Knowledge points of quadratic equation in one variable
Teaching emphasis: the correct understanding and application of the discriminant theorem and inverse theorem of roots
Teaching difficulties: the application of the discriminant theorem and inverse theorem of roots.
The key to teaching is to thoroughly understand the discriminant theorem of roots and the conditions for the use of their inverse theorems. Main knowledge points:
One-dimensional quadratic equation
1, unary quadratic equation: An integral equation with an unknown number whose highest degree is 2 is called unary quadratic equation.
2. General form of quadratic equation in one variable: ax2? bx？ c？ 0(a？ 0), which is characterized in that a quadratic polynomial about the unknown quantity X is added to the left 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 a quadratic equation
1, direct Kaiping method:
Using the definition of square root to find the solution of quadratic equation in one variable 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 0, x? ab，xa？ B, when b
2. Matching method:
The theoretical basis of the matching method is the complete square formula a2? 2ab？ b2？ (a？ B)2。 If A in the formula is regarded as an unknown X and replaced by X, there is x2? 2bx？ b2？ (x？ b)2 .
The steps of the matching method: first move the constant term to the right of the equation, then change the coefficient of the quadratic term into 1, and add the square of half the coefficient of 1 at the same time, and finally make a complete square formula.
3. Formula method
Formula method is a method to solve the quadratic equation of one variable by finding the root formula, and it is a general method to solve the quadratic equation of one variable.
One-variable quadratic equation ax2? bx？ c？ 0(a？ 0) Root formula:
xb？ b？ 4ac
2a2(b？ 4ac？ 0) 2
Steps of formula method: substitute the coefficients of quadratic equation in one variable respectively, where the coefficient of quadratic term is a, the coefficient of linear term is b, and the coefficient of constant term is c.
4, factorization method
Factorization is to find the solution of the equation by factorization. This method is simple and easy to use, 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 0, 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 a product form.
5. Vieta Theorem Understanding Vieta Theorem with Vieta Theorem In a quadratic equation with one variable, the sum of two roots =-b/a and the product of two roots =c/a can also be expressed as x 1+x2 =-b/a and x 1x2 = c/a, and the quadratic equation with one variable can be solved by using Vieta Theorem.
Thirdly, the discriminant of the root of a quadratic equation with one variable.
discriminant
One-variable quadratic equation ax2? bx？ c？ 0(a？ 0)、b2？ 4ac is called unary quadratic equation 22ax? bx？ c？ 0(a？ The discriminant of the root of 0) is usually "?" To express, that is, B. 4ac I when △ > 0, a quadratic equation with one variable has two unequal real roots;
II When △=0, the quadratic equation of one variable has two identical real roots;
Three dang △
Fourthly, the relationship between the roots and coefficients of a quadratic equation with one variable.
What if equation ax2? bx？ c？ 0(a？ 0) is x 1, x2, then x 1? x2
x 1x2？ Kabbah, 1000. That is to say, for any unary quadratic equation with real roots, the sum of the two roots is equal to the square.
The reciprocal of the quotient obtained by dividing the coefficient of the first term of the process by the coefficient of the second term; The product of two roots is equal to the quotient obtained by dividing the constant term by the coefficient of quadratic term.
Five, generally solve a quadratic equation, the most commonly used method is factorization. When factorization is applied, the equation is written in a general form and the quadratic coefficient is turned into a positive number. Direct leveling method is the most basic method.
Formula and collocation are the most important methods. Formula method is suitable for any quadratic equation with one variable (some people call it universal method). When using the formula method, in order to determine the coefficient, the original equation must be transformed into a general form, and before using the formula, the value of the discriminant of the root must be calculated to judge whether the equation has a solution.
Matching method is a tool to derive formulas. After mastering the formula method, we can directly use the formula method to solve the quadratic equation of one variable, so we generally don't need to use the matching method to solve the quadratic equation of one variable. However, collocation method is widely used in the study of 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 number and the highest degree of the unknown number is 2 is called a quadratic equation.
