knowledge points of quadratic equation in one variable

Teaching emphasis: correct understanding and application of discriminant theorem and inverse theorem of roots

Teaching difficulty: application of discriminant theorem and inverse theorem of roots.

teaching key: a thorough understanding of the discriminant theorem of roots and the conditions for using its inverse theorem. Main knowledge points:

1. One-dimensional quadratic equation

1. One-dimensional quadratic equation: An integral equation with an unknown number and the highest degree of the unknown number is 2 is called one-dimensional quadratic equation.

2. General form of quadratic equation in one variable: ax2? bx? c? (a? ), which is characterized in that a quadratic polynomial about the unknown x is added to the left side of the equation, and the right side of the equation is zero, where ax2 is called quadratic term and A is called quadratic term coefficient; Bx is called a linear term, and b is called a linear term coefficient; C is called a constant term.

Second, the solution of the quadratic equation in one variable

1. Direct Kaiping method:

The method of directly squaring the quadratic equation in one variable by using the definition of square root is called direct Kaiping method. The direct Kaiping method is suitable for solving shapes such as (x? a）2? B's unary quadratic equation. According to the definition of square root, x? A is the square root of b, when b? At , x? a? b,x? a? B, when b <; , the equation has no real root.

2. Matching method:

The theoretical basis of matching method is complete square formula a2? 2ab? b2? （a? B)2. If A in the formula is regarded as the unknown X and replaced by X, there is x2? 2bx? b2? （x? b）2。

steps of matching method: first, move the constant term to the right of the equation, then turn the coefficient of the quadratic term into 1, and add the square of half the coefficient of the quadratic term at the same time, and finally get the complete square formula

3. Formula method

Formula method is a method to solve the quadratic equation of one variable by finding the root formula, which is a general method to solve the quadratic equation of one variable.

unary quadratic equation ax2? bx? c? (a? ):

x? b? b? 4ac

2a2(b? 4ac? ) 2

Steps of formula method: Substitute the coefficients of the quadratic equation of one variable respectively, where the coefficient of the quadratic term is a, the coefficient of the primary term is b, and the coefficient of the constant term is c

4. Factorization method

Factorization method is a method to find the solution of the equation by means of factorization. This method is simple and easy, and it is the most commonly used method to solve the quadratic equation of one variable.

steps of factorization: turn the right side of the equation into , and then see if you can extract the common factor, formula method (here refers to the formula method in factorization) or cross multiplication, and if you can, you can turn it into the form of product

5. vieta theorem used vieta theorem to understand that vieta theorem is in a quadratic equation with one variable, and the sum of two roots is =-b/a, and the product of two roots is = Using vieta theorem, we can find out the coefficients in the quadratic equation of one variable. In the topic, we often use

discriminant of roots of cubic and quadratic equations of one variable

discriminant of roots

quadratic equation of one variable ax2? bx? c? (a? ), b2? 4ac is called unary quadratic equation 22ax? bx? c? (a? The discriminant of the root of ) is usually "?" To express, that is? b? 4ac I when △> , the unary quadratic equation has two unequal real roots;

II when △=, a quadratic equation with one variable has two identical real roots;

III when △ <; , the univariate quadratic equation has no real root

The relationship between roots and coefficients of the univariate quadratic equation

If the equation ax2? bx? c? (a? The two real roots of ) are x1 and x2, so x1? x2?

x1x2? caba，。 That is to say, for any unary quadratic equation with real roots, the sum of the two roots is equal to the inverse of the quotient obtained by dividing the coefficient of the primary term of the equation with square

by the coefficient of the secondary term; The product of two roots is equal to the quotient obtained by dividing the constant term by the coefficient of quadratic term.

5. In general, factorization is the most commonly used method to solve a quadratic equation with one variable. When applying factorization, the equation should be written in a general form first, and the quadratic term coefficient should be turned into a positive number. Direct Kaiping method is the most basic method.

Formula method and matching method are the most important methods. The formula method is suitable for any quadratic equation with one variable (some people call it universal method). When using the formula method, we must turn the original equation into a general form in order to determine the coefficient, and before using the formula, we should first calculate the value of the discriminant of the root in order to judge whether the equation has a solution.

