Problem

3. The quadratic equation

$2x^2 - 5x + 7 = 0$

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, giving each answer as a simplified fraction, the value of

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

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

(c) find a quadratic equation that has roots

$\dfrac{1}{\alpha + \beta^2} \text{ and } \dfrac{1}{\beta + \alpha^2}$

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