We want to apply Vieta's formulas, but the given equation is not a polynomial equation, because of the $\frac1x$ term. To turn this equation into an equivalent polynomial equation, we multiply both sides by $x$ and rearrange: \[\begin{aligned} 1+5x^3 &= 6x^2 - 24x \\ 5x^3 - 6x^2 + 24x + 1 &= 0 .\end{aligned}\]Now we can use Vieta: the sum of the roots is $a+b+c=\boxed{\frac65}.$
Question:
Compute $a+b+c,$ given that $a,$ $b,$ and $c$ are the roots of \[\frac{1}{x} + 5x^2 = 6x - 24.\]
Answer:
Let's think step by step.

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\frac65
Compute $a+b+c,$ given that $a,$ $b,$ and $c$ are the roots of \[\frac{1}{x} + 5x^2 = 6x - 24.\]
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