Let's talk strategy here. Many explanations of Quantitative questions focus blindly on the math, but remember: the GMAT is a critical-thinking test. For those of you studying for the GMAT, you will want to internalize strategies that actually minimize the amount of math that needs to be done, making it easier to manage your time. The tactics I will show you here will be useful for numerous questions, not just this one. My solution is going to walk through not just what the answer is, but how to strategically think about it. Ready? Here is the full “GMAT Jujitsu” for this question:
First, this problem requires you to understand positive/negative rules. When adding two numbers together,
(+) + (+) = (+)(-) + (-) = (-)(+) + (+) = ???When multiplying two numbers together,
(+) * (+) = (+)(+) * (-) = (-)(-) * (-) = (+)Next, let’s talk terminology. When a problem refers to the “
roots” of a quadratic equation containing
\(x\), it is talking about the possible
numerical solutions for
\(x\). In this context, the term “root” is completely synonymous with the term “solution.” The solutions of quadratics can often be calculated by breaking down the quadratic into its linear components such that
\(x^2+bx+c=0\)becomes
\((x+\underline{\hspace{2em}})(x+ \underline{\hspace{2em}})=0\)You fill in the blanks here by looking at the pairs of factors of
\(c\) (called “complementary factors”) that when multiplied together equals
\(c\), but when added together they equal
\(b\). Of course, if you need to actually solve for the values, it is important to realize that the solutions (or roots) of the equation are going to be the negatives of these factors. (Think about it, if
\((x+4)=0\), then
\(x\) must be
\(-4\).)
Now that we have the basic rules, down, let’s focus on tactics. This is a “
Yes/No” question – a very common structure for Data Sufficiency problems. The fundamental trap for problems like these is to bait you into thinking that you actually need to solve for every value. You don’t. As soon as you have enough information to conclude that a statement is either sufficient or insufficient, you can move on. For “
Yes/No” questions, if you can think of two situations (or two variable inputs) that are consistent with all of the problem’s constraints but come up with different answers to the question, you know a statement is insufficient. In my classes, I call this strategy “
Play Both Sides.” The problem asks us, “
is \(rs < 0\)?” – in other words, “
when you multiply the two solutions to a quadratic equation, do you get a negative number?” We can start to anticipate how we are going to “
Play Both Sides.” Using the positive/negative rules we showed above, the answer to this question is “Yes” if
one of the solutions is positive and the other is negative. The answer is “No” if either
both solutions are positive or
both solutions are negative.
Statement #1 tells us that
\(b\) is negative. There are a couple of ways this could happen: (1) the complementary factors of
\(c\) could be
both negative, in which case
\(r\) and
\(s\) are both positive, or (2) one complementary factor of
\(c\) is positive and the other is negative (so long as the negative factor is larger in magnitude than the positive factor.) This would mean that we would have one negative solution and one positive one. Since the first possibility gives us a “No” answer to the question while the second option gives us a “Yes” answer, Statement #1 is not sufficient.
Statement #2 tells us that
\(c\) is negative. There is only one possible way this could happen: one complementary factor of
\(c\) is positive and the other is negative. You can only get a negative product if one factor is positive and one is negative. The roots would mirror this. So, Statement #2 can only give us a “Yes” answer to the
Yes/No question. Since we have only one answer, it is sufficient.
The answer is “
B”.
Now, let’s look back at this problem through the lens of strategy. This question can teach us patterns seen throughout the GMAT. Notice that this problem is much more about
logic and
critical-thinking than it is about math. With “
Yes/No” questions involving positive/negative rules, a great tactic that you can often use is to focus conceptually on the
+/- possibilities, using very basic rules. If multiple answers are possible using those rules, you can “
Play Both Sides” and disprove sufficiency. On the other hand, if there is only one answer to the Yes/No question, then you proved sufficiency. If you can analyze the problems conceptually, the actual math involved is often minimal, and you don’t even need to solve for specific values. And that is how you think like the GMAT.
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Aaron PondVeritas Prep Teacher of the Year / Professionally mentoring GMAT students since 2006 Visit me at https://www.veritasprep.com/gmat/aaron-pond/ if you would like to learn even more "GMAT Jujitsu"!