What would the cubic formula be if roots never existed?

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$$ax^{2}+bx+c=0$$ $$x=-\frac{b}{a}+\frac{1}{\frac{b}{c}+\frac{1}{-\frac{b}{a}+\frac{1}{\frac{b}{c}+\frac{1}{...}}}}$$ $$ax^{3}+bx^{2}+cx+d=0$$ $$x=?$$ I know it can't recur like the quadratic continued fraction. Must be arithmetic operations. Got ideas?