"Application of Quadratic Equations of One Variable" PPT courseware

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"Application of Quadratic Equations of One Variable" PPT courseware

What are the steps to solve word problems using equations?

(1) Review: refers to understanding the question, clarifying the meaning of the question, clarifying which quantities are known, which are unknown, and what the question requires;

(2) Find: Find the equivalence relationship, that is, the equality relationship given in the question that can express the full meaning of the word problem;

(3) Assume: refers to assuming elements, that is, assuming unknown numbers;

(4) Column: It is a column of equations. According to the equation of equations and the algebraic expression of the equation, each quantity in the equation is expressed, and then an equation containing unknown numbers is obtained, that is, an equation;

(5) Solution: It is to solve the equation and find the value of the unknown;

(6) Test: When formulating an equation to solve a word problem, the unknowns must be tested. The purpose of the test is twofold: first, to test whether the value of the unknown satisfies the equation; second, to test the value of the unknown. Do the values ​​of the unknowns meet the requirements of the actual problem? The values ​​of the unknowns that are suitable for the equation but not suitable for the actual problem should be discarded;

(7) Answer: Just write the answer, and be careful not to omit the name of the unit when writing.

(1) Problems of area and length in geometry

Example 1: As shown in the figure, there is a rectangular open space with one side against a wall. The wall is 30m long and the other three sides are surrounded by a 35m long wire mesh. It is known that the area of ​​the rectangular open space is 125m2. Find the length and width of the rectangular open space.

Analysis: According to the formula for the area of ​​a rectangle, use length × width = 125 to list the equation to get the answer. The length of the wall of 30m is not directly used in the equation, but when checking the results, it should be noted that the length of one side of the rectangle parallel to the wall cannot exceed 30m, otherwise, the wall will have no value as a chicken farm.

Example 2 As shown in the figure, a 10 m long ladder is leaning against the wall. The distance from the top of the ladder A to the ground is 8 m. If the top of the ladder slides 2 m along the wall, then the bottom of the ladder How far does it slide on the ground?

summary

1. Solution two is simpler than solution one. It uses the principle that "the area of ​​the graph will not change after it is moved". Moving the vertical and horizontal roads can make the equation easier (the purpose is to find the road surface width, as for the actual construction, the road can still be built according to the location of the original map).

2. When some students make equations to solve word problems, they often keep the positive solutions when they see them and discard them when they see the negative solutions. In fact, even the correct solution must be tested according to the conditions of the question, and it should be discarded. In this question, you must pay attention to the condition that the original rectangle is "width is 20 m and length is 32 m", so as to make the correct choice.

Summarize

To solve such problems, one must have good knowledge of geometric concepts and be familiar with formulas such as length, area, volume, etc.

Sometimes it is necessary to solve the problem through translation.

Frequently Asked Questions: How wide to dig a trench, making a box.

(2) Numbers and equations

1. A two-digit number, its tens digit is 3 less than the ones digit, and the square of its ones digit is exactly equal to this two-digit number. Find this two-digit number.

2. There is a two-digit number, and the sum of its tens digits and ones digits is 5. After exchanging the tens digits and ones digits of this two-digit number, we get another two-digit number, two two-digit numbers. The product of is 736. Find the original two-digit number.

(3) Growth rate issue

Example 1: According to the requirements of the "Ninth Five-Year Plan" national economic development plan in Binh Duong, the total social output value in 2003 increased by 21% compared with 2001. Find the average annual growth percentage. (Tip: The base is the total social output value in 2001, which can be regarded as a)

Example 2. In order to solve the problem of citizens’ difficulty in seeing a doctor, a certain city decided to lower the price of medicines. After two consecutive price cuts of a certain drug, the price was reduced from 200 yuan per box to 128 yuan. What is the average price reduction percentage of this drug each time?

Summarize

1. The basic quantitative relationship in the average growth rate problem is

A(1+X)n=B(A is the starting amount, B is the ending amount, n is the number of times of growth, x is the average growth rate)

Similarly, the basic quantitative relationship in the average reduction rate problem is A (1-X) n = B (A is the starting amount, B is the ending amount, n is the number of reductions, and x is the average reduction rate)

2. Regarding the "growth rate" issue, such as population reduction, interest rate reduction, car depreciation, etc., they all decrease from the original base and cannot be confused with general increases and decreases.

practise:

1. The total output of a factory in January this year was 500 tons, and the total output in March was 720 tons. The average monthly growth rate is x, and the equation is ( )

A.500(1+2x)=720 B.500(1+x)2=720 C.500(1+x2)=720 D.720(1+x)2=500

2. A school’s investment in experimental equipment last year was 20,000 yuan, and the total investment this year and next is expected to be 80,000 yuan. If the average growth rate of the school’s investment in experimental equipment this year and next is x, then the equation can be formulated for___________.

[Method summary]

1. Solving practical problems by making equations is generally divided into six steps: reviewing the question, finding equivalent relations, assuming unknowns, making equations, solving the equation, testing, and writing the answer. Although the question review process is carried out on draft paper, this step is very important. Importantly, only by carefully reviewing the questions, distinguishing the known conditions and required quantities, and clarifying the quantitative relationship between quantities, can we accurately find out the equality relationship and list the equations.

2. When formulating quadratic equations to solve practical problems, we should also pay attention to some key words, such as "more", "times", "difference", "ahead of time", "at the same time", "early", "late", "increase" "How many times" etc.

3. When solving complex problems, we can use auxiliary methods such as tables to clarify the quantitative relationships in the problem and list equations.

4. Quadratic equations are an effective model for solving many problems in our daily lives. We must be good at using this mathematical model to solve various problems in real life, and pay attention to interpretation and testing based on practical significance. Understand the thinking methods of mathematical modeling.

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Update Time: 2024-10-18

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