Problem

7. The equation $3x^2 + px - 5 = 0$, where $p$ is a constant, has roots $\alpha$ and $\beta$.

(a) Determine the value of

(i) $\alpha\beta$

(ii) $\left(\alpha + \dfrac{1}{\beta}\right)\left(\beta + \dfrac{1}{\alpha}\right)$

(b) Obtain an expression, in terms of $p$, for

(i) $\alpha + \beta$

(ii) $\left(\alpha + \dfrac{1}{\beta}\right) + \left(\beta + \dfrac{1}{\alpha}\right)$

Given that

$\left(\alpha + \dfrac{1}{\beta}\right)^2 + \left(\beta + \dfrac{1}{\alpha}\right)^2 = 2\left(\alpha + \dfrac{1}{\beta}\right)\left(\beta + \dfrac{1}{\alpha}\right)$

(c) determine the value of $p$.

(d) Using the value of $p$ found in part (c), obtain a quadratic equation, with integer coefficients, that has roots $\left(\alpha + \dfrac{1}{\beta}\right)$ and $\left(\beta + \dfrac{1}{\alpha}\right)$