Algebra.Com's Answer #814445 by ikleyn(52781)![\"\"](\"/images/stars/stars-13.gif\")  
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document.write( "Let ax^2 + bx + c = 0 be a quadratic equation with no real roots and a,b > 0.
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document.write( "Prove that -b/(2a) > H, where H is the harmonic mean of the roots of this quadratic equation.
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document.write( "Let p and q be the roots of the given equation,\r\n" );
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document.write( "The harmonic mean of the roots is\r\n" );
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document.write( " H =  =  .\r\n" );
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document.write( "According to Vieta's theorem, pq =  , p+q =  , so\r\n" );
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document.write( " H = =  =  .\r\n" );
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document.write( "Thus the inequality we need to prove takes the form\r\n" );
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document.write( "  > .\r\n" );
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document.write( "It is equivalent to\r\n" );
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document.write( "  <  \r\n" );
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document.write( "which with positive \"a\" and \"b\" is equivalent to\r\n" );
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document.write( " b^2 < 4ac, or b^2 - 4ac < 0.\r\n" );
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document.write( "The last inequality is equivalent to the condition that the given quadratic equation has no real roots.\r\n" );
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document.write( "Proved and solved.\r
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