Quadratic equation with one variable has four characteristics: (1) contains only one unknown; (2) The sum of the times of the highest unknown term 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, it must be sorted out. If it can be sorted out in the form of AX 2+BX+C = 0 (A ≠ 0), it will be a quadratic equation with one variable. (4) Make the equation into a general form: ax 2.
Basic knowledge explanation:
1. An integral equation with only one unknown and the highest degree of the unknown is 2 is called a quadratic equation with one variable.
That is to say, the unary quadratic equation must satisfy the following three conditions: (1) equation is an integral equation; (2) It contains only one unknown number; (3) The maximum number of unknowns is 2.
2. The general form of unary quadratic equation is: ax2 +bx+c=0(a≠0), and any unary quadratic equation can be simplified 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 knowledge of quadratic equation in one variable.
Interpretation of key and difficult knowledge
Knowledge point 1 the significance of quadratic equation in one variable
An integral equation with only one unknown and the highest degree of the unknown is 2 is called a quadratic equation.
For this definition, we can understand it from the following aspects:
It must be an integral equation.
(2) contains only one unknown number.
(3) After removing brackets, moving items and merging similar items, unknown items will appear at most twice.
Only the equation that satisfies the above three conditions is an unary quadratic equation. For example, x2= 1,
-x2 = x+ 1, (x+ 1) (x-3) = 2, x (x2-1) = x (x+1) (x-2) and so on are all unary quadratic equations;
Knowledge point 2 The general form, quadratic coefficient, linear coefficient and constant term of the quadratic equation of one variable.
Any univariate quadratic equation about X can be transformed into the form of ax2+bx+c=0(d≠0), which is called the general form of univariate quadratic equation, where ax2 is called quadratic term and A is called quadratic term coefficient; Bx is called a linear term, and b is the coefficient of a linear term; C is called a constant term.
In the general form ax2+bx+c=0(a≠0) of the unary quadratic equation, the linear term coefficient b and the constant term c can be arbitrary real numbers, but the quadratic term coefficient a is not equal to zero, because when a = 0, the equation is not an unary quadratic equation. For example, the equations x2 = 0 and x2+x = 0 are both quadratic.
Note: the equation in the form of (1)ax2+bx+c = 0 is not necessarily a quadratic equation. When a≠0, it is a quadratic equation. When a=0 and b≠0, 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 quadratic equation in one variable
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4. What are the knowledge points of a quadratic equation?
Combine parabola graph and analytical formula to understand. The transformation relationship between several forms. The relationship between roots and coefficients.
1. general formula: y = ax 2+bx+c.a > 0, with upward opening, a.
delta=b^2-4ac=a^2(x 1-x2)^2
Greater than 0 means two different real roots (the curve intersects the X axis), equal to 0 means two equal real roots (the curve is tangent to the X axis), and less than 0 means no real roots (the curve does not intersect the X axis).
2.
: y = a (x-h) 2+d.h =-b/(2a), d = c-ah 2 = (4ac-b 2)/(4a), which are directly derived from the general formula.
Vertex is (h, d), a >；; 0 is the minimum value, a.
X=h is the symmetry axis of the curve. If there are two on both sides of the symmetry axis.
Ad<0 has two different real roots, d=0 has two equal real roots, and ad >；; 0 has no real root.
3.
Formula: y=a(x-x 1)(x-x2)
x 1+x2=-b/a，x 1x2=c/a，
The two rules have the same number c/a >; 0, two different numbers are C/A.
Two positive roots-b/a >; 0, the two negative roots are-b/a.
