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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

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.

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    Related: https://en.wikipedia.org/wiki/Integer_relation_algorithm – 2017-01-17
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    For clarification: if given $r = 0.618$, which is approximately $(\sqrt{5}-1)/2$, would a possible output be $a = 1, b = 1, c = -1$ (since that $r$ is a root of $x^2+ x - 1$)? – 2017-01-17
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    @MichaelLugo Yes, that is how it is meant – 2017-01-17

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