Applet showing Lill's method applied to cubic equations
Applet figure showing the right-angle paths that represent a cubic equation with real solutions
The horizontal green line is the x-axis where the point O represents x = 0, and distances are measured in units of a, the fixed distance between points O and SP (starting point). The initially positive coefficients, b, c, and d can be adjusted by moving the points B, C, and D respectively. The point P is constrained to move along the x-axis and the purple line E-P is constrained to be perpendiculat to the purple line P-SP, and the point E can be extended as needed until it crosses the line c or its vertical extension. The point, X, marks the intersection of the purple line P-E and the vertical line c or its extension. The point F can be extended as far as needed. Real solutions to the cubic equation are found as the length of the line P-O whenever the line F-X lies on the point D. If the line P-X intersects the blue circle, then there will be three real solutions to the cubic equation: The right-angle path SP-P-X-D describes a quadratic equation, whose two solutions are also solutions to the cubic equation.
If the mouse cursor is clicked on the applet and then the space bar is pressed, the applet will reset to its original position. The applet can be enlarged by clciking on the applet, pressing the return key, and dragging the lower right corner of the applet to any desired size.