Can we solve $nx^2+2x\sqrt{pn^2+q}+rn+s=0$?

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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 \(s\ge 4+2\sqrt{2}\).