The idea and method of "argument" are widely used in mathematics. The flexible use of method of substitution to solve problems helps to clarify the quantitative relationship, simplify the complex, turn the difficult into the easy, and give simple and ingenious answers.

In the process of solving problems, a formula in the problem, such as f(x), is taken as a new variable Y or a formula in which a variable in the problem, such as X, is replaced by a new variable T, that is, by making f(x)=y or X = g(t) carry out variable substitution, a new problem-solving method with simple structure and easy solution is obtained, which is usually called method of substitution or variable substitution method.

The key to solving problems with method of substitution lies in choosing method of substitution f(x)=y or x=g(t) which can control the complex and turn the difficult into the easy by simple means according to the structural characteristics of the problem. As far as specific substitution forms are concerned, there are various, such as rational substitution, radical substitution, exponential substitution, logarithmic substitution, trigonometric substitution, anti-trigonometric substitution and complex variable substitution. It is advisable to constantly sum up experience and master relevant skills to solve problems.

For example, when designing trigonometric substitution for solving algebraic problems, we should follow the following principles: (1) Give full consideration to the definition, range and related formulas and properties of trigonometric functions; (2) Minimize the number of variables and simplify the problem structure; (3) With the help of known trigonometric formulas, the internal relations between variables can be easily established. Only by comprehensively considering the above principles can we seek suitable triangular substitution.

Method of substitution is an important mathematical method, which is widely used in factorization of polynomials, simplified calculation of algebraic expressions, proof of identities, conditional equations or inequalities, solution of equations, equations, inequalities, inequality groups or mixed groups, derivation of function expressions, definition domains, range or maximum values, coordinate replacement in analytic geometry, and mutual transformation between ordinary equations, parametric equations and polar coordinate equations.

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Second, the elimination method

For multivariable problems, sometimes we can use the problem setting conditions and some known identities (algebraic identities or trigonometric identities) to eliminate some variables through appropriate deformation, so that the problem can be solved. This method of solving problems is usually called elimination method, also known as elimination method.

Elimination method is the basic method to solve equations, and it is also important in deriving conditional equations and transforming parametric equations into ordinary equations.

The elimination method is very skillful in solving problems, and it is often necessary to flexibly choose the appropriate elimination method according to the characteristics of the topic.

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Third, the undetermined coefficient method

According to a certain law, first write the form of the solution of the problem (generally referring to a formula, expression or equation), which contains some values of undetermined coefficients, so as to get the solution of the problem. This method of solving problems is usually called undetermined coefficient method; Among them, the undetermined coefficient is called undetermined coefficient.

There are two common methods to determine the value of undetermined coefficient: comparison coefficient method and special value method.

Fourth, the discrimination method.

One-variable quadratic equation with real coefficients

ax2+bx+c=0 (a≠0) ①

△=b2-4ac discriminant has the following properties:

> 0, if and only if equation ① has two unequal real roots.

△ = 0, if and only if Equation ① has two equal real roots;

< 0, if and only if equation ② has no real root.

For quadratic function

y=ax2+bx+c (a≠0)②

Its discriminant △=b2-4ac has the following properties:

> 0, if and only if the parabola ② and the X axis have two common points;

△ = 0, if and only if parabola ② and X axis have a common point;

< 0, if and only if the parabola ② has no common point with the X axis.

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V. Analysis and synthesis methods

Analysis and synthesis are two thinking methods with opposite thinking directions, which are derived from analysis and synthesis and play a very important role in the process of solving problems.

In mathematics, analysis is regarded as a way of thinking from the result to the cause of this result, while synthesis is regarded as another way of thinking from the cause to the result produced by the cause. The former is usually called analytical method, while the latter is called comprehensive method.

Six, mathematical model method

In the 8th century, there was a Pleg River in Konigsberg, East Prussia. The river has two tributaries, which meet in the city center and flow into the Baltic Sea. There are seven distinctive bridges in the city, connecting the island and the two banks. Every evening or holiday, many residents will come here for a walk and enjoy the beautiful scenery. As I get older, some people will ask this question: Can I start from somewhere, cross each bridge once and only once, and then go back to the starting point?

Mathematical model method refers to a mathematical method that abstracts the practical problems investigated, constructs the corresponding mathematical model, and solves the practical problems through the study of the mathematical model.