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"The relationship between roots and coefficients of quadratic equations" PPT courseware
1. Fill in the form
Equation x1,x2 x1+x2 x1·x2
① x2-3x+2=0 2,1 3 2
② X2-2x-3=0 -1,3 2 -3
③ X2-5x+4=0 1,4 5 4
Question: Did you find the roots and coefficients of these quadratic equations?
What are the rules?
When the coefficient of the quadratic term is 1
The two roots of x2+px+q=0 are x1,, x2
Then we have x1+x2=-P x1·x2=q
guess:
If the quadratic equation
The two roots of ax2+bx+c=0 (a, b, c are constants and a=0 b²-4ac≥0) are x1 and x2, then
The relationship between x1+x2 x1·x2 and coefficients a, b, c.
x1+x2=-b/a x1·x2=c/a
Independent practice and flexible use
1. What is the sum of the two roots and the product of the two roots of the following equations? (Do not solve the equation)
(1)x2-3x+1=0
(2)3x2-2x=2
(3)2x2+3x=0
(4)3x2=1
2. Assume x1.x2 are the two roots of the equation 2x2+4x-3=0. Use the relationship between the roots and coefficients to find the values of the following formulas.
(1) (x1+1) (x2+1) (2) x2/x1 + x1/x2
Example 2:
(1) If one root of the equation 2x2+5x+n=0 about x is -2, find its other root and the value of n.
(2) If one root of the equation x2+kx-6=0 about x is -2, find its other root and the value of k.
1. Key points of knowledge:
1. The general form of a quadratic equation is ax2+bx+c=0 (a≠0).
2. If the two roots of the quadratic equation ax2+bx+c=0 (a≠0) are x1 and x2 respectively, then x1+x2=-b/a, x1x2=c/a.
3. The conditions for solving problems using the relationship between roots and coefficients are (1) a≠0 (2) △≥0
2. Typical examples
Example 1: It is known that the two roots of the equation x2=2x+1 are x1 and x2,
Without solving the equation, find the values of the following equations.
(1) (x1－x2)2 (2)x13x2＋x1x23
Example 3:
Assume x1 and x2 are the two roots of the equation 2x2-3x+m=0,
And 8x1-2x2=7, find the value of m.
Example 4:
It is known that the quadratic equation x2+(2k-1)x+k2=0 about x
There are two unequal real roots, and the sum of the two roots of the equation is greater than the product of the two roots. Find the value of k.
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"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 2:
"The Relationship between the Roots and Coefficients of a Quadratic Equation" PPT Courseware 2 Recall 1. What is the general form of a quadratic equation? ax2+bx+c=0 (a0) 2. What is the formula for finding the root of a quadratic equation? x=-bb-4ac/2a(b-4ac0 3. The situation of the roots of the quadratic equation...