16th century Europe, with the development of mathematics, the cubic equation of one variable has a fixed solution. In many mathematical documents, the formula for finding the root of cubic equation is called "cardano formula", which is obviously to commemorate the first Italian mathematician cardano who published the formula for the root of cubic equation with one variable in the world. So, was the general solution of the univariate cubic equation first discovered by cardano? This is not a historical fact.
In the history of mathematics, the first person to find the general solution of a cubic equation is/kloc-another Italian mathematician in the 6th century, Hotel Nigro fontana. Fontana was born in poverty, lost his father, and there was no condition for him to study at home. However, through hard work, he finally became one of the most accomplished Italian scholars in the16th century. Because Fontana suffers from stuttering, people nicknamed him "Tarta", which means "stuttering" in Italian. Later, in many math books, Feng Tana was directly called "Tarta Riya".
After years of exploration and research, Fontana found a method to find the root of a cubic equation in a general form with a very clever method. This achievement made him win a great victory in several open mathematics competitions and became famous in Europe. But Feng Tana didn't want to make this important discovery public.
Cardano, another Italian mathematician and doctor at that time, was very interested in Feng Tana's discovery. He sincerely visited several times for advice, hoping to get Fontana's roots. But Fontana kept her mouth shut. Although cardano was frustrated many times, he was extremely persistent and tried to "dig the secret" from Feng Tana. Later, Vontana finally "revealed" the solution of the cubic equation to cardano in incantation-like obscure language. Feng Tana thought it was difficult for cardano to break his "magic spell", but cardano's understanding was great. Through the comparative practice of solving cubic equations, he quickly cracked Fontana's secret completely.
Cardano wrote Feng Tana's cubic equation root formula into his academic book Dafa, but did not mention Feng Tana's name. With the advent of European Dafa, people realized the general solution of cubic equation. Because the first person who published the formula for finding the root of cubic equation was really cardano, later generations called this solution "cardano formula".
Cardano stole other people's academic achievements and took them for himself, leaving a disgraceful page in the history of human mathematics. This result is of course unfair to Fontana who has worked hard. However, Feng Tana's insistence on not disclosing his research results is incorrect, at least for the development of human science, and it is an irresponsible attitude.
The formula for finding the root of the univariate cubic equation can't be deduced by ordinary deductive thinking. The standard univariate cubic equation of AX 3+BX 2+CX+D+0 can only be formalized into a special type of X 3+PX+Q = 0 by using a matching method similar to the formula for finding the root of the univariate quadratic equation.
The solution of the solution formula of the univariate cubic equation can only be obtained by inductive thinking, that is, the form of the root formula of the univariate cubic equation is summarized according to the form of the root formula of the univariate quadratic equation and the special higher order equation. The formula for finding the root of a univariate cubic equation in the form of x 3+px+q = 0 should be X = A (1/3)+B (1/3), which is the sum of two exponents. Summarized the form of the root formula of the univariate cubic equation, and the next step is to find the content of the square, that is, to represent a and b by P and Q, as follows:
(1) can get the simultaneous cube of X = A (1/3)+B (1/3).
(2)x^3=(a+b)+3(ab)^( 1/3)(a^( 1/3)+b^( 1/3))
(3) Because X = A (1/3)+B (1/3), (2) can be changed to
X 3 = (a+b)+3 (ab) (1/3) x, transpositions are available.
(4) x 3-3 (ab) (1/3) x-(a+b) = 0. Comparing the univariate cubic equation with the special type x 3+px+q = 0,
(5)-3 (AB) (1/3) = P, -(A+B) = Q, simplified.
(6)A+B=-q,AB=-(p/3)^3
(7) In this way, the roots of the univariate cubic equation are formulated into the roots of the univariate quadratic equation, because A and B can be regarded as the two roots of the univariate quadratic equation, and (6) is Vieta's theorem about the two roots of the univariate quadratic equation in the form of ay 2+by+c = 0, namely.
(8)y 1+y2=-(b/a)，y 1*y2=c/a
(9) Comparing (6) and (8), we can make A = Y 1, B = Y2, Q = B/A,-(p/3) 3 = c/a.
(10) because the formula for finding the root of the unary quadratic equation of type ay 2+by+c = 0 is
y 1=-(b+(b^2-4ac)^( 1/2))/(2a)
y2=-(b-(b^2-4ac)^( 1/2))/(2a)
Can become
( 1 1)y 1=-(b/2a)-((b/2a)^2-(c/a))^( 1/2)
y2=-(b/2a)+((b/2a)^2-(c/a))^( 1/2)
Substitute a = y 1, b = y2, q = b/a, -(p/3) 3 = c/a in (9) into (1 1).
( 12)a=-(q/2)-((q/2)^2+(p/3)^3)^( 1/2)
b=-(q/2)+((q/2)^2+(p/3)^3)^( 1/2)
(13) substitute a and b into X = A (1/3)+B (1/3).
( 14)x=(-(q/2)-((q/2)^2+(p/3)^3)^( 1/2))^( 1/3)+(-(q/2)+((q/2)^2+(p/3)^3)^( 1/2))^( 1/3)
The equation (14) is only the real root solution of the univariate cubic equation. According to the cubic equation of Vieta's theorem, there should be three roots, but according to the cubic equation of Vieta's theorem, only one of them is needed, and the other two roots are easy to find.