The collocation method is a tool to derive formulas. After mastering the formula method, you can directly use the formula method to solve the quadratic equation of one variable, so it is generally not necessary to use the collocation method to solve the quadratic equation of one variable. However, matching method is widely used in learning other mathematical knowledge, and it is one of the three important mathematical methods required to be mastered in junior high school, so we must master it well. Three important mathematical methods: method of substitution, collocation method and undetermined coefficient method.

2. knowledge points of quadratic equation with one variable

definition: in an equation, an integral equation with only one unknown and the highest degree of the unknown is 2 is called quadratic equation with one variable.

The quadratic equation with one variable has four characteristics: (1) It contains only one unknown; (2) The sum of the times of the highest term of the unknown is 2; (3) It is an integral equation. To judge whether an equation is an unary quadratic equation, we should first look at whether it is an integral equation. If it is, we should sort it out. If it can be sorted out in the form of AX 2+BX+C = (a ≠ ), it will be an unary quadratic equation. (4) Make the equation into a general form: AX 2.

that is, an unary quadratic equation must meet the following three conditions: (1) the equation is an integral equation; (2) It contains only one unknown number; (3) The highest number of unknowns is 2.

2. The general form of the unary quadratic equation is: ax2 +bx+c=(a≠), and any unary quadratic equation can be reduced to a general form, in which ax2 is called quadratic term, A is called quadratic term coefficient, bx is called linear term, B is called linear term coefficient, and C is called constant term.

3. Detailed explanation of the key and difficult points of the unary quadratic equation

Interpretation of the key and difficult points

Knowledge point 1 The meaning of the unary quadratic equation

An integral equation with only one unknown and the highest degree of the unknown is 2 is called the unary quadratic equation.

For this definition, We can understand it from the following aspects:

(1) It must be an integral equation.

(2) It contains only one unknown.

(3) After removing brackets, shifting items and merging similar items, the maximum number of times of containing unknown items is 2.

An equation is a quadratic equation with one variable only when the above three conditions are met at the same time.

knowledge point 2: the general form of the unary quadratic equation and the quadratic term coefficient, the linear term coefficient and the constant term

any unary quadratic equation about x can be changed into the form of: ax2+bx+c=(d≠), which is called the general form of the unary quadratic equation. Ax2 is called the quadratic term, and A. Bx is called a linear term, and b is a linear term coefficient; C is called a constant term.

In the general form of unary quadratic equation ax2+bx+c=(a≠), the linear term coefficient B and the constant term C can be any real number, but the quadratic term coefficient A is a real number not equal to zero, because when a=o, the equation is not an unary quadratic equation. For example, the equation X2 = , X2. When a= and b≠, it is a linear equation.

(2) When writing quadratic coefficient, linear coefficient and constant term, don't leave out the previous symbols.

Knowledge point 3 Solution of linear quadratic equation

You can see the details://munication/XDetailx? Id=1998

4. What are the knowledge points of quadratic equation with one variable

Understand it by combining parabola graph and analytical formula. The transformation relationship between several forms. The relationship between roots and coefficients.

1. general formula: y = ax 2+bx+c.a > , the opening is upward, a <;

delta = b 2-4ac = a 2 (x1-x2) 2

greater than , 2 different real roots (the curve intersects the X axis), equal to , 2 equal real roots (the curve is tangent to the X axis), and less than , no real roots (the curve has no intersection with the X axis).

2.

: y = a (x-h) 2+d.h =-b/(2a), d = c-ah 2 = (4ac-b 2)/(4a), which is derived from the general formula directly.

the vertex is (h, d), a >; is the minimum value, a <; is the maximum value

x=h is the symmetry axis of the curve. If there are two

ad <: has 2 different real roots, d= has 2 equal real roots, ad >; has no real root.

