Find \(C\)

Algebra Level 4

\(E\) and \(F\) are the roots of the following quadratic equation

\[A(\sqrt{3}-\sqrt{2})x^2+\frac{B}{(\sqrt{3}+\sqrt{2})}x+C=0\]

where \(A= \sqrt [4] {49+20\sqrt{6}}\) and \(B=8\sqrt{3}+\frac{8\sqrt{6}}{\sqrt{3}}+\frac{16}{\sqrt{3}} + \cdots\)

If \(\left|E-F\right|=(6\sqrt{6})^k\), where \(k=\log _6 \left(10\right)-2\log _6\left(\sqrt{5}\right)+\log _6\sqrt{\log _6\left(18\right)+\log _6\left(72\right)}\)

Find the value of \(C\).

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