Interchanging "if" and "then"
Starting at the bottom, your last sentence won't work.
Here are the standard phrases I've seen:
- There exists a unique solution of 0=a+b x+c x2 if and only if c is zero and b is nonzero, or b2=4 a c.
- c is zero and b is nonzero, or b2=4 a c, if and only if 0=a+b x+c x2 has a unique solution.