If $p,q\in \{1,2,3,4,5\}$  , then number of equations of the form  $p^2{x}^2+q^{2}x+1=0$ having real roots is 

For real roots we get  $q^4-4p^2\ge 0\Rightarrow q^2\ge 2p$
If   $p=1\Rightarrow q=2,3,4,5$ $p=2\Rightarrow q=2,3,4,5$

$$\begin{gathered} p=3\Rightarrow q=3,4,5 \\ p=4\Rightarrow q=3,4,5 \\ p=5\Rightarrow q=4,5 \end{gathered}$$
Total number of different possibilities = 16