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"Discriminant of Roots of Quadratic Equation" PPT courseware
A. Introduced by solving the equation:
Solve the equation: ①x²+x-1=0 b²-4ac=1+4=5
②x²-4x+4=0 b²-4ac=16-16=0
③2x²+3x+4=0 b²-4ac=9-32＜0 This equation has no real roots,
It can be seen that the roots of the equation can be judged from the value of b²-4ac.
B. New lesson:
1. Discriminant
1. The discriminant of the roots of the equation ax²+bx+c=0(a≠0) is: △=b²-4ac.
(1) △=b²-4ac>0 The equation has two unequal real roots
(2) △=b²-4ac=0 The equation has two equal real roots
(3) △=b²-4ac<0 The equation has no real roots.
2. Application of discriminant
(1) Directly determine the roots of a quadratic equation of one variable;
(2) Based on the roots of the quadratic equation given in the question, find the values or value ranges of the undetermined coefficients in a, b, and c.
2. Root relationship
1. The relationship between the two roots x1, x2 and the coefficients a, b, c of the equation ax2+bx+c=0 (a≠0, b2-4ac≥0) about x is: x1+x2=-b/a; x1·x2=c/a
Note: The prerequisite for applying the root system relationship is a≠0 and △≥0
2. Application of root system relationship:
(1) Given one root of the equation, find the value of the other root and the letter coefficient.
(2) Knowing the relationship between the two roots, determine the value of the letter coefficient in the equation.
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.
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"Discriminant of Roots of Quadratic Equations" PPT Courseware 2:
"Discriminant of the Roots of a Quadratic Equation" PPT Courseware 2 Knowledge Review 1. What is the formula for finding the roots of a quadratic equation? Generally speaking, for the quadratic equation of one variable ax+bx+c=0 (a0), when b-4ac0, its root is x=-bb-4ac/2a 2. Use the formula method to solve the equation of one variable..