# A strange equation III

Algebra Level 5

The three real roots of the polynomial $$P(x)=x^{3}+3x^{2}-51x+54\sqrt{3}-107$$ can be expressed as:

$x_{1}=-a\sqrt{b}\cos(\alpha)-c , x_{2}=a\sqrt{b}\cos(\beta)-c , x_{3}=a\sqrt{b}\cos(\theta)-c,$

where $$x_{1}<x_{2}<x_{3}$$, $$0°<\alpha<90°$$, $$0°<\beta<90°$$, $$0°<\theta<90°$$, such that $$a$$, $$b$$ and $$c$$ are positive integers; and $$\alpha$$, $$\beta$$ and $$\theta$$ are integer degress. Find $$\dfrac{\alpha+\beta+\theta-1}{a+b+c}$$.

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