GMAT Quantitative Concept: Disguised Quadratics

If you’ve been studying for the GMAT, you’ve probably encountered situations that require knowledge of quadratic equations and how they work. In so doing, you’ve also probably memorized the following three equivalencies:

(x + y)² = x² + 2xy + y²

(x – y)² = x² – 2xy + y²

(x – y)(x + y) = x² – y²

Knowledge of these three equations will, without a doubt, come in handy on several questions on the GMAT. If, for example, you’re presented with the expression (x + 2)² , you can quickly bypass foiling and instead immediately arrive at the expression: x² + 4xy + 4.

But if you’re shooting for a 700+, the real skill you’ll need to develop is the ability to identify tricky expressions that conform to these rules. To take a GMAT Prep question as a sample:

What is the value of

1001² – 999^2
_________
101² – 99²

If you’re like most test-takers, you’ll recognize that actually doing out these calculations is NOT the appropriate method for solving this expression. The issue, though, is how to simplify this expression so that you can minimize the rote arithmetic you do. This is where quadratics come in. In both the numerator and denominator, we see that we’re subtracting two squares. In the numerator, we’re subtracting the square of 999 from the square of 1001. In the denominator, we’re subtracting the square of 99 from the square of 101. What do you have when you’re subtracting two squares? A difference of squares! In both the numerator and denominator, we have expressions that are in the form of x² – y². To see this, think of the 1001 in the numerator as your x and the 999 as your y. So, instead of 1001² – 999², we have x² – y². From our rules, we know that x² – y² always factors out to: (x + y)(x – y). So, we can re-write the numerator as (1001 + 999)(1001 – 999). Applying the same logic to the denominator, we get: (101 + 99)(101 – 99).

Now, we have a much easier fraction to deal with:

(1001 + 999)(1001 – 999)
(101 + 99)(101 – 99)

At this point, it’s a matter of simple arithmetic to arrive at the fraction:

(2000)(2)
(200)(2)

Reduce and you arrive at the answer: 10.

The key take-away from all of this is that whenever you encounter a quadratic expression that seems unwieldy or difficult to manipulate, you should always consider whether the expression conforms to one of the three major quadratic equations.