A more complete answer: All of the discussion so far (including my response of almost two years ago) assumes the leading coefficient is 1 which will guarantee the slope of the tangent line in question is positive. However if the cubic has leading coefficient of A (not equal to zero) the real part of the complex solutions remains the first coordinate of the intersection point but the imaginary parts are +/- the square root of m/A where m is the slope of the tangent line.

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