Khayyam's work on cubic equations

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!)

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.