Problem

The equation $5x^2 - 4x + 2 = 0$ has roots $\dfrac{1}{p}$ and $\dfrac{1}{q}$

(a) Without solving the equation,

(i) show that $pq = \dfrac{5}{2}$

(ii) determine the value of $p + q$

(b) Hence, without finding the values of $p$ and $q$, determine a quadratic equation with roots

$\frac{p}{p^2 + 12} \quad \text{and} \quad \frac{q^2}{q + 1}$

giving your answer in the form $ax^2 + bx + c = 0$ where $a$, $b$ and $c$ are integers.