Consider the quadratic equation \[\begin{equation*} nx^2+2x\sqrt{pn^2+q}+rn+s=0, \tag{$*$} \label{eq:star} \end{equation*}\]

where \(p>0\), \(p\ne r\) and \(n=1\), \(2\), \(3\), \(\dots\).

  1. For the case where \(p=3\), \(q=50\), \(r=2\), \(s=15\), find the set of values of \(n\) for which equation \(\eqref{eq:star}\) has no real roots.

  2. Prove that if \(p<r\) and \(4q(p-r)>s^2\), then \(\eqref{eq:star}\) has no real roots for any value of \(n\).

  3. If \(n=1\), \(p-r=1\) and \(q=s^2/8\), show that \(\eqref{eq:star}\) has real roots if, and only if, \(s\le 4-2\sqrt{2}\) or \(s\ge 4+2\sqrt{2}\).