Presentation on the topic of equation and its roots. Summary and presentation of the lesson "the whole equation and its roots"

Is the equation quadratic? a) 3.7 x x + 1 = 0 b) 48 x 2 – x 3 -9 = 0 c) 2.1 x x - 0.11 = 0 d) x = 0 e) 7 x = 0 f) - x 2 = 0

Determine the coefficients of the quadratic equation: 6 x x + 2 = 0 a = 6 b = 4 c = 2 8 x 2 – 7 x = 0 a = 8 b = -7 c = 0 -2 x 2 + x - 1 = 0 a = -2 b = 1 c = -1 x 2 – 0.7 = 0 a = 1 b = 0 c = -0.7

Write quadratic equations: abc

0, has two roots: Proof: Let's move d to the left side of the equation: x 2 - d = 0 Since by condition d > 0, then by definition of an arithmetic square root Therefore the equation can be rewritten" title="Equation x 2 = d Theorem. The equation x 2 = d, where d > 0, has two roots: Proof: Move d to the left side of the equation: x 2 - d = 0 Since by condition d > 0, then by definition of the arithmetic square root Therefore the equation you can rewrite it" class="link_thumb"> 10 !} Equation x 2 = d Theorem. The equation x 2 = d, where d > 0, has two roots: Proof: Let's move d to the left side of the equation: x 2 - d = 0 Since by condition d > 0, then by definition of the arithmetic square root Therefore, the equation can be rewritten as follows: 0, has two roots: Proof: Let's move d to the left side of the equation: x 2 - d = 0 Since by condition d > 0, then by definition of an arithmetic square root Therefore, the equation can be rewritten "> 0, has two roots: Proof: Let's move d to the left side of the equation: x 2 - d = 0 Since by condition d > 0, then by definition of the arithmetic square root Therefore, the equation can be rewritten as follows: "> 0, has two roots: Proof: Let's move d to the left side of the equation: x 2 - d = 0 Since by condition d > 0, then by definition of the arithmetic square root Therefore the equation can be rewritten" title="Equation x 2 = d Theorem. The equation x 2 = d, where d > 0, has two roots: Proof: Let's move d to the left side of the equation: x 2 - d = 0 Since by condition d > 0, then by definition of an arithmetic square root Therefore the equation can be rewritten"> title="Equation x 2 = d Theorem. The equation x 2 = d, where d > 0, has two roots: Proof: Let's move d to the left side of the equation: x 2 - d = 0 Since by condition d > 0, then by definition of the arithmetic square root Therefore, the equation can be rewritten"> !}

Definition If in a quadratic equation ax 2 + bx + c=0 at least one of the coefficients b or c is equal to 0, then such an equation is called an incomplete quadratic equation. Types: If b = 0, then the equation is ax 2 + c=0 If c = 0, then the equation is ax 2 + bx =0 If b = 0 and c = 0, then the equation is ax 2 =0

Assignment: Write: 1) a complete quadratic equation with the first coefficient 4, free term 6, second coefficient (-7); 2) incomplete quadratic equation with the first coefficient 4, free term (-16); 3) a reduced quadratic equation with a free term, a second coefficient (-3). 4 x 2 -7 x + 6 = o 4 x = o

Task: Classify quadratic equations x 2 + x + 1 = 0; x 2 – 2 x = 0; 7 x – 13 x = 0; x 2 – 5 x + 6 = 0; x 2 – 9 = 0; x 2 – 9 x = 0; x x = 4 x x – 4.

Task: Transform the equations into the following: 2 x x – 4 =0 18 x 2 – 12 x + 6 = 0 4 x 2 – 16 x + 5 = 0 4 x 2 – 12 x = 0 Hint: divide all terms of the equation by the leading one coefficient.

