Omar Khayyam's approach to solving certain cubic equations

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 [math]x^3+cx=d[/math], where [math]c[/math] and [math]d[/math] are given positive constants. To solve the given cubic equation using Khayyam's method, let |AB| = [math]\sqrt{c}[/math] and |BC| = [math]\frac{d}{c}[/math]. Construct the semi-circle on diameter BC and the parabola [math]y=\frac{x^2}{\sqrt{c}}[/math], as shown, intersecting at D. Complete the rectangle BEDZ. Then [math]x=|BE|[/math] 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: [i]A History of Mathematics - an introduction [/i]by VG Katz (1993). Find out more about Khayyam on the web.