"The relationship between roots and coefficients of quadratic equations" PPT courseware 2

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"The relationship between roots and coefficients of quadratic equations" PPT courseware 2

remember

1. What is the general form of a quadratic equation?

ax2+bx+c=0（a≠0）

2. What is the formula for finding the root of a quadratic equation?

x=-b±√b²-4ac/2a(b²-4ac≥0

3. How to determine the roots of a quadratic equation?

△=b²-4ac

△> 0, the equation has two unequal real roots,

△= 0, there are two equal real roots,

△<0, there are no real roots.

If the two roots of the quadratic equation ax2+bx+c=0 are x1 and x2, then:

x1+x2=-b/a x1·x2=c/a

This is the relationship between the roots and coefficients of a quadratic equation, also called Vedic theorem.

1. It is known that the two roots of x²-2x-1=0 of the quadratic equation are x1x2, then: x1+x2=___,x1·x2____

2. It is known that the two roots of 3x²+x=6 of the quadratic equation are x1x2, then: x1+x2=___,x1·x2____

3. It is known that one root of the quadratic equation 3x²-9x+m=0 is 1, then the other root of the equation is ___, m=___:

4. It is known that the two roots of x²+px+q=0 of the quadratic equation are -2 and 1 respectively, then: p =__; q=__

basic exercises

1. If -1 is a root of the equation 2X2-X+m=0, then the other root is ___, m =____.

2. Suppose X1 and X2 are the two roots of the equation X2-4X+1=0, then

X1+X2 = ___,X1X2 = ____,

X12+X22 = (X1+X2)2 - ___ = ___

(X1-X2)2 = ( ___ )2 - 4X1X2 = ___

3. Determine whether it is right or wrong:

The equation with 2 and -3 as roots is X2-X-6=0 ( )

4. It is known that the sum of two numbers is 1 and the product is -2, then the two numbers are _____.

Extension: 1. If ax2+bx+c=0 (a≠0 △≥0)

(1) If the two roots are opposite numbers to each other, then b=0;

(2) If the two roots are reciprocals of each other, then a=c;

(3) If one root is 0, then c=0;

(4) If one root is 1, then a+b+c=0;

(5) If one root is -1, then a-b+c=0;

(6) If a and c have different signs, the equation must have two real roots.

Summary

1. What is the relationship between the roots and coefficients of a quadratic equation?

2. When applying the relationship between the roots and coefficients of a quadratic equation, the known equation must first be transformed into a general form.

3. When applying the relationship between the roots and coefficients of a quadratic equation of one variable, special attention should be paid to the condition that the equation has real roots, that is, in junior high school algebra, the relationship between the roots and coefficients can be applied if and only if b²-4ac≥0.

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For more information about the "Relationship between Roots and Coefficients of Quadratic Equations" PPT courseware, please click on the PPT tag "Relationship between Roots and Coefficients of Quadratic Equations".

"The relationship between roots and coefficients of quadratic equations" PPT courseware 3:

"The Relationship between Roots and Coefficients of Quadratic Equations" PPT Courseware 3 If the two roots of ax2+bx+c=0 (a0) are x1 and x2, then x1+x2=________, x1x2=________. Two conditions should be noted when applying the relationship between roots and coefficients: (1)____________..

"The relationship between roots and coefficients of quadratic equations" PPT courseware:

"The relationship between roots and coefficients of quadratic equations" PPT courseware 1. Fill in the table equation x1x2 x1+x2 x1x2 ① x2-3x+2=0 21 3 2 ② X2-2x-3=0 -13 2 -3 ③ X2- 5x+4=0 14 5 4 Question: Did you find that the roots and coefficients of these quadratic equations have...

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Update Time: 2024-07-29

This template belongs to Mathematics courseware Qingdao Edition Ninth Grade Mathematics Volume 1 industry PPT template

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