Omar Khayyam (al-Khayyami, 1048-1131), mathematician, philosopher and poet laid down the gauntlet for the algebraic solution of cubic equations. Here we see how he tackled , where and are given positive constants.
To solve the given cubic equation using Khayyam's method, let |AB| = and |BC| = . Construct the semi-circle on diameter BC and the parabola , as shown, intersecting at D. Complete the rectangle BEDZ. Then satisfies the given cubic.

Explore the problem by adjusting the sliders.
Investigate why Khayyam's method works.
Compare his method with those of other mathematicians, such as Cardano.
Reference: A History of Mathematics - an introduction by VG Katz (1993).
Find out more about Khayyam on the web.