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Authoritative PPT Summary
"Quadratic Equation of One Variable" PPT courseware 6
teaching objectives:
1. Experience and explore the concept of quadratic equations, and understand the quadratic terms, linear terms, and constant terms in quadratic equations.
2. Understand the general form of quadratic equations and convert quadratic equations into general forms.
3. Cultivate students' awareness of active participation and cooperation, and improve their self-confidence in learning mathematics.
Do it
The area of a football field is 7140m2 and the perimeter is 346m. Find the length and width of the football field.
As shown in the figure, the three sides of a right triangle are all integers. The length of its hypotenuse is 11cm. The difference between the two right-angled sides is 7cm. Find the length of the two right-angled sides.
collaborative inquiry
1. If the above equation is deformed, that is, analogous to the general form of a linear equation of one variable ax+b=o(a≠0), how do you think the equation will be deformed?
2. Can you organize these equations into a form where the right side is 0 and the left side is arranged in descending powers of x?
Observe what the three equations have in common?
Review and Thoughts
From the above three questions, we can get three equations:
(173-x)x=7140 i.e. x2 - 173x +7140 = 0.
x2＋(x+7)2＝112 That is, x2 +7x -36＝0.
x2=1-x i.e. x2 +x -1 =0.
What do the above three equations have in common?
The above equations are all integral equations containing only one unknown number x, and they can be transformed into the form ax2+bx+c=0 (a, b, c are constants, a≠0). Such equations are called quadratic equations of one variable.
Let ax2+bx+c=0 (a, b, c are constants, a≠0) called the general form of a quadratic equation, where ax2, bx, c are called quadratic terms, linear terms and constant terms respectively, a, b are respectively They are called quadratic term coefficients and linear term coefficients.
Explore and think
Which of the following equations are quadratic equations of one variable?
(1)7x2－6x＝0
(2)2x2－5xy＋6y＝0
(3)2x2－1/3x－1＝0
(4)y2 =0
(5)x2+2x-3＝1+x2
connotation and denotation
1. The equation about x (k-3)x2 + 2x-1=0. When k≠3, it is a quadratic equation of one variable.
2. The equation about x (k2-1)x2 + 2 (k-1) x + 2k + 2=0. When k≠ 1, it is a quadratic equation of one variable. , when k=-1, it is a linear equation of one variable.
Practice in class
1. Once upon a time, a drunkard entered the house with a bamboo pole. He couldn't get in even if he held it horizontally or vertically. It was 4 feet wider than the door frame horizontally and 2 feet taller than the door frame vertically. Another drunkard taught him to follow the two opposite sides of the door. Holding the pole at an angle, the drunkard tried it and got in just right. Do you know how long the bamboo pole is? Please formulate an equation based on this problem.
Solution: Suppose the length of the bamboo pole is x feet, then the width of the door is (x-4) feet and the length is (x-2) feet. According to the question, the equation is obtained:
(x-4)2+ (x-2)2＝x2
That is, x2-12 x +20 = 0
2. Convert the equation (3x+2)2=4(x-3)2 into the general form of a quadratic equation, and write down its quadratic term coefficients, linear term coefficients and constant terms.
Solution: Simplify the original equation to:
9x2+12x+4＝4(x2－6x+9)
9x2+12x+4＝4x2-24x+36
9x2-4x2+12x+24x+4-36=0
5x2＋36x－32＝0
Summary Expand
What new knowledge did you learn in this lesson?
1. Learn what a quadratic equation is, its general form ax2+bx+c=0 (a, b, c are constants, a≠0) and related concepts, such as quadratic terms, linear terms, constant terms, quadratic term coefficients, First term coefficient.
2. Be able to use quadratic equations to express quantitative relationships in real life
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