Khayyam's Solution of a Cubic: All Roots
Drag sliders a, b, and c to manipulate the figures. Note that the points of intersection of the circle with the hyperbola have the same x-coordinates as the roots of the cubic.
This applet demonstrates an extension of Khayyam's original geometric construction, which generated the positive root of a given cubic. Khayyam's semicircle is extended to a circle, and both part of the hyperbola are used to generate the negative roots (when they exist). Although such solutions would not have made sense in Khayyam's mathematical context, the applet demonstrates the generalizability of his method.