Problem

The quadratic equation

$4x^2 - 3x + k = 0$

where $k$ is an integer, has roots $\alpha$ and $\beta$

(a) Write down, in terms of $k$ where appropriate, the value of $\alpha + \beta$ and the value of $\alpha\beta$

(b) Determine, in simplest form in terms of $k$, the value of $\dfrac{\alpha}{\beta^2} + \dfrac{\beta}{\alpha^2}$

(c) Determine a quadratic equation which has roots

$\frac{\alpha}{\beta^2} \quad \text{and} \quad \frac{\beta}{\alpha^2}$

giving your answer in the form $px^2 + qx + r = 0$ where $p$, $q$ and $r$ are integer values in terms of $k$