Show that if p,q,r,sp,q,r,s are real number and pr=2(q+s)pr=2(q+s) then at-least one of the equations x^2+px+q=0x2+px+q=0 and x^2+rx+s=0x2+rx+s=0 has real roots?

1 Answer
Jan 8, 2018

Please see below.

Explanation:

The discriminant of x^2+px+q=0x2+px+q=0 is Delta_1=p^2-4q

and that of x^2+rx+s=0 is Delta_2=r^2-4s

and Delta_1+Delta_2=p^2-4q+r^2-4s

= p^2+r^2-4(q+s)

= (p+r)^2-2pr-4(q+s)

= (p+r)^2-2[pr-2(q+s)]

and if pr=2(q+s), we have Delta_1+Delta_2=(p+r)^2

As sum of the two discriminants is positive,

at least one of them would be positive

and hence atleast one of the equations x^2+px+q=0 and x^2+rx+s=0 has real roots.