Let all four roots are imaginary. Then roots of both equation \( \Large P \left(x\right)=0 \)

\( \Large Q \left(x\right)=0 \) are imaginary thus \( \Large b^{2}-4ac<0;\ d^{2}-4ac<0,\ so\ b^{2}+d^{2}<0 \) which is impossible unless b = 0, d = 0.

So if b ≠ 0 or d ≠ O at least two roots must be real, if b = 0, d = 0, are have the equations

\( \Large P \left(x\right)=ax^{2}+c=0 \)

and \( \Large Q \left(x\right)=-ax^{2}+c=0 \)

or \( \Large x^{2}=-\frac{c}{a};\ x^{2}=\frac{c}{a} as\ one\ of\ -\frac{c}{a} and\ \frac{c}{a} \)

must be positive so two roots must be real.