(...) 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?