Let $$p,q \in \left\{ {1,2,3,4} \right\}.$$   The number of equations of the form $$p{x^2} + qx + 1 = 0$$    having real roots is

Solution :
For the equation $$p{x^2} + qx + 1 = 0$$    to have real roots
$$\eqalign{ & D \geqslant 0\,\,\,\,\,\,\,\,\,\, \Rightarrow \,\,{q^2} \geqslant 4p \cr & {\text{If }}p = 1\,\,{\text{then }}{q^2} \geqslant 4 \cr & \Rightarrow \,\,q = 2,3,4 \cr & {\text{If }}p = 2\,\,{\text{then }}{q^2} \geqslant 8 \cr & \Rightarrow \,\,q = 3,4 \cr & {\text{If}}\,{\text{ }}p = 3\,\,{\text{then }}{q^2} \geqslant 12 \cr & \Rightarrow \,\,q = 4 \cr & {\text{If }}p = 4\,\,{\text{then }}{q^2} \geqslant 16 \cr & \Rightarrow \,\,q = 4 \cr & \therefore \,\,{\text{No}}{\text{. of req}}{\text{. equations}} = 7. \cr} $$