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

[Physicsissuef]

$$X^3+PX+Q=0$$

$$X_0 = A + B$$

$$(A+B)^3 + P(A+B) + Q = 0$$

$$A^3 + B^3 + (3AB+P)(A+B) + Q = 0$$

The next step is 3AB+P=0

What make Cardano drop 3AB+P?

Why it is zero?

 

[tiny-tim]

$$X^3+PX+Q=0$$

$$X_0 = A + B$$

$$(A+B)^3 + P(A+B) + Q = 0$$

$$A^3 + B^3 + (3AB+P)(A+B) + Q = 0$$

The next step is 3AB+P=0

What make Cardano drop 3AB+P?

Why it is zero?

Hi Physicsissuef!

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!

 

[Physicsissuef]

Ok, thank you very much Timmy.

 

[kennysmith39]

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

 

[blue_raver22]

Cardan's cubic equation is a mathematical equation that is used to solve for the roots of a cubic polynomial. It was first discovered by Italian mathematician Gerolamo Cardano in the 16th century. The equation is in the form of X^3+PX+Q=0, where P and Q are constants. In order to solve this equation, Cardano introduced a new variable, X_0, which is represented as A + B. This substitution helps to simplify the equation and make it easier to solve.

The next step in solving the equation is to expand (A+B)^3 and substitute it into the original equation. This results in A^3 + B^3 + (3AB+P)(A+B) + Q = 0. The term 3AB+P is then isolated and set equal to zero in order to solve for A and B.

Cardano's method of solving the cubic equation is based on the concept of finding two numbers, A and B, whose sum and product are equal to the coefficients of the equation. This is known as the Vieta's formulas. By setting 3AB+P=0, Cardano is essentially finding two numbers whose product is equal to the constant term, Q, in the equation.

The reason 3AB+P is set to zero is because it simplifies the equation and allows for easier solving. By setting this term to zero, the equation becomes A^3 + B^3 + Q = 0, which is a simpler form to work with. It also follows the principle of finding two numbers whose product is equal to Q, as mentioned before.

Overall, Cardan's cubic equation is a significant contribution to mathematics and has been used for centuries to solve cubic equations. The dropping of 3AB+P in the equation is a crucial step in solving for the roots and is based on the concept of finding two numbers whose product is equal to the constant term in the equation.