Show that if `p,q,r` and `s` are real numbers and `pr=2(q+s)`, then atleast one of the equations `x^(2)+px+q=0` and `x^(2)+rx+s=0` has real roots.

Explanation

Let `D_(1)` and `D_(2)` be the discriminants of the give equations `x^(2)+px+q=0` and `x^(2)+rx+s=0`, respectively.
Now `D_(1)+D_(2)=p^(2)-4q+r^(2)-4s=p^(2)+r^(2)-4(q+s)`
`=p^(2)+r^(2)-2pr` [given `pr=2(q+s)`]
`=(p-r)^(2)ge0` [ `:'p` and `q` are real]
or `D_(1)+D_(2)ge0`
Hence, atleast one of the equations `x^(2)+px+q=0` and `x^(2)+rx+s=0` has real roots.