Quadratic Equations

Every now and then when I am surfing I see these mathematical “challenges” passing by that are supposedly part of entry exams at one or the other famous university. The most recent one I saw is the one below

As usual the claim was that only a few people could solve this. Most examples ask to factor a polynomial that requires some tricks, substitutions and lengthy calculations. Because of the length alone I can imagine most people don’t have the stamina to see the calculation through even if they know the tricks and substitutions to apply. But for this particular case I don’t see the challenge, unless the claim is that very few people know about complex numbers and how to take the square root of a negative number. I attacked this problem by multiplying the top equation with x (or y, you get the same equation out) and then substituting xy by 36. You end up with a standard quadratic equation.

The “problem” is that the discriminant is negative, so there are no real solutions but if you allow complex solutions this quite standard fare if you remember high school algebra.

If you apply this formula you will get a negative discriminant and a complex solution:

Not too difficult. It was serendipity that I came across this problem just now because it made me remember quadratic equations and 2 weeks later my son came to ask me to help him with preparing for his math test dealing with quadratic equations.