Find root of implicit function in Mathematica

Question

Find root of implicit function in Mathematica

I have an implicit function, for example:

f(x,y) = x^3 + x*y + y^2 - 36

I want to find the

Answer 1

A $.02 contribution on Verbeia's answer:

Remember to check both x(y) and y(x) solutions as one of them could be cleaner than the other.

In your example: