6x + 14\)?
A. 0
B. 1
C. 5
D. 6
E. 14
The standard way to solve questions like this (without formulas you'd never need on the GMAT) is by "completing the square". If we first look only at this part of the quadratic:
x^2 - 6x
we can make this into a perfect square by adding 9:
x^2 - 6x + 9 = (x - 3)^2
We see how to do that just by dividing the "-6" by 2. So using that in the original question:
x^2 - 6x + 14 = x^2 - 6x + 9 + 5 = (x - 3)^2 + 5
and since (x-3)^2 is a square, its minimum value is 0 (when x = 3), so the minimum value of (x - 3)^2 + 5 is equal to 5.
I can't recall a single official question where I've needed to use this technique, though.
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