First, the knowledge structure of this chapter

1, knowledge structure combing

2. Theorem formula summary

The general form of (1) unary quadratic equation is ax2+bx+c=0(a≠0).

(2) The root formula of unary quadratic equation ax2+bx+c=0(a≠0):

(3) The discriminant of the root of the quadratic equation in one variable: △=b2-4ac.

* Originally published in The Whole Course of Textbooks: Synchronization and Exquisiteness (Quan Yi Junior Three Algebra Book), China Children's Publishing House, May 2004.

(4) the discriminant theorem of the roots of quadratic equation in one variable:

△ > 0 equation has two unequal real roots;

△ = 0 equation has two equal real roots;

△ < 0 equation has no real root;

△≥0 equation has two real roots.

(5) Theorem of relationship between roots and coefficients of quadratic equation in one variable:

If the two real roots of the unary quadratic equation ax2+bx+c=0(a≠0) are x 1, x2, then

x 1+x2=

Inference 1: If the two real roots of equation x2+px+q=0 are x 1, x2, then x 1+x2=-p, x1x2 = q. 。

Inference 2: The quadratic equation with root x 1 and x2 (quadratic coefficient is 1) is x2-(x1+x2) x+x1x2 = 0.

(6) Quadratic trinomial factorization formula: ax2+bx+c=a(x-x 1)(x-x2). Where x 1 and x2 are two roots of the unary quadratic equation ax2+bx+c=0(a≠0).

(7) Find the values of two symmetric formulas of the unary quadratic equation x 1, x2. The commonly used formulas are:

①x 12+x22 =(x 1+x2)2-2x 1x 2；

②(x 1-x2)2 =(x 1+x2)2-4x 1x 2

Second, the summary of mathematical laws

1. The equations and equations we have learned include integral equations (one-dimensional linear equations and one-dimensional quadratic equations), fractional equations, two-dimensional linear equations and two-dimensional quadratic equations, all of which belong to rational equations in algebraic equations.

Among the equations we have studied, the one-dimensional linear equation and the one-dimensional quadratic equation are the most basic knowledge and skills for solving equations (groups). It is the key and premise to solve algebraic equations (groups) skillfully by solving linear equations and quadratic equations in one variable. Therefore, we must learn this part of knowledge well.

2. This chapter introduces four methods to solve the quadratic equation of one variable-direct Kaiping method, collocation method, formula method and factorization method. Among them, the formula method is suitable for any quadratic equation with one variable, and it is a general method to solve quadratic equations with one variable. To master the method of finding the roots of quadratic equation by formula method, the key is to correctly understand the specific derivation process of formula (that is, collocation method), fully understand the production process and context of this knowledge, and then firmly remember the form, structure and connotation of formula. When the formula is used to find the root of the equation, it is to use the knowledge of quadratic roots to find the values of two quadratic roots. However, when solving a quadratic equation with one variable, it is necessary to analyze the characteristics of the equation in detail and choose appropriate methods to simplify the problem-solving process.

Generally speaking, the choice order of solving a quadratic equation with one variable is: first consider the direct Kaiping method, then consider the factorization method, and finally consider the formula method. Matching method is a tool to find the root formula. After mastering the formula method, we can directly use the formula method to solve the quadratic equation of one variable. Therefore, collocation method is generally not used to solve the quadratic equation of one variable (except that the problem requires collocation method to solve the equation), but collocation method is widely used in other mathematical contents except learning the root formula of quadratic equation of one variable, so collocation method is a very important mathematical method. We must correctly understand the intention of the formula, master the method of the formula and learn this part of knowledge well.

3. Conditions for the decomposition of quadratic trinomial ax2+bx+c in the real number range.

b2-4ac≥0

4. The condition that the unary quadratic equation ax2+bx+c=0 has a real root b2-4ac≥0.

Third, the summary of thinking methods

1, change your mind

In this chapter, the idea of "transformation" runs through like a red line. Solving a quadratic equation with one variable needs to be transformed into a linear equation with one variable; Solving fractional equation needs to be transformed into integral equation; Solving binary quadratic equation needs to be transformed into binary linear equation or univariate quadratic equation. The factorization of quadratic trinomial in the real number range needs to be transformed into the problem of solving the corresponding quadratic equation with one variable, and the determination of the letter coefficient in the equation can also be solved by transforming it into the problem of solving the equation. The specific conversion process and conversion method are shown in the following figure:

Factorization reduction

Denominator algebraic expression

Substitution elimination

Factorization reduction

Conversion thought is a common mathematical thought in junior high school mathematics, and it is widely used. When solving mathematical problems, we often use transformation ideas to turn complex problems into simple problems and unfamiliar problems into familiar ones.

