The continued-fraction-method allows to calculate a linear approximation of a real number with a table calculator.
But I do not know an analogue method for a quadratic approximation.
Given a real number $r$ with $0<r<1$ , how can I find integers $a,b,c$ , lets say with $|a|,|b|,|c|\le 100$ , such that $r$ is near a root of $ax^2+bx+c$ ?
I only heard about the LLL-Method (if I remember the name right), but I did not find a concrete algorithm. It seems that this method is a kind of optimzation algorithm, perhaps simplex, which is probably too complicated for a calculation with a table calculator.
Another possibility would be to transform the continued fraction to a periodic continued fraction near the given one, but usually this will lead to a quadratic equation with coeffcients with large absolute value. Moreover, it is not easy to derive the quadratic equation from the periodic continued fraction.
A method which can be used to find approximative linear dependency's of real numbers, like $e$ and $\pi$, as well, would be best.