Think algebra would be easy?

Geometry Level 5

\(\boxed{\large\text{Challenge}~\sim~\normalsize\text{solve using pure Geometry only!}}\)

In the given \(\bigtriangleup ABC\), \(AB = 4\), \(AC = 7\) and \(\widehat{A} = 60^{\circ}\), and if \(x_1,~x_2\) and \(x_3\) are roots of the equation

\(x^3 - (4R + r)x^2 + s^2x - s^2r = 0\)

where \(R\) is the circum-radius, \(r\) is the in-radius and \(s\) is the semi perimeter of \(\bigtriangleup\!\!\mathrm{ABC}\). Find the value of \(x_1^3 + x_2^3 + x_3^3\) rounded to nearest thousandths.


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