SOLUTION: Let ax^2 + bx + c = 0 be a quadratic equation with no real roots and a,b > 0. Prove that -b/(2a) > H, where H is the harmonic mean of the roots of this quadratic equation.

Let p and q be the roots of the given equation,


The harmonic mean  of the roots is


    H =  = .


According to Vieta's theorem,  pq = ,  p+q = ,  so


    H =  =  = .


Thus the inequality we need to prove takes the form


     > .


It is equivalent to


     < 


which with positive "a" and "b" is equivalent to


    b^2 < 4ac,   or   b^2 - 4ac < 0.


The last inequality is equivalent to the condition that the given quadratic equation has no real roots.