guess is not math method.
suppose f(x)=A(x-5)^2+b, is just a little smart. But would not give the solution easily when the given condition was f(8)=0, in place of f(2)=0. Worse, one might guess that f(5)=8, which would give the wrong anwser f(0)=-128/9, as you did.
The key to the original or my revised problem is that f(5) is unique, because ff(x) has unique root. and then you can choose the value of f(5) between the two roots of f(x).
when you realise this point, no smart supposition, nor reasonless guess is needed, let alone the stupid calcul.