7th grade Municipal budgetary educational institution “Secondary school No. 32 with in-depth study of aesthetic subjects”, Ussuriysk, Ussuri city district Mathematics teacher Dyundik Vera Petrovna “I hear, and I forget, I see, and I remember, I do, and I understand” Chinese proverb 1. How to find an unknown term? Stage of repetition of theoretical material 2. How to find an unknown minuend? 3.How to find an unknown subtrahend? 4. How to find an unknown factor? a) Y + 32 = 152, b) X – 38 = 142, Y = 152 + 32, X = 142 + 38, Y= 184. X = 180. Answer: 184 Answer: 180 c) X – 25 = 125, d) 518 – Z = 400, X = 125 – 25, Z = 518 – 400, X = 120. Z = 118. Answer: 120 Answer: 118 Find errors in the equations a) Y + 32 = 152, b) X – 38 = 142, Y = 152 + 32, error X = 142 + 38, Y = 184. 120 X = 180. Answer: 120 Answer: 180 c) X – 25 = 125, d) 518 – Z = 400, X = 125 – 25, error Z = 518 – 400, X = 120. 150 Z = 118. Answer: 150 Answer: 118 Find errors in equations When you solve an equation, my friend, You must find ……………. It’s not difficult to check the meaning of a letter. Substitute it into the equation carefully. If you achieve the correct equality, then call that hour......meaning. Guess the word 1. Solve the equation x + 1 = 6 2. Is the number 7 the root of the equation a) 3 – x = - 4; b) 5 + x = 4. Orally transfer a term from one part of the equation to another, changing its sign to the opposite; both sides are multiplied or divided by the same number other than zero. From this equation, an equivalent equation is obtained if: Properties of equations Solve the equation 4 + 16 x = 21 – (3 + 12x). Solve equation 1. The root of the equation is the value ……….. at which the equation becomes …………… numerical equality. 2. Equations are called equivalent if they have ………. or have no roots. 3. In the process of solving equations, they always try to replace this equation with a simpler equation that is equivalent to it. In this case, the following properties are used: 1) from this equation an equivalent equation is obtained if ……………. term from one part of the equation to another, …………… its sign; 2) from this equation an equivalent equation is obtained if both parts are multiplied or divided by ………………………... Test 1. The root of an equation is the value of a variable (1 point) at which the equation becomes correct (1 point) numerical equality. 2. Equations are called equivalent if they have the same roots (1 point) or have no roots. 3. In the process of solving equations, they always try to replace this equation with a simpler equation that is equivalent to it. In this case, the following properties are used: 1) from this equation an equivalent equation is obtained if we move (1 point) a term from one part of the equation to another, changing (1 point) its sign; 2) from this equation an equivalent equation is obtained if both parts are multiplied or divided by the same number other than zero (2 points). Key to the test Test scoring system “2” 0 – 3 points “3” 4 – 5 points “4” 6 points “5” 7 points Test scoring system Summary I II III I listened and I forgot. I don't like this kind of communication. I saw and I remembered. But I wasn’t always comfortable. I did it, and I understood. I liked it very much. How many roots can an equation have? x + 1 = 6 (x – 1)(x – 5)(x – 8) = 0 x = x + 4 Z(x + 5) = 3x + 15

Lesson topic: “The whole equation and its roots.”

Goals:

educational:

• consider a way to solve an entire equation using factorization;

developing:

educational:

Class: 9

Textbook: Algebra. 9th grade: textbook for educational institutions / [Yu.N. Makarychev, N.G. Mindyuk, K.I. Neshkov, S.B. Suvorov]; ed. S.A. Telyakovsky.- 16th ed. – M.: Education, 2010

Equipment: computer with projector, presentation “Entire equations”

During the classes:

Organizing time.

Watch the video “Everything is in your hands.”

There are times in life when you give up and it seems like nothing will work out. Then remember the words of the sage “Everything is in your hands:” and let these words be the motto of our lesson.

Oral work.

2x + 6 =10, 14x = 7, x 2 – 16 = 0, x – 3 = 5 + 2x, x 2 = 0,

Message of the lesson topic, goals.

