Inspired by Daniel Liu

Geometry Level 5

f(x)=8x312x2+bx+d\Large f\left( x \right) ={ 8x }^{ 3 }-{ 12x }^{ 2 }+bx+d

For variables bb and dd independent of xx, we are given the function f(x)f(x) as described above to have roots 1α,β,γ1 -1\le\alpha ,\beta ,\gamma \le 1 and that

α+β+γ=1+4sincos1(α)2×sincos1(β)2×sincos1(γ)2 \alpha +\beta +\gamma =1+4\sin { \frac { \cos ^{ -1 }{ (\alpha ) } }{ 2 } } \times \sin { \frac { \cos ^{ -1 }{ (\beta) } }{ 2 } } \times \sin { \frac { \cos ^{ -1 }{ (\gamma ) } }{ 2 } }

Evaluate b+db+d.

This problem is original.
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