From the two equations: \(\alpha + \beta = p\), \(\alpha\beta = r\), and \(\frac{\alpha}{2} + 2\beta = q\). (The product \(\frac{\alpha}{2} \cdot 2\beta = r\) agrees with \(\alpha\beta = r\).)

Put \(\alpha = p - \beta\) into the third: \(\frac{p - \beta}{2} + 2\beta = q \Rightarrow \beta = \frac{2q - p}{3}\).

Then \(\alpha = p - \beta = \frac{4p - 2q}{3} = \frac{2(2p - q)}{3}\).

\(r = \alpha\beta = \frac{2}{9}(2p - q)(2q - p)\)

Answer: (D)