| pq ≠ 0 | |
| Quantity A | Quantity B |
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The issue of quadratics can be boiled down to one principle: know how to FOIL well. Many questions concerning quadratics hinge on your ability to FOIL factored expressions correctly.
Quadratics can appear either in one or both of the quantities or in the common information. Where it is will determine how you approach the question.
If the quadratic expressions appears in the quantities, then your goal is to FOIL and eliminate common terms to make a direct comparison.
| pq ≠ 0 | |
| Quantity A | Quantity B |
| (2p + q)(p + 2q) | p2 + 5pq + q2 |
To make a meaningful comparison between the two quantities, you have no choice but to FOIL Quantity A.
You get:
First = 2p · p = 2p2
Outside = 2p · 2q = 4pq
Inside = q · p = pq
Last = q · 2q = 2q2
The expression on the left equals 2p2 + 5pq + 2q2. Both quantities contain the term 5pq, which you can safely subtract. The comparison becomes:
| pq ≠ 0 | |
| Quantity A | Quantity B |
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|
|
The information at the top tells you that neither p nor q can be 0, and you know that p2 and q2 will both be positive, so you can now definitively say that Quantity A is larger than Quantity B. To answer this question correctly, you had to do two things: 1) FOIL Quantity A (the faster the better), and 2) eliminate common terms from both quantities and compare the remaining terms. The correct answer is (A).
As QC questions involving quadratic expressions get more difficult, they can make either FOILing or simplifying more difficult. Try this example problem:
| r > s | |
| Quantity A | Quantity B |
| (r + s)(r − s) | (s + r)(s − r) |
This problem now requires you to FOIL two expressions, not just one (you can’t simply divide out (r + s) from each side, because (r + s) might be negative). However, this is where knowledge of special products can save you some time. Each of these expressions is a difference of squares:
| r > s | |
| Quantity A | Quantity B |
| (r + s)(r − s) = r2 − s2 | (s + r)(s − r) = s2 − r2 |
Now you need to be able to compare these expressions. You know r is greater than s, so it might be tempting to conclude that Quantity A is greater than Quantity B. Plug in r = 3 and s = 2:
| r > s | |
| Quantity A | Quantity B |
|
|
Quantity A is greater than Quantity B. 
But there’s a problem. You know r is greater than s, but you don’t know the sign of either variable. Remember to check negative possibilities!
Now plug in r = −2 and s = −3:
| r > s | |
| Quantity A | Quantity B |
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Here, you get the opposite conclusion, that Quantity B is greater than Quantity A. Because you can’t arrive at a consistent conclusion, the answer is (D).
Questions that contain quadratic equations in the common information will present different challenges. For example:
| x2 − 6x + 8 = 0 | |
| Quantity A | Quantity B |
| x2 | 2x |
The first thing to note here is that there will be two possible values for x. But you should not jump to conclusions and assume the answer will be (D). To make sure you get the right answer, you need to solve for both values of x and plug them both into the quantities.
First, solve for x by factoring the equation so that it reads (x − 2)(x − 4) = 0.
That means that x = 2 or x = 4. Start by plugging in 2 for x in both quantities:
| Quantity A | Quantity B |
| (2)2 = 4 | 2(2) = 4 |
When x = 2, the quantities are equal. 
Now try x = 4:
| Quantity A | Quantity B |
| (4)2 = 16 | 2(4) = 16 |
Even though there are two possible values for x, both of these values lead to the same conclusion: the quantities are equal. The correct answer is (C).