Bunuel wrote:

Given that (P + 2Q) is a positive number, what is the value of (P + 2Q)?

(1) Q = 2

(2) P^2 + 4PQ + 4Q^2 = 28

Kudos for a correct solution.

MAGOOSH OFFICIAL SOLUTION:The prompt tells us that (P + 2Q) is a positive number, and we want to know the value of P. Remember

number properties! We don’t know that (P + 2Q) is a positive integer, just a positive number of some kind.

Statement #1: Q = 2

Obvious, by itself, this tells us zilch about P. Alone and by itself, this statement is completely insufficient.

Statement #2: P^2 + 4PQ + 4Q^2 = 28

Now, this may be a pattern-recognition stretch for some folks, but this is simply the “Square of a Sum” pattern. It may be clearer if we re-write it like this:

P^2 + 2*P*(2Q) + (2Q)^2 = 28

This is now the “Square of a Sum” pattern, with P in the role of A and 2Q in the role of B. Of course, this should equal the square of the sum:

P^2 + 2*P*(2Q) + (2Q)^2 = (P + 2Q)^2 = 28

All we have to do is take a square root. Normally, we would have to consider both the positive and the negative square root, but since the prompt guarantees that (P + 2Q) is a positive number, we need only consider the positive root:

(P + 2Q) = sqrt{28}

This statement allows us to determine the unique value of (P + 2Q), so this statement, alone and by itself, is sufficient.

Answer = B

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