3.

formula: y = a (x-x1) (x-x2)

x1+x2 =-b/a, x1x2 = c/a,

two identical numbers are c/a> , two different numbers are c/a <;

two positive roots -b/a> , two negative roots are-b/a <;

5. Knowledge points of junior high school mathematics quadratic equation

1. Basic concepts of quadratic equation of one variable 1. Constant term of quadratic equation of one variable 3x2+5x-2= is -2. 2. The coefficient of the primary term of quadratic equation of one variable 3x2+4x-2= is 4, and the constant term is -2. 3. The constant term is -7. 4. Transform the equation 3x(x-1)-2=-4x into the general formula 3x2-x-2 = .2. The basis for solving the equation-the property of the equation 1. A = b ←→ A+C = b+C2. A = b ←→ AC =

2. Solution of linear equation system: ⑴ Basic idea: "elimination" ⑴ Method: ⑴ Substitution ② Addition and subtraction ⑴ Quadratic equation of one variable 1. Definition and general form: 2. Solution: ⑴ Direct Kaiping method (pay attention to features) ⑴ Matching method (pay attention to steps-knocking down the root formula) 𖲔Formula method: 𖲔Factorization method (features: left = ). 5. Common equations: 5. Equation that can be transformed into a quadratic equation with one variable 1. Fractional equation 1. Definition 2. Basic idea: 3. Basic solution: 1. Denominator removal 2. method of substitution (such as) 4. Root checking and method 2. Unreasonable equation 1. Definition 2. Basic idea: 3. Basic solution: 1. Multiplication method (pay attention to skills! ! (2) method of substitution (example,) (4) Root test and method 3. Simple binary quadratic equations, which consist of a binary linear equation and a binary quadratic equation, can be solved by substitution method.

VI. Solving application problems with equations (groups) An overview of solving application problems with equations (groups) is an important aspect of integrating mathematics with practice in middle schools. The specific steps are as follows: (1) Examining questions.

understand the meaning of the question. Find out what is the known quantity, what is the unknown quantity, and what is the equal relationship between the problem and the problem.

(2) the argument (unknown). ① Direct unknowns ② Indirect unknowns (often both).

generally speaking, the more unknowns, the easier the equation is to list, but the more difficult it is to solve. ⑶ Use algebraic expressions containing unknowns to express related quantities.

(4) Find the equality relation (some are given by the topic, others are given by the equality relation involved in this problem), and list the equations. Generally, the number of unknowns is the same as the number of equations.

(5) Solving equations and testing. [6] answer.

to sum up, the essence of solving application problems by column equations (groups) is to first transform practical problems into mathematical problems (setting elements and column equations), and then solve practical problems (column equations and writing answers) by solving mathematical problems. In this process, the column equation plays the role of connecting the past with the future.

Therefore, the column equation is the key to solving practical problems. Second, the commonly used equality relationship 1. Travel problem (uniform motion) Basic relationship: S = vt (1) Meeting problem (starting at the same time):+=; (2) Catch up with the problem (start at the same time): If A starts t hours later, B starts and then catches up with A at B, then (3) sail in the water: 2. batching problem: solute = solution * concentration solution = solute+solvent 3. growth rate problem: 4. engineering problem: basic relationship: workload = work efficiency * working time (often looking at workload as unit "1").

5. Geometric problems: Pythagorean theorem in common use, area and volume formulas of geometric bodies, similar shapes and related proportional properties, etc. Pay attention to the interaction between language and analytical expressions, such as "more", "less", "increased", "increased to", "at the same time", "expanded to", ... Another example is a three-digit number, with a hundred digits of A, a hundred digits of B and a single digit of C.

Fourth, pay attention to writing the equality relationship from the language narrative. For example, if X is greater than Y by 3, then x-y=3 or x=y+3 or X-3 = Y.

as another example, if the difference between x and y is 3, then x-y=3. Pay attention to the conversion of units, such as "hours" and "minutes"; The consistency of s, v and t units, etc.

VII. Application Examples (omitted) Chapter VI One-dimensional linear inequalities (groups) Focus on the properties and solutions of one-dimensional linear inequalities 1. Definition: a> B, a2. One-dimensional linear inequality: ax> B, ax3. One-dimensional linear inequality group: 4. The nature of inequality: (1) a > b←→a+c> b+c ⑵a> b←→ac> bc(c> ) ⑶a> b←→ac b,b> c→a> c ⑸a> b,c> d→a+c> B+D.5. The solution of one-dimensional linear inequality and the solution of one-dimensional linear inequality.