2. Equal thinking

When solving mathematical calculations, equations or equations are often established through the connection between known and unknown, and the values of unknown quantities are obtained by solving equations or equations, thus solving problems. This mathematical idea, which links the known with the unknown through a series of equations, is usually called equation thought.

In this chapter, the idea of equation is mainly embodied in solving practical problems by listing equations (groups), determining undetermined coefficients (letter coefficients) in a quadratic equation with one variable by using discriminant and Vieta theorem, decomposing quadratic trinomial, and solving "II-I" equation by using the relationship between roots and coefficients.

3. Develop and classify discussions.

In mathematics, for those "shaping" mathematical problems that have rules to follow, we always

I hope to find a formula. When solving a problem, we can get the result (conclusion) of the problem by substituting the known number, so as to achieve the purpose of solving the "module" accurately and quickly. For example, the formula of circular area, the formula of trapezoidal area, the formula of finding distance from time and speed S=Vt and so on. Under the guidance of this idea, we have worked out the formula for finding the root of the quadratic equation x= in one variable through the formula.

But in our formula, there is a quadratic root, and its root number is △=b2-4ac. When △≥0, the root is meaningful and the formula can be established and applied. Then when △ < 0, the formula cannot be used. At this time, where is the problem? Does the equation ax2+bx+c=0(a≠0) have no roots? This can't be said, because the derivation process of the formula has shown that only when △≥0 can we get the root formula, that is to say, only when △≥0 can we use the root formula to solve the quadratic equation of one variable and seek truth from several roots. If △ < 0, the number can't be found by the formula, maybe it can be found by other methods. This raises a question: can we judge whether the equation has a real root without solving the equation? Through careful analysis of the formula process, the role of "△" in distinguishing the real root of a quadratic equation with one variable is finally clarified:

For the equation ax2+bx+c=0(a≠0), let △=b2-4ac, then:

(1) If △ > 0, the equation has unequal real roots;

(2) If △=0, the equation has two equal real roots;

(3) If △ < 0, the equation has no real root.

vice versa

Because △ > 0, △=0 and △ < 0 are complete classifications of △ values, which correspond to equations with unequal, equal and no real roots (also complete classifications of the roots of equations), which lays a foundation for classification discussion. In addition, when encountering equations with letters, we should discuss the division of letter coefficients, and then study, explore and solve them according to the knowledge of various situations. (Examples have been given earlier)

Classified discussion is an important way of thinking in mathematics. We must pay attention to this way of thinking and accumulate our own mathematical literacy.

⑷ The mathematical methods used in this chapter mainly include: ① substitution elimination method; (2) factorization reduction method; ③ substitution method; 4 matching method.

The substitution elimination method and decomposition reduction method are mainly embodied in solving binary quadratic equations; Method of substitution is mainly embodied in the fractional equation whose solution can be transformed into a quadratic equation and the factorization of quadratic trinomial. The matching method is mainly embodied in solving the quadratic equation of one variable by using the matching method, the derivation of the root formula of the quadratic equation of one variable, the application of the discriminant of the root of the quadratic equation of one variable, and the application of the relationship between the root and the coefficient of the quadratic equation of one variable.

Fourth, the guidance of problem-solving methods.

1, the thinking method of observation and analysis

"Observation" and "analysis" are the basic ways of thinking widely used in solving mathematical problems. No matter the solution of quadratic equation, fractional equation and binary quadratic equation, it is inseparable from in-depth observation and analysis.

Observation and Analysis of (1) Solving a Quadratic Equation with One Variable

① To solve the equation with the shape of (x-m)2=n(n≥0), we should choose the direct Kaiping method. First, according to the meaning of the square root, x-m = positive or negative, and then move the term to get the roots of the equation as x 1 = m+, x-m= m-.

② An equation in the form of (x-m)(x-n)=0 can be transformed into x+m=0 or x-n=0 according to the situation that the product of several factors is zero, so at least one of these factors is zero, and the roots of the equation are x 1=m, x2=n m, x2 = ..

(3) In addition to the above two forms, the non-general quadratic equation should be converted into the general form ax2+bx+c=0(a≠0) before being solved. For easy factorization, use factorization method to solve, for difficult factorization, use formula method to solve.

For example, choose an appropriate method to solve the following equation.

⑴(x+2)2=4，⑵3(x+ 1)(-x)=0，⑶3x(x-)= 1，⑷2x2+3x( 1-x)+2=0。

Analysis: By observing and analyzing the equations provided, we can easily draw the following conclusions: (1) It is simpler to use direct leveling method; (2) If the product of two factors is 0, it can be directly written as X 1 =- 1, X2 =；; (3) If it is converted into the general form of 3x2-2x- 1=0, it can be solved by simple factorization; (4) The general form is x2-3x-2=0. Because it is not easy to decompose, it can be solved by formula method.

