1, construction equation method
Constructing equations is usually to construct some special equations, such as quadratic equations with one variable. Because the unary quadratic equation itself has some extension contents, if the equation has real roots, the discriminant is greater than or equal to zero; There is a very special relationship between its roots and coefficients-Vieta theorem; The equation has real roots in the interval, which can correspond to functions, images and so on. By constructing equations, some "equal relations" can be transformed into "unequal relations" or "unequal relations" can be transformed into "equal relations".
Example 1 is a real number, which satisfies the range of finding.
Analysis: It is obtained from known conditions, so a quadratic equation with one variable can be constructed according to Vieta's theorem.
This equation has two real roots, and its discriminant is not less than zero, that is, there is
The range of values thus obtained is.
What needs to be explained here is that the equation to be constructed in a specific problem depends on the demand of the specific problem, but when it comes to "the sum of two numbers or the product of two numbers", it should be considered that the equation can be constructed by Vieta's theorem, and when it comes to the relationship similar to the discriminant structure, it should also be considered that the corresponding equation can be constructed.
Example 2 Any point of the circumscribed circle (lower arc) is known to be positive, which proves that:
Example 3 determines all integer solutions of the equation, and the equation is
Analysis: This problem is a high-order system of equations, which is very difficult, but it can be easily solved by solving it, so that the knowledge related to the equation can be used.
2. Constructor method
Function is the most important thought in mathematics. In elementary mathematics, problems such as numbers, formulas, inequalities, series and curves are related. The constructor is to construct a new function from the characteristics of the problem itself, and then use the properties of the function to find the solution of the problem.
Example 4 is a real number known to satisfy. Try to determine the maximum value.
3. Structural graphic method
Example 6 Find the range of functions.
Analysis: This relationship reflects the slope of a straight line passing through two points, which is a point on the unit circle. Therefore, considering the slope range of a straight line moving on the unit circle, it is easy to get the slope range as follows.
It should be noted that first of all, we should consider the correspondence between some basic algebraic expressions and geometric figures, such as algebraic expressions of equations and the properties of lines, circles, ellipses, hyperbolas, parabolas and some basic figures, such as sine and cosine theorems of trigonometric functions.
4. Build a model
The relationship between conditions and quantity in the problem is realized and explained on the constructed model, thus realizing the proof of the problem. In the process of solving the problem, some models can be obtained from the conditions of the problem itself, and some models are exquisitely constructed.
Example 7 proves that the sum of cosine of an acute triangle whose vertex is on the unit circle is less than half the circumference of the triangle.
5. The method of constructing inequality
In some problems, especially the maximum value of functions, the conditions or composition of the function arrangement system often imply some restrictive conditions, such as some basic inequalities when the equation has a solution. Make full use of them to form inequalities and solve problems.
(National High School Mathematics League)
6. Structural distance method
Example 10 sets and finds the minimum value of.
Analysis: can be converted into. Where is the square of the distance between points, and these two points are summed on a straight line respectively. According to the position of two straight lines, it is not difficult to know that the shortest distance between points on two straight lines is. Therefore, the known minimum value is 6.
7. Build a corresponding relationship
The so-called structural correspondence means that one thing corresponds to another, which is often used in dealing with some counting problems. Because it may be difficult to meet some requirements directly, it may be easier to consider another type of element corresponding to it.
Example 1 1 How many positive integer solutions does the equation have?
Analysis: We can construct such a correspondence: if 2002 identical balls are arranged in a row, they will have 200 1 intervals, and 1000 plates are inserted in these 200 1 intervals (only one plate can be inserted in each interval), then obviously each group of interpolation methods will correspond to each group of solutions of the original equation.
The application of structural method is of great significance for flexible examination in examinations and competitions, as well as for cultivating ability and inspiring thinking. The above are just common centralized construction methods, and there are many types of construction, such as constructing complex numbers, constructing equivalence propositions, constructing sequences, constructing identities, constructing conclusions, constructing complex numbers and so on. In mathematical construction, according to different types of questions, the conditions or conclusions in the questions are skillfully used to solve problems. This unique method can often broaden the thinking of solving problems and reduce unnecessary troubles in the process of solving problems. But at the same time, the construction method is a flexible method, and different types of problems should be solved in different ways. In a word, the construction method is a flexible mathematical problem-solving method, which requires the solver to have solid basic knowledge, keen observation ability and rich imagination, so as to get twice the result with half the effort in the process of doing the problem.