Today we will get acquainted with a new type of equations - these are whole equations. Let's learn how to solve them.

Let's write down in a notebook the number, class work and the topic of the lesson: “The whole equation, its roots.”

2.Updating basic knowledge.

Solve the equation:

Answers: a)x = 0; b) x =5/3; c) x = -, ; d) x = 1/6; - 1/6; e) there are no roots; e) x = 0; 5; - 5; g) 0; 1; -2; h)0; 1; - 1; i) 0.2; - 0.2; j) -3; 3.

3.Formation of new concepts.

Conversation with students:

What is an equation? (equality containing an unknown number)

What types of equations do you know? (linear, square)

3. How many roots can a linear equation have?) (one, many and no roots)

4.How many roots can a quadratic equation have?

What determines the number of roots? (from discriminant)

In what case does a quadratic equation have 2 roots? (D0)

In what case does a quadratic equation have 1 root? (D=0)

In what case does a quadratic equation have no roots? (D0)

Whole equation is an equation of the left and right sides, which is an entire expression. (read aloud).

From the considered linear and quadratic equations, we see that the number of roots is not greater than its degree.

Do you think it is possible to determine the number of its roots without solving an equation? (possible children's answers)

Let's get acquainted with the rule for determining the degree of an entire equation?

If an equation with one variable is written in the form P(x) = 0, where P(x) is a polynomial of standard form, then the degree of this polynomial is called the degree of the equation. The degree of an arbitrary integer equation is the degree of an equivalent equation of the form P(x) = 0, where P(x) is a polynomial of standard form.

The equationn Ouch degree has no moren roots.

The whole equation can be solved in several ways:

ways to solve entire equations

factorization graphical introduction of new

variable

(Write the diagram in a notebook)

Today we will look at one of them: factorization using the following equation as an example: x 3 – 8x 2 – x +8 = 0. (the teacher explains on the board, students write down the solution to the equation in a notebook)

What is the name of the factorization method that can be used to factor the left side of an equation? (grouping method). Let's factorize the left side of the equation, and to do this we will group the terms on the left side of the equation.

When does the product of factors equal zero? (when at least one of the factors is zero). Let us equate each factor of the equation to zero.

Let's solve the resulting equations

How many roots did we get? (write in notebook)

x 2 (x – 8) – (x – 8) = 0

(x – 8) (x 2 – 1) = 0

(x – 8)(x – 1)(x + 1) = 0

x 1 = 8, x 2 = 1, x 3 = - 1.

Answer: 8; 1; -1.

4.Formation of skills and abilities. Practical part.

work on textbook No. 265 (write in notebook)

What is the degree of the equation and how many roots does each equation have:

Answers: a) 5, b) 6, c) 5, d) 2, e) 1, f) 1

№ 266(a)(solution at the board with explanation)

Solve the equation:

5. Lesson summary:

Consolidation of theoretical material:

What equation with one variable is called an integer? Give an example.

How to find the degree of an entire equation? How many roots does an equation with one variable of the first, second, nth degree have?

6.Reflection

Evaluate your work. Raise your hand who...

1) understood the topic perfectly

2) understood the topic well

I'm still experiencing difficulties

7.Homework:

clause 12 (p. 75-77 example 1) No. 267 (a, b).

