BrentGMATPrepNow wrote:
What is the minimum possible value of 2x² - 2xy + y² - 2x + 3?
A) 0
B) 1
C) 2
D) 3
E) 4
Strategy: When I examine the given expression, 2x² - 2xy + y² - 2x + 3, I see that the expression "2x² - 2xy + y²" closely resembles the special product x² - 2xy + y², which conveniently factors to become (x - y)².
To see whether this special product is part of our solution, I'm going to explore whether there is any benefit to rewriting 2x² as x² + x². Given:
2x² - 2xy + y² - 2x + 3Rewrite as follows:
x² + x² - 2xy + y² - 2x + 3Rearrange to get:
(x² - 2xy + y²) + x² - 2x + 3Factor the bracketed expression to get:
(x - y)² + x² - 2x + 3Strategy: Now that I've expressed part of the expression as (x - y)², I recognize that x² - 2x + 3 looks a lot like x² - 2x + 1, which can be factored to get (x - 1)²So, let's rewrite the expression as follows:
(x - y)² + (x² - 2x + 1) + 2Factor the bracketed expression to get:
(x - y)² + (x - 1)² + 2Important property: something² is always greater than or equal to 0. So, the smallest possible value of (x - y)² is zero, and the smallest possible value of (x - 1)² is also 0.
In fact, when x = 1 and y = 1, (x - y)² = 0, and (x - 1)² = 0
Of course, the expression we want to minimize is
(x - y)² + (x - 1)² + 2So, when x = 1 and y = 1, we get:
(x - y)² + (x - 1)² + 2 = (1 - 1)² + (1 - 1)² + 2 = 0² + 0² + 2 = 2This means 2 is the smallest possible value of the given expression (and this minimum value occurs when x = y = 1)
Answer: C