Problem

5. The quadratic equation

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

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

(i) $\alpha^2 + \beta^2$

(ii) $\alpha^3 + \beta^3$

(c) find a quadratic equation which has roots

$(\alpha^3 - \beta) \text{ and } (\beta^3 - \alpha)$

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