Let $p,q,r,s$ be real numbers such that $pr=2(q+s)$. Consider the equations $x^2+px+q=0$ and $x^2+rx+s=0$. Then
The discriminant of $x^2+px+q=0$ is $Δ_1=p^2−4q$
and that of $x^2+rx+s=0$ is $Δ_2=r^2−4s$
and if $pr=2(q+s)$, we have $Δ_1+Δ_2=(p+r)^2$
Since, the sum of two discriminants is positive, so at least one of them has to be positive.
Therefore, at least one the equation $x^2+px+q = 0 $ and $x^2+rx + s = 0 $ has the real roots.