I Have Seen Sine, Cosine, What About Tangent?

Geometry Level 5

Suppose \(a, b, c\) are sides of a triangle \(\Delta ABC\) and the roots of the following equation. \[x^3-24x^2+180x-420=0.\] If \(\tan A+\tan B+\tan C=-\dfrac{p}{q}\), where \(p\) and \(q\) are coprime positive integers, find the value of \(p+q\).


Inspiration.

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