If P(x)=ax2+bx+c and Q(x)=-ax2+dx+c where ac≠0, then P(x).Q(x)=0 has at least

Let all four roots are imaginary. Then roots of both equations P(x)=0 and Q(x)=0 are imaginary.Thus b2-4ac<0; d2+4ac<0, so b2+d2<0, which is impossible unless b=0, d=0.So, if b≠0 or d≠0 at least two roots must be real.If b=0, d=0, we have the equations.P(x)=ax2+c=0 and Q(x)=-ax2+c=0or x2=-ca; x2=ca as one of ca and -ca must be positive, so two roots must be real.