Problem

The quadratic equation

$3x^2 - 2x + 5 = 0$

has roots $\alpha$ and $\beta$.

Without solving the equation,

(a) write down the value of $(\alpha + \beta)$ and the value of $\alpha\beta$

(b) determine the value of $\alpha^2 + \beta^2$

(c) determine a quadratic equation that has roots

$\alpha + \frac{1}{\alpha}\quad \text{and}\quad \beta + \frac{1}{\beta}$

giving your answer in the form $px^2 + qx + r = 0$ where $p, q$ and $r$ are integers.