Review question

# Can we solve $nx^2+2x\sqrt{pn^2+q}+rn+s=0$? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R6246

## Question

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 $s\ge 4+2\sqrt{2}$.