Yet another question 5

Algebra Level 5

Let a quadratic equation be represented as

\[A(\sqrt{3} - \sqrt{2})x^2 + \dfrac {Bx}{(\sqrt{3} + \sqrt{2})} + C=0, \]

where \(A = (49 + 20\sqrt 6)^{1/4}\) and \(B= 8\sqrt 3 + \dfrac{8\sqrt6}{\sqrt 3} + \dfrac{16}{\sqrt3} + ....\).

Let \(\alpha\) and \(\beta\) be the roots of the above equation which are related to the constraint \[( \displaystyle | \alpha - \beta | = (6\sqrt6)^k, \]

where \(\displaystyle k = \log_{6}10 - 2\log_6 \sqrt5 + \log_6\sqrt{\log_{6} 18 + \log_{6} 72}\).

Find the value of \(C\).

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