cma,where have you been? Following is my solution for you,

Too much answers in web,maybe confuse you.

1) f(x)=a(x-5)^2+b

2) f(2)=0, a=-b/9.b not 0

3) f(f(5))=-b/9(f(5)-5)^2+b=-b/9(-b/9(5-5)+b-5)^2+b)=0,...(b-5)^2=9,b=2 or 8.

4)b=2,f(f(x)) has only 5 one real root,b=8, it has 3 real roots: 5 and 5+(or-)3SQUT3/2,so b can't be 8.

5)f(0)=-32/9.