$\text{ If }p,q\in \{1,2,3,4\},\text{ the number of equations of the form }p{x}^2+qx+1=0\text{ having real roots is }$

For real roots, we must have 

$\begin{array}{l}\text{ Disc }\ge 0\Rightarrow {\mathrm{q}}^2-4\mathrm{p}\ge 0\Rightarrow {\mathrm{q}}^2\ge 4\mathrm{p}\\ \text{ If }\mathrm{p}=1\text{, then }{\mathrm{q}}^2\ge 4\mathrm{p}\Rightarrow {\mathrm{q}}^2\ge 4\Rightarrow \mathrm{q}=2,3,4\\ \mathrm{p}=2\text{, then }{\mathrm{q}}^2\ge 4\mathrm{p}\Rightarrow {\mathrm{q}}^2\ge 8\Rightarrow \mathrm{q}=3,4\\ \mathrm{p}=3\text{, then }{\mathrm{q}}^2\ge 4\mathrm{p}\Rightarrow {\mathrm{q}}^2\ge 12\Rightarrow \mathrm{q}=4\\ \mathrm{p}=4\text{, then }{\mathrm{q}}^2\ge 4\mathrm{p}\Rightarrow {\mathrm{q}}^2\ge 16\Rightarrow \mathrm{q}=4\end{array}$

Thus, we see that there are 7 cases.