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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.
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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.
Proved and solved.