Find \(C\)

Algebra Level 4

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

If \(E\) and \(F\) are real roots of the quadratic equation above, where \(A= \sqrt [4] {49+20\sqrt{6}}\) and \(B=8\sqrt{3}+\dfrac{8\sqrt{6}}{\sqrt{3}}+\dfrac{16}{\sqrt{3}} + \cdots\) and that \(\left|E-F\right|=(6\sqrt{6})^k\), 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 \(C\).