Strange Square Root Problem

Ok, I need some kind of algebraic method to:

- Determine which values of r give a real answer for t, and which values of r give a complex answer for t

http://i190.photobucket.com/albums/z...athscthing.jpg

Now i have been solving this easy enough using the quadratic formula

(let x= etc

then x^2=r+http://i190.photobucket.com/albums/z...athscthing.jpg(without the t=)

x^2-x=r

then x^2-x-r=0, solve using quadratic formula).

Now my teacher said that my method of trialling different values works, but isn't quite what he is after, he wants some kind of algebraic method to show which values produce a complex or real answer, like just shove in a value and it spits out an answer.

Thanks any help is appreciated

By the way - This is rather urgent.:)