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Find the harmonic mean

  1. Nov 23, 2012 #1


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    1. The problem statement, all variables and given/known data
    The harmonic mean of the roots of the equation [itex](5+\sqrt{2})x^2-(4+\sqrt{5})x+8+2\sqrt{5}=0[/itex]

    2. Relevant equations

    3. The attempt at a solution

    I know this question is easy but the main problem arises in finding the roots of the above equation. When I use the quadratic formula I get some complicated terms which is not easy to solve. What should I do?
  2. jcsd
  3. Nov 23, 2012 #2


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    No, "finding the roots of the equation" is not the hard part because you don't need to find the roots! The first thing I would do is divide the entire equation by [itex]5+\sqrt{2}[/itex] to make the leading coefficient 1. Such a quadratic equation can be written as [itex](x- a)(x- b)= x^2- (a+b)x+ ab= 0[/itex] where a and b are the roots. You can read both a+ b and ab directly from the equation and use them to find the harmonic mean.
  4. Nov 26, 2012 #3
    Substitute x = 1/y. Then the roots of the quadratic equation for y are the reciprocals of the roots of the equation for x. In the quadratic equation for y, -b/a is the sum of the roots for y, and is also equal to the sum of the reciprocals of the roots for x.
  5. Nov 28, 2012 #4


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