1, solve the absolute value problem
It mainly includes simplification, evaluation, equations, inequalities, functions and other issues. The basic idea is to turn a problem with absolute value into a problem without absolute value.
The specific transformation method is as follows:
① Classification discussion method: according to the positive, zero and negative scores of numbers or formulas in absolute value symbols, the absolute value is removed.
② Zero-point subsection discussion method: it is suitable for the case that a letter has multiple absolute values.
③ Two-sided flat method: it is suitable for equations or inequalities with non-negative edges.
④ Geometrical meaning method: It is suitable for cases with obvious geometric meaning.
2. Factorization
It is an important skill to choose the method according to the number of terms and follow the general steps. The general steps of factorization are:
Extract common factor
Selection formula
Cross multiplication
Group multiplication
Method for splitting items and adding items
3. Matching method
Using the complete square formula to change a formula or part into a complete square is the matching method, which is an important method and skill in mathematics. The main basis of the matching method is:
4. Alternative methods
Method of substitution is used to solve some complicated special equations. The general steps of solving equations by substitution method are:
Set Yuan → Exchange Yuan → Share Yuan → Return Yuan.
5, undetermined coefficient method
The undetermined coefficient method is a method to find an object under the condition of knowing its shape. It is suitable for solving some important problems, such as coordinates, resolution function and curve equation. The steps to solve the problem are as follows: ① Set 2 columns, 3 solutions and 4 writing.
6. Complex algebraic equations
Skills of using equality conditions in complex algebra: zero is on the left and deformation is on the right.
① Factorization type:
(-) (-) = 0 Both cases are OR type.
(2) Formula:
(-) 2+(-) 2 = 0 is a kind of union.
7. The two greatest problem-solving ideas in mathematics
The idea of (1) evaluation lists the equations or equations with the letters you want.
(2) The idea of evaluation domain is the inequality or inequality group of letters in evaluation domain.
8. Simplify the secondary roots
The basic idea is to turn √m into a completely flat way. Namely:
9. Observation
10, algebraic evaluation
These methods are:
(1) direct substitution method
(2) Simplified substitution method
(3) Appropriate deformation method (sum product substitution method)
Note: when the algebraic expression evaluated is a symmetrical expression of letters, it can usually be converted into the form of "sum and product" of letters, so it can be evaluated by "sum and product substitution method"
1 1, and solve the parameter equation.
Except for unknowns, other letters contained in the equation are called parameters, and this equation is called parametric equation. Usually, the parameter equation is solved by "classified discussion method", and its principle is:
(1) Solve by Type
(2) Discuss as needed.
(3) Write conclusions by category
12, a useful condition for the identity to hold.
(1)ax+b=0 holds for any X. The equation ax+b=0 has countless solutions, and a=0 and b=0.
(2) AX2+BX+c=0 holds for any X, and the equation AX2+BX+c=0 has countless solutions: a=0, b=0, and c=0.
13, the conditions for the establishment of identity inequality
From the conclusion that the solution set of the unary quadratic inequality is r, it is easy to get the following conditions for the establishment of identity inequality:
14, translation method
The law of image motion is an important method to study complex variable functions. The law of translation is:
15, image method
An important method to discuss the nature of function is the image method-looking at the image and finding the nature.
Define the corresponding part of the domain image on the x-axis.
The corresponding part of the range image on the y-axis.
Monotonicity is from left to right, and the interval corresponding to the continuous rise on the X axis is an increasing interval; From left to right, the interval corresponding to the continuous decline on the X axis is the subtraction interval.
The largest image point has a value and the lowest image point has a minimum value.
Parity is an even function symmetric about Y axis and a odd function symmetric about the origin.
16, the important relationship among functions, equations and inequalities
equation root
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Abscissa of intersection point between function image and X axis
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Endpoint of inequality solution set
17, the solution of one-dimensional quadratic inequality
One-dimensional quadratic inequality can be transformed into two-dimensional linear inequality group by factorization, but it is more complicated; Its simple and practical solution is to use the image of quadratic function according to the relationship between "three quadratic functions", and the specific steps are as follows:
Turn quadratic into positive
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Identify and find the root cause
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Draw a schematic diagram
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On the horizontal axis of the solution set
Discussion on the root of 18 and the quadratic equation of one variable
Using the discriminant of roots and the relationship between roots and coefficients, we can solve the symbolic problem or M-type problem of the roots of a quadratic equation, but the problems of roots in general, especially the problems of interval roots, should be solved by using the images of quadratic functions according to the relationship of "three quadratic". The general idea of "mirror method" to solve the root problem of quadratic equation is:
Title meaning
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Quadratic function image
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Inequality system
The inequality group includes: the symbol of a; Delta situation; The position of the symmetry axis; Symbol of interval endpoint function value.
19, the range of the basic function on the interval
The named functions we have learned, such as linear function, inverse proportional function and quadratic function, are all basic functions. There are two situations in which the basic function calculates the domain or maximum value:
(1) When the domain is not particularly limited-memory method or conclusion method;
(2) When the domain has special restrictions-image truncation method, the general idea is:
Draw an image
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suspend
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come to a conclusion
20. The most valuable solution to applied problems
In the application problem, the problem involving "when one variable takes what value, the other variable takes the value or the minimum value" is the maximum value application problem. The basic idea of solving the most valuable application problems is the functional thinking method, and its solving steps are as follows:
Set variable
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Column function
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Find the maximum value
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Write a conclusion
2 1, threading method
Threading method is a method to solve higher inequality and fractional inequality. The general idea is:
The first standardization
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Find the root and mark the root.
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Right upper perforation
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Odd wear and even return
Note: ① Higher-order inequalities should be transformed into the form of "left product and right zero" by shifting terms and factorization. (2) Fractional inequality can not be solved by multiplying the denominators on both sides. It should be transformed into "quotient zero" by shifting terms, dividing and merging, and factorization, and solved by thread method.