catalogue
1 historical process
2 cardan formula
3 other methods
Factorization method
Alternative methods.
Derivative solution method
Jinsheng formula method
Jinsheng theorem
4 examples of solving problems
5 Correct solution to the problem
1 historical process editing
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 Feng Tana suffers from stuttering, people nicknamed him "Nicolo Tartaglia", 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 this important discovery to be known to the world, because at that time, Italian mathematical competitions prevailed, and Feng Tana took his secret of solving cubic equations as a magic weapon, which was his sword to win the competition.
At that time, another Italian mathematician and doctor, Cardin (also known as Cardin and cardano in some literatures), 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, Fontana finally "revealed" the solution of the cubic equation to Cardin in a language as obscure as a spell. 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. Cardin 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 Carl Dan, later generations called this solution "Carl Dan formula", and some materials were also called "Carl Dan formula". Kadan plagiarized 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.
Karl Dan was the first mathematician to write negative numbers in quadratic roots, and thus introduced the concept of imaginary numbers. Later, after the efforts of many mathematicians, it developed into a complex number theory. In this sense, Caldan formula has made great contributions to the development of mathematics. Historically, Caldan formula is a great formula.
Solving the univariate cubic equation is a famous, complex and interesting problem in the history of world mathematics. The introduction of the concept of imaginary number and the establishment of complex number theory originated from the solution of cubic equation problems. Cubic equation with one variable is widely used, such as power engineering, water conservancy engineering, construction engineering, mechanical engineering, power engineering, mathematics teaching and other fields. Solving a univariate cubic equation with root sign, although there is a famous Caldan formula and the corresponding discrimination method, it is more complicated and lacks intuition to solve the problem with Caldan formula. In the 1980s, Shenjin Fan, a middle school math teacher in China, made a deep research and exploration on solving the univariate cubic equation, invented a new root-seeking formula-Jinsheng formula, which is more practical than Kadan formula, and established a simple, intuitive and practical new discriminant-Jinsheng discriminant. At the same time, he put forward the Golden Sage Theorem, which clearly answered the doubts about understanding cubic equations, which was very interesting. The characteristic of Jinsheng formula is the simplest multiple root discriminant A = B2-3ac；; B = 9adC = c 2-3bd BC and the total discriminant δ = b 2-4ac, which embodies the beauty of order, symmetry, harmony and conciseness in mathematics, is concise and easy to remember, and is intuitive, accurate and efficient in solving problems, especially when δ = b 2-4ac = 0, the golden formula is 3: x (1) =-. X = x =-k/2, where K=B/A, (A≠0), its expression is very beautiful, and there is no square root (there is still a square root in Kadan formula at this time), so the efficiency of manual problem solving is higher. Jinsheng formula 3 is called super simple formula. Jinsheng formula, discriminant method and theorem constitute a complete, concise, practical and mathematically beautiful theoretical system for solving cubic equations. This universal system method, which was created by Shenjin Fan, has contributed to the study of solving higher-order equations and improving the efficiency of solving cubic equations.
Qin, a mathematician in the Southern Song Dynasty, discovered the formula for finding the root of the cubic equation of one yuan at the latest in 1247, which took Europeans more than 400 years to discover, but this formula is still named after that European in China's textbooks. (Nine chapters of mathematics, etc. )
2 Cartan formula editing
Caldan formula method
Special univariate cubic equation x 3+px+q = 0 (p, q∈R).
Discriminant δ = (q/2) 2+(p/3) 3.
Caldan formula
x 1=(y 1)^( 1/3)+(y2)^( 1/3)；
x2 =(y 1)^( 1/3)ω+(y2)^( 1/3)ω^2；
x3=(y 1)^( 1/3)ω^2+(y2)^( 1/3)ω，
Where ω = (-1+i3 (1/2))/2;
Y( 1，2)=-(q/2)((q/2)^2+(p/3)^3)^( 1/2)。 )
The standard unary cubic equation AX 3+BX 2+CX+D = 0, (A, B, C, d∈R, a≠0).
Let x = y-b/(3a) be substituted into the above formula.
It can be transformed into a special one-dimensional cubic equation y 3+py+q = 0, which is suitable for directly solving Kardan formula.
Cardin discriminant method
When δ = (q/2) 2+(p/3) 3 >; 0, the equation has a real root and a pair of * * * yoke imaginary roots;
When δ = (q/2) 2+(p/3) 3 = 0, the equation has three real roots, one of which is a multiple root;
When δ = (q/2) 2+(p/3) 3
3 Other editing methods
In addition to the Cartan formula above, there are other solutions to the univariate cubic equation, which are listed as follows:
Factorization method
Factorization does not apply to all cubic equations, but only to some simple cubic equations. For most cubic equations, only by finding its root can factorization be carried out. Of course, some simple cubic equations can be solved by factorization, which is very convenient to solve and directly reduces cubic equations.
For example, solve the equation x 3-x = 0.
Left factorization, x(x+ 1)(x- 1)=0, three roots of the equation: x1= 0; x2 = 1； x3=- 1 .
Alternative methods.
For the general cubic equation, the equation is first transformed into a special type of x 3+px+q = 0.
Let x=z-p/3z, and substitute it for simplification to get: z 3-p/27z+q = 0. Let z=w and substitute it, and we get: w 2+p/27w+q = 0. This is actually a quadratic equation about W. Solve W, and then solve Z and X in turn.
Derivative solution method
By using derivative, the maximum value, minimum value, monotone increasing and decreasing interval of the function are obtained, and the function image is drawn, which is beneficial to the approximate solution of the equation and can quickly get the number of solutions of the equation. This method is very suitable for solving high school math problems.
For example, f (x) = x 3+x+1,x 3+x =-1is a transposable term, y1= x 3+x, y2 =- 1,
The derivative of y 1' = 3x 2+ 1 shows that y 1' is always greater than 0, and y 1 monotonically increases on R, so there is only one solution to the equation. When y 1=- 1, x is. 0 formula, infinite approximation, get a more accurate solution.
Jinsheng formula method
Cubic equation is widely used. Solving a univariate cubic equation with root sign, although there is a famous Caldan formula and the corresponding discrimination method, it is more complicated and lacks intuition to solve the problem with Caldan formula. Shenjin Fan derived a set of new general formulas for finding the root of a cubic equation directly expressed by A, B, C and D- Jinsheng formulas, and established a new discriminant method-Jinsheng discriminant method.