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Authoritative PPT Summary
"Graphics of quadratic functions and quadratic equations of one variable" PPT courseware
learning target
1. Explore the relationship between the abscissa of the intersection of the parabola and the x-axis and the roots of a quadratic equation, and understand the close relationship between equations and functions;
2. Learn to use the image method to find approximate roots of quadratic equations of one variable;
Observation and Thinking (1)
Observe the parabola y=x²-2x-3 and think about the following questions:
(1) How many common points do the parabola and the x-axis have? What are the coordinates of the common points?
The parabola and the x-axis have two common points (-1,0), (3,0).
(2) When x takes what value, the value of function y=x²-2x-3 is 0?
When x=-1,x=3, the value of function y is 0. That is, x²-2x-3=0.
(3) Does the quadratic equation x²-2x-3=0 have roots? If it has roots, what are its roots?
The roots of the quadratic equation x²-2x-3=0 are x1=-1, x2=3,
(4) What is the relationship between the root of the quadratic equation x²-2x-3=0 and the abscissa of the common point of the parabola y=x²-2x-3 and the x-axis?
Example 1 Use the graphical method to discuss the roots of the quadratic equation x²-3x-2=0 (accurate to 0.1)
(1) Draw a parabola y=x²-3x-2.
(2) It can be seen from the image that there is a root between -1 and 0 and between 3 and 4.
Calculate the function values of x=0, x=-1, and x=-0.5 respectively. The list is as follows:
Since when x=-1, y>0, when x=-0.5, y<0, the roots of the equation are between -1 and -0.5.
Example 2 Use the graphical method to discuss the roots of the quadratic equation x²-2x+3=0.
(1) Draw the parabola y=x²-2x+3
(2) Since the graph has no common point with the x-axis, the quadratic equation x²-2x+3=0 has no real roots.
Discriminant of Roots of Quadratic Equation
For a quadratic equation of one variable
ax²+bx+c=0 (a, b, c are constants, a≠0), ①
Since the number of roots of a quadratic equation is determined by the sign of the algebraic expression b²-4ac, b²-4ac is called the discriminant of the roots of a quadratic equation, usually represented by the Greek letter △, that is, △=b²-4ac
Specifically, there are three situations for the roots of quadratic equations of one variable:
(1) When △>0, equation ① has two unequal real roots;
(2) When △=0, equation ① has two equal real roots;
(3) When △<0, equation ① has no real roots.
On-site inspection:
1. The two roots of the quadratic equation x²+x-6=0 are x1=-3 and x2=2. Then the coordinates of the graph of the quadratic function y=x²+x-6 and the common point on the x-axis are _______.
2. If the quadratic equation x²-2x+m=0 about x has two equal real roots, then m=_______. At this time, the parabola y=x²-2x+m and the x-axis have _______ common points .
3. Use the graphical method to discuss the roots of the quadratic equation 3/4x²-3x+3=0.
4. Use the graphical method to discuss the roots of the quadratic equation 1/2x²-4x+3=0 (accurate to 0.1).
Homework assignment:
(1) Exercise 5.9 The second and third questions
(2) What we are learning today is to use the graphical method to find approximate solutions to quadratic equations of one variable. It uses the mathematical ideas of combining numbers and shapes and approximation. It is similar to the dichotomy method to find approximate solutions to equations in the field of mathematics. For those who are interested in this class You can check the information online to understand what is the dichotomy?
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"Graphics of quadratic functions and quadratic equations of one variable" PPT courseware 2:
"Graphics of Quadratic Functions and Quadratic Equations" PPT Courseware 2 Learning Objectives 1. Experience the process of exploring the relationship between quadratic functions and quadratic equations, and understand the connection between equations and functions; 2. Use the image method to find Approximate roots of quadratic equations. New lesson introduction questions..