Khayyam's work on cubic equations Omar Khayyam is known for his significant progress in solving cubic polynomial equations. For example, his biography on www-history.mcs.st-andrews.ac.uk says

(...) This problem in turn led Khayyam to
solve the cubic equation x^3 + 200x =
20x^2 + 2000 and he found a positive
root of this cubic by considering the
intersection of a rectangular
hyperbola and a circle.
(...) Indeed Khayyam did produce such a work, the Treatise
on Demonstration of Problems of
Algebra  which contained a complete
classification of cubic equations with
geometric solutions found by means of
intersecting conic sections.

But I still can't see the big picture of those days.  I'm possibly omitting something about the idea of geometric solutions of algebraic equations, but why were they trying hard to find intersections of conic sections, and building large classification schemes for it?. If the idea was to get a numerical value out of these constructions by measuring lengths on paper,  they could just have prepared a careful template for the function $y = x^3$, and then solved all the cubic equations by intersecting it with a parabola, like in the figure below for the mentioned equation.
I would appreciate answers that would clarify my confusion.  Was it that they did not conceive $y=x^3$ as a curve, if they were interested in getting a numerical value? Or was it a conceptual challenge to show that all cubic equations can be represented as an intersection of two conic sections?

 A: I don't think they had the idea of $y=x^3$ as a curve. For one thing, there was no analytic geometry until Descartes. With coordinates, some problems became much simpler. 
A: There's a brief note in this book on how Khayyam bumped into having to solve a cubic.
I'll only make the note that you should remember the context of the time: there was no concept of negative, much less complex, solutions. Corresponding to our current Cartesian system, Khayyam only looked at intersections in the first quadrant.
Another note should be made that the curves of the time were constructed with geometric tools (straightedge, compass, and a bunch of other contraptions), and $y=x^3$ isn't really a sort of curve that easily lends itself to such a construction (but is now easily constructed thanks to our current knowledge of coordinate geometry).
Here is a more explicit mention of the hyperbola-circle intersection problem Khayyam studied and was mentioned in the OP.
Here is a (more or less) complete table of all the intersection cases Khayyam studied. (The book has an appendix containing a (translated) section of Khayyam's work.)
Here is yet another reference.
(I'll keep updating this answer as I comb through more books; watch this space! As an aside, it's funny that my attempts to look for answers to this question are leading me to references for this question!)
