Cubic Roots-Fit a Quadratic Between a Turning Point And Midpoint!

Author(s): Greg Oliver

Originally published on Towards AI.

A Root Approximation Tool Kit Mixing and Matching Polynomial ArchitecturesGenetic Cubic Architectural Dimensions

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This post presents a novel Cubic-Quadratic function matchup for finding Cubic roots. It exploits the little publicised fact that the Midpoint between 2 adjacent roots of a reduced Cubic when multiplied by -2 gives us the 3rd root!

This is related to the sum of the factors = Coefficient B of x². In the example B=0 being a reduced Cubic.

Besides being graphically intuitive the adopted Quadratic function greatly simplifies Cubic function redesign with varying Constants D, because it’s a lot easier to find Quadratic roots with changing Constants c than Cubic roots with changing Constants D.

This post assumes math at the year 12 level.

Before doing a couple of examples, let’s do a brief recap on genetic Cubic architecture.

Cubic Architecture Recap

The header graph shows reduced Cubic y=Ax³+Cx+D and its genetic dimensions shown in black. It is rotationally symmetrical about its Inflection Point Ip(0, y)=Constant D; (Imaginary propellor shaft ?):):). It has y=Ip(y) intercepts as follows:

Int A(x)= – SqRt[-C/A] and Int B(x)= + SqRt[-C/A] with Midpoints:

Midpoint (Int A : Ip(0, D)=Int A(x)-SqRt[-C/4A] and +SqRT[-C/4A] (not shown)

And Turning Points Tp(x)=+-SqRt[-C/3A]

Roots Rt 1, Rt 2 with Root Midpoint; Mid Point (Rt… Read the full blog for free on Medium.

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Published via Towards AI