$$\begin{gathered} If the roots of the equation x^2-5x+16=0 are \alpha , \beta and \\ the roots of the equation x^2+px+q=0 are {\alpha }^2+{\beta }^2, \frac{\alpha \beta }{2} \end{gathered}$$

$$\begin{gathered} \alpha +\beta =5,\alpha \beta =16 \\ {\alpha }^2+{\beta }^2+\frac{\alpha \beta }{2}=-p \\ (\alpha +\beta {)}^2-2\alpha \beta +\frac{\alpha \beta }{2}=-p \\ 25-2(16)+8=-p \\ 33-32=-p \\ p=-1 \\ ({\alpha }^2+{\beta }^2)\left(\frac{\alpha \beta }{2}\right)=q \\ [(\alpha +\beta {)}^2-2\alpha \beta ]\left(\frac{\alpha \beta }{2}\right)=q \\ (25-32)\left(8\right)=q \\ p=-1, q=-56 \end{gathered}$$