☆ Summary ☆
I. Basic concepts
1. equation, its solution (root), its solution, its solution (group)
2. Classification:
Second, the basis of solving equation-the nature of equation
1.a=b←→a+c=b+c
2.a=b←→ac=bc (c≠0)
Third, the solution
1。 Solution of linear equation with one variable: removing denominator → removing brackets → moving terms → merging similar terms →
The coefficient becomes 1→ solution.
2. Solution of linear equations: ① Basic idea: "elimination method" ② Method: ① Replacement method.
② addition and subtraction
Fourth, a quadratic equation
1。 Definition and general form:
2。 Solution: (1) direct leveling method (pay attention to characteristics)
(2) Matching method (pay attention to the step-inferring the root formula)
(3) Formula method:
(4) factorization method (feature: left =0)
3。 Discriminant of roots:
4。 The relationship between root and coefficient top;
Inverse theorem: If, then the quadratic equation with one root is:.
5。 Common equation:
5. Equations that can be transformed into quadratic equations
1。 fractional equation
(1) definition
(2) Basic ideas:
⑶ Basic solution: ① Denominator removal ② Substitution method (such as).
(4) Root test and method
2。 irrational equation
(1) definition
(2) Basic ideas:
(3) Basic solution: ① Multiplication method (pay attention to skills! ! (2) substitution method (example), (4) root test and method.
3。 Simple bivariate quadratic equation
A binary quadratic equation consisting of a binary linear equation and a binary quadratic equation can be solved by method of substitution.
Six, column equation (group) to solve application problems
summary
Solving practical problems by using 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. Travel problem (uniform motion)
Basic relationship: s=vt
(1) Meeting problem (at the same time):
+ = ;
(2) Follow-up questions (start at the same time):
If A starts in t hours, B starts, and then catches up with A at B, then
(3) sailing in the water:
2. batching problem: solute = solution × concentration
Solution = solute+solvent
3。 Growth rate problem:
4。 Engineering problem: basic relationship: workload = working efficiency × working time (workload is often considered as "1").
5。 Geometric problems: Pythagorean theorem, area and volume formulas of geometric bodies, similar shapes and related proportional properties.
Third, pay attention to the relationship between language and analytical formula.
Such as more, less, increase, increase to (to), at the same time, expand to (to), ...
Another example is a three-digit number, where A has 100 digits, B has 10 digits and C has one digit. Then this three-digit number is: 100a+ 10b+c, not abc.
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, and if the difference between X and Y is 3, then x-y=3. Pay attention to unit conversion
Such as the conversion of "hours" and "minutes"; Consistency of s, v and t units, etc.