A GeoGebra Rendition of One of Omar Khayyam's Solutions for a Cubic Equation - Combining Khayyam's Construction with a Modern Graph

Notably, Khayyam's construction generates a single line segment \(\overline{LB}\) which solves the cubic equation of the form \(x^3 + C_1x^2 + C_2x = C_3,\) where \(C_1,C_2,C_3>0.\) In Figure 6 and Figure 7, for an example, we have fixed the coefficients to be \(x^3 + 7.9x^2 + 1.3^2 x = 2.1^3,\) and in Figure 10, below, our example is \(x^3 + 4.8x^2 + 1.4^2 x = 1.7^3.\) In Figure 10, we have reflected Khayyam's construction about the line \(\overline{HB},\) the vertical asymptote of the hyperbola. Then, superimposing a Cartesian graph of a cubic shows clearly that \(BL\) corresponds exactly to the positive root of the cubic.

Figure 10. A reflection of Khayyam's construction for \(x^3 + 4.8x^2 + 1.4^2 x = 1.7^3,\) superimposed with a Cartesian graph of the equivalent cubic function. We take line segment \(\overline{HB},\) an asymptote for the hyperbola, as a vertical axis and line segment \(\overline{BL}\) to have length \(x\), which is exactly the value of a root to the cubic polynomial.

The slider bars in the GeoGebra applet in Figure 11 (below) enable students to explore how Khayyam's construction works for cubics with a range of coefficients. This dynamic exploration facilitates an appreciation of the general nature of the proof – Khayyam's method is not simply constructing one root to one specific polynomial. His construction technique constructs one root for any polynomial of the form \(f(x) = x^3 + C_1x^2 + C_2x - C_3,\) where \(C_1, C_2, C_3 > 0.\) The applet also presents in an accessible visual form the relationship between Khayyam's line segment \(\overline{LB}\) and a root of the cubic graph familiar to many students.

Figure 11. A reflection of Khayyam's construction for \(x^3 + ax^2 + b^2 x = c^3,\) with various positive values of \(a,b,\) and \(c,\) superimposed with a Cartesian graph of the equivalent cubic function. We take line segment \(\overline{HB},\) an asymptote for the hyperbola, as a vertical axis and line segment \(\overline{BL}\) to have length \(x\), which is exactly the value of a root to the cubic polynomial.

In Figures 10 and 11, superimposing a Cartesian graph of a cubic shows clearly that \(BL\) corresponds exactly to the positive root of the cubic. But what of the other roots?