Testing values is overrated for solving QC questions

Many students rely too much on testing values when answering QC questions.  While this approach has some merits, I don’t think it deserves the attention it gets. To help understand why I’m not a fan, consider the following question:

QUANTITY A: 2x² + 2x - 2

QUANTITY B: x² - 4x - 11

We’ll start by testing some quick and easy numbers.

Try x = 0 to get:

QUANTITY A: 2(0)² + 2(0) - 2 = -2

QUANTITY B: (0)² - 4(0) - 11 = -11

In this case, Quantity A is greater.

Aside: I often see students test ONE value before concluding that the correct answer is A. This is never enough. After testing x = 0, all we know for certain is that the correct answer is either A or D. This is the nice feature of testing values: after testing just 1 value, we’re immediately down to a 50-50 situation.

That said, we still need to test more values if we want to know the correct answer.

So, let's try x = 1 to get:

QUANTITY A: 2(1)² + 2(1) - 2 = 2

QUANTITY B: (1)² - 4(1) - 11 = -14

Once again, Quantity A is greater.

Try x = -2 to get:

QUANTITY A: 2(-2)² + 2(-2) - 2 = 2

QUANTITY B: (-2)² - 4(-2) - 11 = 1

Once again, Quantity A is greater.

Hmm, maybe the correct answer is, indeed, A. Should we keep going? Sure, let’s test two more values.

Try x = 10 to get:

QUANTITY A: 2(10)² + 2(10) - 2 = 218

QUANTITY B: (10)² - 4(10) - 11 = 49

Once again, Quantity A is greater.

Try x = -10 to get:

QUANTITY A: 2(-10)² + 2(-10) - 2 = 178

QUANTITY B: (-10)² - 4(-10) - 11 = 129

Once again, Quantity A is greater.

Now what?

At this point, we’ve already spent significant time testing values. The correct answer SEEMS to be A, but there's no way to be 100% sure, since we haven't tested every possible value of x.

Herein lies the problem with testing values: we can never be absolutely certain of the correct answer, unless we get two conflicting outcomes (i.e., in one case, Quantity A is greater, and in another case, Quantity B is greater), and this is precisely why I think it’s an overrated strategy. 

The truth of the matter is that there is one x-value for which Quantity A is not greater than Quantity B.

If we test x = -3, we get:

QUANTITY A: 2(-3)² + 2(-3) - 2 = 10

QUANTITY B: (-3)² - 4(-3) - 11 = 10

In this case, the two quantities are equal, which means the correct answer is D.

So, even when the correct answer is D, the strategy of testing values can still be ineffective.

This isn't to say that testing values is never a good idea. I'm just saying that, many students don't recognize the limitations of this strategy.

In my opinion, testing values is best used when:

  • you have a feeling the correct answer is D
  • you can't think of another way to solve the question

In my next article, I’ll discuss a super useful (yet underrated) strategy for solving QC questions.

 

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