"Equations" Equality and Inequality PPT (the solution set of a quadratic equation in lesson 2 and the relationship between its roots and coefficients)

"Equations" Equality and Inequality PPT (the solution set of a quadratic equation in lesson 2 and the relationship between its roots and coefficients)

Part One: Learning Objectives

Understand the relationship between the value of the discriminant Δ and the number of roots of a quadratic equation, and be able to apply the relationship between the roots and coefficients of a quadratic equation

Perform calculations and evaluations and find the value range of parameters

Equation PPT, part 2 content: independent learning

Problem guide

Preview the contents of textbook P47-P50 and think about the following questions:

1. How to determine the solution of the quadratic equation ax2+bx+c=0 (a≠0) through the discriminant Δ?

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

A preliminary exploration of new knowledge

1. Solution set of quadratic equation of one variable

Generally, Δ=b2-4ac is called the discriminant of the quadratic equation ax2+bx+c=0 (a≠0).

(1) When Δ>0, the solution set of the equation is ____________________________;

(2) When Δ=0, the solution set of the equation is __________;

(3) When Δ<0, the solution set of the equation is _____.

2. The relationship between the roots and coefficients of a quadratic equation

If x1 and x2 are two roots of the quadratic equation ax2+bx+c=0 (a≠0), then x1+x2=_____, x1x2=_____.

■Instructions from famous teachers

When applying the relationship between the roots and coefficients of a quadratic equation, the following deformations often occur:

①(x1＋1)(x2+1)＝x1x2＋(x1＋x2)+1;

②x2x1+x1x2=(x1+x2)2-2x1x2x1x2;

③|x1-x2|＝(x1＋x2)2-4x1x2.

self-test

Judge whether it is true or false (mark “√” if it is correct and “×” if it is wrong)

(1) If the quadratic equation ax2+bx+c=0 (a≠0) has two unequal real roots, then b2-4ac>0.()

(2) The quadratic equation x2+ax+a-1=0 has real roots. ()

Among the following quadratic equations of one variable, the equation with two unequal real roots is ()

A． x2+1＝0　B. x2－2x＋1＝0

C. x2＋2x+4＝0 D． x2－x－3＝0

If the quadratic equation x2+4x+k=0 about x has real roots, then the range of k is ________.

Equation PPT, the third part of the content: interactive teaching and practice

Judgment and Application of the Number of Roots of Equations

It is known that the quadratic equation 3x2-2x+k=0 about x is known. According to the following conditions, find the range of k respectively.

(1) The equation has two unequal real roots;

(2) The equation has two equal real roots;

(3) The equation has real roots;

(4) The equation has no real roots.

Track training

1. Without solving equations, determine the number of real roots of the following equations.

(1)2x2-3x+1=0;

(2)4y2+9=12y;

(3)5(x2+3)-6x=0.

2. It is known that the equation x2+kx+1=0 (k>0) has real roots, find the value range of the function y=k2+2k-1.

Directly apply the relationship between roots and coefficients for calculations

If x1 and x2 are the two roots of the equation x2+2x-2 007=0,

Find the values ​​of the following expressions:

(1)x21+x22;

(2)1x1＋1x2;

(3)(x1-5)(x2-5);

(4)|x1-x2|.

regular method

When finding the value of an algebraic expression containing two roots of a quadratic equation, using the relationship between the roots and coefficients to solve the problem can play a role in making difficult problems easy and complex problems simple. When calculating, we must first find the sum of two roots and the product of two roots based on the original equation, then transform the algebraic expression into a form that locally contains the sum of two roots and the product of two roots, and then substitute it into the calculation.

Equation PPT, Part 4: Feedback on Compliance

1. The root of the equation x2-23kx+3k2=0 is ()

A． has a real root

B． There are two unequal real roots

C. There are two equal real roots

D. no real roots

2. If the equation mx2+(2m+1)x+m=0 about x has two unequal real roots, then the value range of the real number m is ()

A． m<14　B． m>－14

C. m<14 and m≠0 D. m＞-14 and m≠0

3． It is known that x1 and x2 are the two roots of the equation x2+bx-3=0 about x, and satisfy x1+x2-3x1x2=5, then the value of b is ()

A． 4B． -4

C. 3D． -3

4. It is known that the equation x2+tx+1=0, according to the following conditions, find the value range of t respectively.

(1) Both roots are greater than 0;

(2) Both roots are less than 0;

(3) One root is greater than 0 and the other root is less than 0.

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"Equations" Equations and Inequalities PPT Lesson (Lesson 2: The solution set of a quadratic equation and the relationship between its roots and coefficients):

"Equations" Equations and Inequalities PPT Lesson (Lesson 2: The solution set of a quadratic equation and the relationship between its roots and coefficients) Part 1 content: Learning objectives 1. Understand the definition of a quadratic equation of one variable, and be able to find the solution of a quadratic equation of one variable The set of solutions to a quadratic equation. (Key points) 2. Master one dollar..