Why is 3AB+P Set to Zero in Cardan's Cubic Equation?

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Discussion Overview

The discussion revolves around the reasoning behind setting the term 3AB + P to zero in Cardano's cubic equation. Participants explore the implications of this step in the context of solving cubic equations, specifically through the lens of Cardano's and Del Ferro's methods.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Homework-related

Main Points Raised

  • One participant presents the equation X^3 + PX + Q = 0 and derives the expression (A + B)^3 + P(A + B) + Q = 0, leading to the term 3AB + P.
  • Another participant suggests that by choosing a ratio such that AB = -P/3, it simplifies the equation, potentially reducing it to a quadratic form.
  • There is a request for an example to illustrate the application of Cardano's and Del Ferro's methods, indicating a desire for practical understanding.

Areas of Agreement / Disagreement

Participants express different perspectives on the reasoning behind setting 3AB + P to zero, with some proposing specific values for AB while others seek clarification through examples. The discussion remains unresolved regarding the broader implications of this choice.

Contextual Notes

Participants do not fully explore the assumptions behind the choice of AB = -P/3, nor do they clarify the implications of this choice on the overall solution process. The discussion does not resolve the mathematical steps involved in applying Cardano's method.

Who May Find This Useful

Readers interested in the historical methods of solving cubic equations, particularly those studying Cardano's and Del Ferro's techniques, may find this discussion relevant.

Physicsissuef
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[tex]X^3+PX+Q=0[/tex]

[tex]X_0 = A + B[/tex]

[tex](A+B)^3 + P(A+B) + Q = 0[/tex]

[tex]A^3 + B^3 + (3AB+P)(A+B) + Q = 0[/tex]

The next step is 3AB+P=0

What make Cardano drop 3AB+P?

Why it is zero?
 
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Physicsissuef said:
[tex]X^3+PX+Q=0[/tex]

[tex]X_0 = A + B[/tex]

[tex](A+B)^3 + P(A+B) + Q = 0[/tex]

[tex]A^3 + B^3 + (3AB+P)(A+B) + Q = 0[/tex]

The next step is 3AB+P=0

What make Cardano drop 3AB+P?

Why it is zero?

Hi Physicsissuef! :smile:

AB can be anything we like, so long as A + B = X.

So we choose a ratio such that AB = -P/3.

That eliminates one of the terms in the equation, and makes it much easier … a quadratic equation (in A^3, I think).

And we already know how to solve a quadratic! :wink:
 
Ok, thank you very much Timmy.
 
could someone just show me with an example on how to work the cardano and Del ferro method from start to finish say using
x^3-6x^2+2x-1
 

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