(2) Observation and analysis of solving fractional equation

"Transformation" is the basic idea of solving complex equations such as fractional order equations and higher order equations. So, how to achieve transformation? This requires us to observe and analyze the formula structure of the fractional equation provided and find a more suitable solution.

The method of transforming fractional equation into integral equation is "denominator removal method", and the operation flow of this method is to multiply both sides of the original equation by the least common multiple of each denominator. However, the transformed equation is sometimes a higher order equation. So far, we haven't found a general method to solve higher-order equations. Therefore, according to the "formula structure" characteristics of the fractional equation, we can choose a special solution-method of substitution. When solving fractional equations with method of substitution, it should be noted that some equations have obvious method of substitution characteristics, while others do not. Appropriate deformation is needed to display the alternative features, such as the following equation:

⑴ ⑵

⑶ ⑷2(x2+)-9(x+)+ 14=0

Analysis: In the above equation, (1) and (1) have the condition of direct substitution, where (1) = can be converted into Y2-Y-2 = 0; ⑵ Set y=, which can be converted into y+;

⑶ Convert 3x2+9x in the equation to 3(x2+3x), and let y=x2+3x, the original equation can be changed to 3y-; (4) Pay attention to the relationship between x2+ and x+. If y=x+, y2=x2+, ∴x2+, the original equation becomes: 2(y2-2)+y+ 14=0.

(3) Observation and analysis of solving binary quadratic equation

The bivariate quadratic equations we study can be divided into two categories: the first category "II-I" refers to an equation composed of a bivariate linear equation and a bivariate quadratic equation, and the general solution of this kind of equation is "substitution elimination method"; the second category "II-II" refers to an equation composed of two bivariate quadratic equations, and the general solution of this kind of equation is "decomposition reduction method". Therefore, when solving the binary quadratic equation, we must carefully observe and analyze the types and characteristics of each equation in the equation, and adopt the strategy of "suit the remedy to the case" to find the most suitable solution. For example, the following equation:

⑴ ⑵

⑶ ⑷

Analysis: (1) belongs to "Ⅱ-Ⅰ" and is solved by substitution elimination method; ⑵ If it belongs to "II-I" type, it can be solved by substitution elimination method, but ⑵ 2 1 equation can be factorized and then solved by "decomposition reduction method" (note that "II-I" type can sometimes be solved by "decomposition reduction method"); (3) If it belongs to "II-II" type, it can be solved by "decomposition and reduction", and of course it can also be "split" into four binary linear equations by bilateral Kaiping method; (4) It belongs to "II-II" type, but only the first equation can be factorized, so it can only be solved by "decomposition reduction" method and can be "split" into two "II-I".

2. The thinking method of analysis and construction.

Vieta theorem method can be used to solve equations, that is, X and Y are regarded as two roots of quadratic equation z2-az+b=0, and the solution of equations can be obtained by solving this quadratic equation. Its strategy is to skillfully construct a univariate quadratic equation according to the formula structure, and then solve the equations by solving this univariate quadratic equation.

Using Vieta's theorem to solve some bivariate quadratic equations with the above special forms can make the problems difficult and simplify them. Observe the following equation:

⑴ ⑵

⑶ ⑷

These equations can be solved by Vieta theorem, some can be directly constructed, and some need to be properly deformed before constructing a quadratic equation.

For example, the system of equations (1) can directly construct the unary quadratic equation z2-7z+12 = 0 with x and y as roots; Firstly, the equation set (2) is transformed into, and then a quadratic equation Z2-11z+18 = 0 is constructed with x and -y as roots. Equation set (3) is firstly transformed into equation Z2-17z+60 = 0 with x and y as roots; Equation set (4), firstly deform and reconstruct equation z2-5z+4=0 with x2 and y2 as roots, or deform or reconstruct equation z2-3z+2=0 with x and y as roots, or z2+3z+2=0.

3. Analysis of comprehensive thinking methods

"Analysis" and "synthesis" are two basic thinking methods, which play a particularly important role in mathematics.

"Analysis" is to break down the whole thing into several parts and examine each part separately; "Synthesis" is to connect all parts of things into a whole and study them as a whole. "Analyze first and then synthesize" is the basic way for people to understand things, and it is also a common means to solve mathematical problems.

Mathematical comprehensive problem can be regarded as a "big problem" composed of several interrelated "small problems". When solving a mathematical comprehensive problem, we should first "analyze" the comprehensive problem-break it down into several interrelated "small problems" and answer them one by one, and then "synthesize" the results obtained from "analysis" to get the comprehensive problem. For example, it is known that two solutions of a system of equations are X 1 and X2 is two unequal positive numbers. (1) Find the range of a; (2) If x12+x22-3x1x2 = 8a2-6a-11,find the value of a.