5. Knowledge points of quadratic equation of one variable in junior high school mathematics
Knowledge point 1: the basic concept of quadratic equation in one variable 1. The constant term 3x2+5x-2=0 of a quadratic equation with one variable is -2. 2. The principal term coefficient of the unary quadratic equation 3x2+4x-2=0 is 4, and the constant term is -2. 3. Univariate quadratic equation 3x2-5x-0. The constant term is -7. 4. Transform the equation 3x(x- 1)-2=-4x into the general formula 3x2-x-2=0. Second, the basis for solving the equation-the properties of the equation 1. A = b ←→ A+C = B+C2。 A.
2. Solution of linear equations: ⑴ Basic idea: "elimination" ⑴ Method: ⑴ Substitution method ② Addition and subtraction method 4. The definition and general form of quadratic equation 1 2. Solution: ⑴ Direct Kaiping method (pay attention to features) ⑴ Matching method (pay attention to steps-tapping Formula) Formula method: Factorial decomposition method (features: left = 0) 5. Common equation: 5. Equation 1 that can be transformed into a quadratic equation with one variable. Definition of fractional equation (1) (2) Basic idea: (3) Basic solution: (1) Denominator removal method (2) Substitution method (such as) (4) Root test and method (2) Definition of irrational equation (1) (2) Basic idea: (2) ! (2) substitution method (example), (4) root test and method 3. A simple binary quadratic equation composed of a binary linear equation and a binary quadratic equation can be solved by substitution method.
Solving application problems with column equations (groups) for intransitive verbs-An overview Solving application problems with column equations (groups) is an important aspect of integrating mathematics with practice in middle schools. The specific steps are as follows: (1) Review the questions.
Understand the meaning of the question. Find out what is a known quantity, what is an unknown quantity, and what is the equivalent relationship between problems and problems.
⑵ Set an element (unknown). ① Direct unknowns ② Indirect unknowns (often both).
Generally speaking, the more unknowns, the easier it is to list the equations, but the more difficult it is to solve them. ⑶ Use algebraic expressions containing unknowns to express related quantities.
(4) Find the equation (some are given by the topic, some are related to this topic) and make the equation. Generally speaking, 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 the solutions of practical problems (column equations and writing answers) are caused by the solutions of mathematical problems. In this process, the column equation plays a role of connecting the past with the future.
Therefore, the column equation is the key to solve the application problem. Two commonly used equality relations 1. The basic relationship of travel problem (uniform motion): S = vt (1) encounter problem (simultaneous departure):+=; (2) Catch up with the problem (set out at the same time): If A sets out in t hours, B sets out, and then catches up with A in B, then (3) Sail in the water: 2. Ingredients: Solute = Solution * Concentration Solution = Solute+Solvent 3. The growth rate problem: 4. Engineering problem: basic relationship: workload = work efficiency * working time (often see unit "1").
5. Geometric problems: Pythagorean theorem, area and volume formulas of geometric bodies, similar shapes and related proportional properties. Pay attention to the interaction between language and analytical expressions, such as "many", "few", "increase", "increase to", "simultaneously" and "expand to" ... Another example is a three-digit number, a hundred-digit number, a hundred-digit number, a hundred-digit number and a single-digit number.
Fourth, pay attention to writing equal relations 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.
Another example is that the difference between x and y is 3, so x-y=3. Pay attention to the conversion of units, such as "hours" and "minutes"; Consistency of s, v and t units, etc.
Seven. Application examples (omitted) Chapter VI Key properties and solution summary of unary linear inequalities 1. Definition: a>b, a2. One-dimensional linear inequality: ax>b, ax3. One-dimensional linear inequality group: 4. The essence of inequality: (1) a > b ←→ a+c > b+ c⑵a & gt； b←→AC & gt； (c>0) BC (3) a > b←→ac b，b & gtc→a & gt； ⑸a & gt； b，c & gtd→a+c & gt； B+D.5. Solution of linear inequality of one variable, solution of linear inequality of one variable 6. The solution of one-dimensional linear inequality group, the solution of one-dimensional linear inequality group (representing the solution set on the number axis) 7. Application examples (omitted).