1.38: Relationships between roots

$f(x)=x^2+px+7$, $p\in\mathbb R$.

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

(a) Find, in terms of $p$ where necessary,

(i) $\alpha^2+\beta^2\qquad$ (ii) $\alpha^2\beta^2$ (4)

Given that $7(\alpha^2+\beta^2)=5\alpha^2\beta^2$,

(b) find the possible values of $p$. (2)

Using the positive value of $p$ found in part (b) and without solving the equation $f(x)=0$,

(c) form a quadratic equation with roots $\displaystyle\frac{2p}{\alpha^2}$ and $\displaystyle\frac{2p}{\beta^2}$. (5)