Cardin's theorem

Cardin's formula was used to determine the root of a cubic equation. Cardin's formula could be obtained by replacing the general cubic equation and introducing variables. To be more specific, through substitution and the introduction of new variables, a general cubic equation could be transformed into a new equation. Then, by solving this new equation, one could get the conclusion of Cardin's formula. Cardin's formula gave the expression of the root of a cubic equation, which involved some parameters and calculations. However, the specific calculation process and reduction method were not given in the provided search results. Therefore, the search results did not provide a clear answer on how to simplify the Cardin formula to solve the complex cubic equation.

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Cardin's formula, also known as Cardano's formula, was used to solve cubic equations. It gave the three solutions of the cubic equation x^3 +px+q=0 as x1=u+v, x2=uw+ vw^2, x3= uw^2 +vw. The Cardin formula was first discovered by the Italian scholar Tattaglia in 1541, but it was not publicly published. Later, Cardano published this result in his 1545 book, The Great Law, so this formula was called the Cardano formula. Through Cardin's formula, one could solve cubic equations with any complex coefficient. The derivation process of Cardin's formula involved the idea of variable substitution and reduction.

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The focal ratio theorem of the conical curve was a theorem related to the polar coordinate equation of the conical curve. According to the given polar coordinate equation of the conical curve, p =ep/(1-e* cos0), and the straight line, 0 =c or 0 = Pi +c, where c is a constant, the focal ratio theorem can be derived as:| 1-e*cosc)/(1+e*cosc)|.The specific derivation process is as follows: Consider the intersection of the conical curve and the straight line. The coordinates of the intersection are (ep/(1-e*cosc), c) and (ep/(1+e*cosc), Pi +c). According to the definition of focal radius, the focal radius length was the distance from the focal point to the intersection point. Therefore, the ratio of focal radius to length is| 1-e*cosc)/(1+e*cosc)|.This was the derivation process of the focal ratio theorem for conical curves.

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Cardin's formula was a formula used to solve cubic equations. It could solve any type of cubic equation and was the universal formula for such equations. The process of solving Cardin's formula mainly included the following steps: 1. The cubic equation to be solved was converted into the standard form, which was in the form of x^3 +px+q=0. 2. By performing a variable substitution, the unknown x was replaced with a new variable y, so that the equation became y^3 +py+q=0. 3. Using Cardin's formula, he calculated the three solutions of y according to the equations 'p and q. 4. Substitute the three solutions of y back to the variable x to obtain the three solutions of the original equation. It should be noted that the process of solving Cardin's formula may involve complex numbers, so the solution may include real numbers and complex numbers. In addition, the calculation process of Cardin's formula might be rather complicated, requiring multiple replacements and calculations. In short, the Cardin formula was a general formula for solving cubic equations. Through variable substitution and calculation, three solutions could be obtained.