Analysis: This is a comprehensive problem, which involves not only the equation, but also the discriminant of the roots of the quadratic equation in one variable and the relationship between the roots and the coefficients. Can be broken down into the following three "small topics":

⑴ What kind of quadratic equation can equations be transformed into?

(2) If x 1 and x2 are two transformed quadratic equations, x 1≠x2, find the value range of the parameter to be found;

(3) If x 1 and x2 are two quadratic equations transformed into a variable, x12+x22-3x1x2 = 8a2-6a-11,find the value of the parameter to be found.

As can be seen from the above examples, the process of solving mathematical comprehensive problems is usually a process of "analyzing first and then synthesizing".

A Comprehensive Case Study of verb (abbreviation of verb)

Example1(Beijing senior high school entrance examination in 2003) It is known that the two real roots of the equation x2-2mx+3m=0 about x are x 1, x2 and (x 1-x2)2= 16. If another equation x2-2mx+ about

Analysis: firstly, according to the condition that x2-2mx+3m=0 is two x 1 and x2 satisfies (x 1-x2)2= 16, the value of parameter m is obtained, then the roots of two equations are obtained respectively according to the value of m, and finally a judgment is made.

[Solution] ∫x 1, x2 is the two real roots of the equation x2-2mx+3m=0.

∴x 1+x2=2m,x 1 x2=3m

∫(x 1-x2)2 = 16 ∴(x 1+x2)2-4x 1x2= 16

∴4m2- 12m= 16, the result is m 1=- 1, m2=4.

(1) When m=- 1,

Equation x2-2mx+3m=0 is x2+2x-3=0, then x 1=-3, x2= 1,

Equation x2-2mx+6m-9=0 is x2+2x- 15=0, then x 1 =-5 and x2 = 3.

∫-5,3 is not between -3 and 1 ∴m=- 1 is irrelevant, and it is discarded.

When m=4, the equation x2-2mx+3m=0 is x2-8x+ 12=0.

Then x 1=2, x2=6.

Equation x2-2mx+6m-9=0 is x2-8x+ 15=0.

Then x 1'=3 x2'=5.

∵2 0∴a < 0, then a=- 16 and b=62.

[Comment] (1) This question is a comprehensive one with strong comprehensive skills. Mathematical knowledge such as absolute value, the discriminant theorem of roots, the relationship between roots and coefficients, the sum of internal angles of triangles, Pythagorean theorem, transformation ideas, equation ideas, transformation propositions and so on should be used in solving. Transformation skills are also used in many places, from which we can learn the essence, expand our own problem-solving ideas and improve our problem-solving ability.

(2) In this question, the letters A and B are not the sides of a triangle, so keep in mind when solving the problem.

Example 6 (1998 senior high school entrance examination in Xuzhou City, Jiangsu Province) A tricycle was driving on a straight road. An athlete on a motorcycle shoots at the target every once in a while, and a referee stands in the middle of the road. When the motorcycle came to him, he heard that every two adjacent shots were separated by 5.7s When the motorcycle passed by him, he heard that every two adjacent shots were separated by 6.3s If the speed of sound is 350m/s, what is the speed of the motorcycle per second?

[Analysis] This is a trip problem, involving two aspects of the trip. First, when the motorcycle came to the referee, the time changed; Secondly, when a motorcycle passes behind the referee, it is not difficult to solve this problem if the relationship between the sounds of two adjacent shots is clarified.

[Solution] Suppose that every time an athlete shoots an xs, the speed of the motorcycle is ym/s, which depends on the meaning of the question.

①+② 2x= 12，

∴x=6.

Substitute x=6 into ① to get y= 17.5.

∴

A: The athletes in the car shoot every 6 seconds, and the speed of the motorcycle is17.5 m/s. 。

[Comment] This is a practical system of binary quadratic equations. The difficulty lies in understanding the meaning of the question and listing the equations. When the motorcycle comes to the referee, the sound of two adjacent shots is closer to that of the previous one, so it is rarely used. When the motorcycle passes the referee, the sound of two adjacent shots is farther than the previous one, so it can be heard by the referee. This is a good topic combined with sound propagation in physics.

Key points of learning "quadratic equation of one variable"

First, the knowledge structure of this chapter

1, knowledge structure combing

2. Theorem formula summary

The general form of (1) unary quadratic equation is ax2+bx+c=0(a≠0).

(2) The root formula of unary quadratic equation ax2+bx+c=0(a≠0):

(3) The discriminant of the root of the quadratic equation in one variable: △=b2-4ac.