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

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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)

Part One: Learning Objectives

1. Understand the definition of a quadratic equation and be able to find the solution set of a quadratic equation. (emphasis)

2. Master the discriminant of the roots of a quadratic equation and be able to use it to determine the number of roots. (emphasis)

3． Master the relationship between the roots and coefficients of a quadratic equation, and be able to use it to find the values ​​of some algebraic expressions about the two roots of the equation. (main difficulty)

core competencies

1. Cultivate mathematical literacy in mathematical abstraction and logical reasoning through learning the solution set of quadratic equations and the relationship between roots and coefficients.

2. Improve your mathematical literacy by finding the solution set of a quadratic equation.

Equation PPT, part 2 content: independent preview to explore new knowledge

A preliminary exploration of new knowledge

1. Definition of quadratic equation of one variable

An equation of the form ax2+bx+c=0 is a quadratic equation, where a, b, c are _______, and a≠0.

2. Solution to quadratic equation of one variable

(1) Direct square root method: The method of directly taking square root to find the solution to a quadratic equation using the definition of square root is called the direct square root method.

(2) Matching method: Through a simple deformation of the equation, match the left side into a completely square form containing unknown numbers. If the right side is a non-negative constant, you can use the direct square root method to solve it. This method of solving quadratic equations of one variable is called Preparation method.

(3) Formula method: Substitute the values ​​of the coefficients a, b, and c in the quadratic equation into the formula x=-b±b2-4ac2a to find the root of the equation. This method of solving the quadratic equation is It's called the formula method.

3． Discriminant of Roots of Quadratic Equation

The formula b2-4ac is called the discriminant of the roots of the quadratic equation ax2+bx+c=0 (a≠0). It is usually expressed by Δ, that is, Δ=b2-4ac. When Δ>0, the quadratic equation ax2+bx+c=0(a ≠0) has two real roots of ____; when Δ=0, the quadratic equation ax2+bx+c=0 (a≠0) has two real roots of ____; when Δ<0, the quadratic equation ax2+bx+c=0 (a≠0) has two real roots of ____; when Δ<0, the quadratic equation ax2+bx+c=0 The equation ax2+bx+c=0(a≠0)____real roots.

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

If the two roots of ax2+bx+c=0 (a≠0) are x1 and x2, then x1+x2=-ba, x1·x2=ca, that is, the sum of the two roots is equal to the inverse of the ratio of the coefficient of the linear term to the coefficient of the quadratic term, The product of two roots is equal to the ratio of the constant term to the coefficient of the quadratic term.

First try

1. The solution set of the quadratic equation x2-16=0 is ()

A． {-8,8}B. {-4}

C. {4} D. {-4,4}

2. Use the matching method to solve the equation x2-8x+5=0 and change it into the form of (x+a)2=b. The correct one is ()

A． (x+4)2=11 B. (x+4)2=21

C. (x-8)2=11 D． (x-4)2＝11

3． When solving the equation 6x-8=5x2 using the formula method, the values ​​of a, b, and c are ()

A． 5, 6, -8 B. 5, -6, -8

C. 5, -6, 8 D. 6, 5, -8

Equation PPT, the third part of the content: cooperative exploration to improve literacy

Solution to quadratic equation of one variable

Angle 1: Direct square root method

[Example 1] Use the direct square root method to find the solution set of the following quadratic equation of one variable:

(1)4y2-25=0; (2)3x2-x=15-x.

[Ideas Enlightenment] The equation can be transformed into the form of x2=p (p≥0). Then take the square root of both sides and reduce the degree, turning it into a linear equation of one variable.

regular method

The main steps of applying the direct square root method to find the solution set of a quadratic equation of one variable

1 is transformed into the form x2=pp≥0;

2 Direct square root;

3. Solve two linear equations of one variable and write the two roots of the equation;

4 Summarize and write it in the form of a solution set.

Track training

1. Use the direct square root method to find the solution set of the following quadratic equation of one variable.

(1)(x+1)2=12;

(2)(6x-1)2-25=0.

Angle 2 matching method

[Example 2] Use the combination method to find the solution set of the following equations.

(1)x2+4x-1=0;

(2)4x2＋8x+1＝0.

regular method

Use the combination method to solve the quadratic equation ax2+bx+c=0a≠0. First change the coefficient of the quadratic term to 1, that is, divide both sides of the equation by a, then move the constant term to the right side of the equation, and then add both sides of the equation. The square of half the coefficient of the linear term is used to formulate one side of the equation into a perfect square, the other side into a non-negative number, and then use the direct square root method to solve If the other side is a negative number, then this equation has no real roots.

The discriminant of the roots of a quadratic equation

[Example 4] Without solving the equation, determine the solution set of the following quadratic equation.

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

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

(3)4x－x2＝x2＋2.

[Solution](1)∵Δ=(-2)2-4×3×(-1)=16>0, the ∴ equation has two unequal real roots. ∴The solution set of the equation has two elements.

(2)∵Δ=(-1)2-4×2×1=-7<0, the ∴ equation has no real roots. ∴The solution set of the equation is the empty set.

(3) The equation is organized as x2-2x+1=0, ∵Δ=(-2)2-4×1×1=0, and the ∴ equation has two equal real roots. ∴There is one element in the solution set of the equation.

regular method

The discriminant of the roots of the quadratic equation ax2+bx+c=0a≠0 is Δ=b2-4ac. When Δ>0, the equation has two unequal real roots; when Δ=0, the equation has two equal roots. The real roots of ; when Δ<0, the equation has no real roots.

Class summary

1. Solutions to quadratic equations of one variable: (1) direct square root method; (2) combination method; (3) formula method.

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

If the two roots of ax2+bx+c=0 (a≠0) are x1 and x2, then x1+x2=-ba, x1x2=ca. Using this relationship, we can find some questions about the algebraic values ​​of the two roots of the equation.

Note: The prerequisites for the relationship between the roots and coefficients of a quadratic equation are: ①a≠0; ②Δ≥0.

Equation PPT, the fourth part of the content: reaching the standard in class and solidifying the double base

1. The solution set of the quadratic equation x2-9=0 is ()

A． {3}B. {-3}

C. {-3,3} D． {-9,9}

2. The solution set of the quadratic equation x2=3x is ()

A． {0}　B． {3} C. {-3} D. {0,3}

3． The solution set of the quadratic equation 4x2+1=4x is ()

A． Is the empty set B. only one element

C. There are two elements D. Unable to determine the number of elements

4. Change the equation x2-2x=3 into the form of (x-m)2=n, then m and n are ________ respectively.

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

"Equations" Equality and Inequality PPT (Lesson 2: The solution set of a quadratic equation and the relationship between its roots and coefficients) Part 1 content: Learning objectives Understand the relationship between the value of the discriminant and the number of roots of a quadratic equation The relationship between and be able to apply the roots of the quadratic equation and...

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Update Time: 2024-07-01

This template belongs to Mathematics courseware People's Education High School Mathematics Edition B Compulsory Course 1 industry PPT template

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