Today’s SAT Question of the Day is a great algebra question that can be solved in two different ways.

If x + y = 3 and x – y = 5, then x² – y² = ??

First, let’s solve this medium-difficulty problem the quickest way: by factoring.

x² – y² is a “difference of the squares” situation, so we know it factors to

(x + y)(x – y)

Conveniently, we are given the values of those expressions in the setup of the problem!

(3)(5) = 15

Done!

Obviously, the testmaker is hoping that you will catch that, apply this trick, and move on.  If you don’t catch it, you can still solve the problem by determining the values of x and y…so we’re on to way #2 of solving this problem.

First, we have to notice something strange: when we subtract y from x, x gets bigger than when we add y to x.  That tells us that y is negative.

When we add y, x becomes 3.  When we subtract it, x becomes 5.  The difference between 5 and 3 is 2, so y has the ability to move the number line by one unit…in other words, it’s equal to negative 1.  x – (-1) = 5, so x = 4.

4² – (-1)² = 16 – 1 = 15

This way is not difficult (it shouldn’t be – the problem is only rated “medium”), but it takes a little more time than the first way.