Just a quick test tip on this one, folks. A problem like this demonstrates the importance of being able to quickly "Reverse FOIL" a quadratic equation. This is a great technique for anybody that wants to solve this without introducing new variables. The technique, in brief, is as follows.
Let's take the equation X^2 + 5X + 6 = 0 as an example.
1) Set up your factored form as ( X + __ ) ( X + __ ) = 0
2) Then simply fill in the blanks with constants that ADD UP to 5 (the X coefficient) and have a PRODUCT of 6 (the constant). In this simple example, they will be 2 and 3.
3) The roots are the OPPOSITE of each of those constants (because the equation will be true when each linear factor equals 0). So the solutions to this example are -2 and -3.
This takes bit of trial and error, but most quadratics that appear on the GMAT can be "Reverse FOILed" pretty easily. They just seem to enjoy testing us on this skill.
Note that these values CAN BE and often are negative.
(You'll notice, also, that this would become much harder if there is a coefficient greater than one on the squared term. It can be done, but is a bit trickier, so I will not go down that road here.)
Now ... for this question, you know that your factored form is going to be ( X - 3 ) ( X + __ )
Determine what your other constant will be, based on each answer choice. To do this, divide the constant by -3, and then see if that number minus 3 gives you the X coeffiecient.
A) -5 works
B) -6 works
C) non-integer, therefore can't reverse foil easily.
D) 4 works
E) 18/-3 gives you 6, but you end up with the wrong X coeffiecient. -3 must not be a solution.
Since the second root is the OPPOSITE of the constant we've come up with, and must be a negative even integer, the only answer choice that works out for us is D.
Hope that's useful!
SP
Stephen
GMAT Instructor
Knewton Inc.
Featured Course
