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# What is the harmonic mean of the two roots of the following equation $(5+\sqrt2)x^2-(4+\sqrt5)x+8+2\sqrt5=0$

$\begin{array}{1 1} 2 \\ 4 \\ 6 \\ 8 \end{array}$

$Explanation: Let\;the\;two\;roots\;of\;the\;quadratic\;equation\;be\;\alpha\;\xi\;\beta$
$\alpha+\beta=\large\frac{4+\sqrt5}{5+\sqrt2}$
$\alpha\;\beta=\large\frac{8+2\sqrt5}{5+\sqrt2}$
$HM=\large\frac{2\alpha\beta}{\alpha+\beta}$
$=\large\frac{2(8+2\sqrt5)}{4+\sqrt5}$
$=\large\frac{4(4+\sqrt5)}{4+\sqrt5}=4.$
edited Jan 24, 2014