1.49: Values of a parameter for real roots

The roots of the equation $f(x)=0$ are $\alpha$ and $\beta$.

(a) Without solving the equation $f(x)=0$, form an equation, with integer coefficients, which has

(i) roots $\dfrac\alpha\beta$ and $\dfrac\beta\alpha$ (6)

(ii) roots $2\alpha+\beta$ and $\alpha+2\beta$ (5)

(b) Express $f(x)$ in the form $A(x+B)^2+C$, stating the values of the constants $A,B$ and $C$. (3)

(c) Hence, or otherwise, show that the equation $f(x)=-8$ has no real roots. (2)