1.44: Relationships between roots

The equation $2x^2+px-3=0$, where $p$ is a constant, has roots $\alpha$ and $\beta$.

(a) Find the value of

(i) $\alpha\beta$

(ii) $\left(\alpha+\dfrac1\beta\right)\left(\beta+\dfrac1\alpha\right)$ (4)

(b) Find, in terms of $p$,

(i) $\alpha+\beta$

(ii) $\left(\alpha+\dfrac1\beta\right)+\left(\beta+\dfrac1\alpha\right)$ (4)

Given that

(c) find the value of $p$. (1)

(d) Using the value of $p$ found in part (c), find a quadratic equation, with integer coefficients, which has roots $\alpha+\dfrac1\beta$ and $\beta+\dfrac1\alpha$. (2)