Classic question types for quadratic equations of one variable

Classic question types of quadratic equations:

Example question: In order to attract citizens to organize group tours to Tianshui Bay Scenic Area, Spring and Autumn Travel Agency introduced the charging standard in the dialogue shown in Figure 1.

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A certain unit organized employees to travel to Tianshui Bay Scenic Area, and *** paid 27,000 yuan to Spring and Autumn Travel Agency for travel expenses. May I ask how many employees of this unit went to Tianshui Bay Scenic Area this time?

District tourism?

Explanation: Suppose the unit has x employees traveling to Tianshui Bay Scenic Area this time. Because 1000×25=25000<

27000, the number of employees must exceed 25 People.

According to the meaning of the question, we get [1000-20(x -25)] x = 27000.

After sorting, we get x 2 -75 x +1350=0, solution From this equation, x 1 = 45, x 2 = 30.

When x = 45, 1000-20 (x - 25) = 600 < 700, so x 1 is discarded;

When x 2 = 30, 1000-20 (x -25) = 900>700, which is in line with the meaning of the question.

Answer: This unit *** has 30 employees going to Tianshui Bay this time Scenic Area Tourism.

The idea of ??solving quadratic equations mainly includes the following steps:

1. Understand the equation: First, we need to understand the basic form of quadratic equations. , that is, ax^2 + bx + c = 0. The key to understanding equations is to understand the meaning and role of quadratic terms, linear terms, and constant terms.

2. Observe the situation of the roots: By observing the value of the discriminant of the equation (i.e. b^2 - 4ac), we can judge the situation of the roots of the equation. If the discriminant is greater than 0, the equation has two different real roots; if the discriminant is equal to 0, the equation has two identical real roots; if the discriminant is less than 0, the equation has no real roots.

3. Choose the appropriate solution method: There are three main methods for solving quadratic equations of one variable: the combination method, the formula method and the factoring method. Which method to choose mainly depends on the characteristics of the equation and the needs of the actual problem.

4. Practical applications: Quadratic equations are widely used in solving practical problems, such as parabolic motion in physics, growth rate problems in economics, and periodic phenomena in engineering. wait. By converting practical problems into mathematical models, we can use quadratic equations to analyze and solve problems.