“student checklist”

Student checklist

Stages of work Grade						Total
Verbal counting	Solve the equation		Solving Quadratic Equations	Solving cubic equations

Student checklist

Class______ Last name First name ___________________

Stages of work Grade						Total
Verbal counting	Solve the equation	What is the degree of familiar equations	Solving Quadratic Equations	Solving cubic equations

Student checklist

Class______ Last name First name ___________________

Stages of work Grade						Total
Verbal counting	Solve the equation	What is the degree of familiar equations	Solving Quadratic Equations	Solving cubic equations

View document contents
"Handout"

1.Solve the equations:

a) x 2 = 0 e) x 3 – 25x = 0

a) x 2 = 0 e) x 3 – 25x = 0
b) 3x – 5 = 0 g) x(x – 1)(x + 2) = 0
c) x 2 –5 = 0 h) x 4 – x 2 = 0
d) x 2 = 1/36 i) x 2 –0.01 = 0.03
e) x 2 = – 25 j) 19 – c 2 = 10

3. Solve the equations:

x 2 -5x+6=0 y 2 -4y+7=0 x 2 -12x+36=0

4. Solve the equations:

I option II option III option

x 3 -1=0 x 3 - 4x=0 x 3 -12x 2 +36x=0

"test"

Hello! Now you will be offered a 4-question math test. Click on the buttons on the screen under the questions that, in your opinion, have the correct answer. Click the "next" button to start testing. Good luck!

1. Solve the equation:

3x + 6 = 0

Correct

No answer

Roots

Correct

No answer

Roots

4. Solve the equation: 0 x = - 4

Roots

A lot of

roots

View presentation content
"1"

• Solve the equation:

• ORAL WORK

Goals:

educational:

• generalize and deepen information about equations; introduce the concept of a whole equation and its degree, its roots; consider a way to solve an entire equation using factorization.
• generalize and deepen information about equations;
• introduce the concept of a whole equation and its degree, its roots;
• consider a way to solve an entire equation using factorization.

developing:

• development of mathematical and general outlook, logical thinking, ability to analyze, draw conclusions;
• development of mathematical and general outlook, logical thinking, ability to analyze, draw conclusions;

educational:

• cultivate independence, clarity and accuracy in actions.
• cultivate independence, clarity and accuracy in actions.

• Psychological attitude

• We continue to generalize and deepen information about equations;
• get acquainted with the concept of the whole equation,

with the concept of degree of equation;

• develop skills in solving equations;
• control the level of material assimilation;
• In class we can make mistakes, have doubts, and consult.
• Each student sets his own guidelines.

• What equations are called integers?
• What is the degree of an equation?
• How many roots does an nth degree equation have?
• Methods for solving equations of first, second and third degrees.

• Lesson Plan

a) x 2 = 0 e) x 3 – 25x = 0 c) x 2 –5 = 0 h) x 4 –x 2 = 0 d) x 2 = 1/36 i) x 2 –0,01 = 0,03 e) x 2 = – 25 k) 19 – s 2 = 10

Solve the equations:

For example:

X²=x³-2(x-1)

• Equations

If the equation is with one variable

written as

P(x) = 0, where P(x) is a polynomial of standard form,

then the degree of this polynomial is called

degree of this equation

2x³+2x-1=0 (5th degree)

14x²-3=0 (4th degree)

For example:

What is the degree of acquaintance equations for us?

• a) x 2 = 0 e) x 3 – 25x = 0
• b) 3x – 5 = 0 g) x(x – 1)(x + 2) = 0
• c) x 2 – 5 = 0 h) x 4 –x 2 = 0
• d) x 2 = 1/36 i) x 2 – 0,01 = 0,03
• e) x 2 = – 25 k) 19 – s 2 = 10

• Solve the equations:

• 2 ∙x + 5 =15
• 0∙x = 7

How many roots can an equation of degree 1 have?

No more than one!

0, D=-12, D x 1 =2, x 2 =3 no roots x=6. How many roots can an equation of degree I (quadratic) have? No more than two!" width="640"

• Solve the equations:

• x 2 -5x+6=0 y 2 -4y+7=0 x 2 -12x+36=0
• D=1, D0, D=-12, D

x 1 =2, x 2 =3 no roots x=6.

How many roots can an equation of degree I have? (square) ?

No more than two!

Solve the equations:

• I option II option III option

x 3 -1=0 x 3 - 4x=0x 3 -12x 2 +36x=0

• x 3 =1 x(x 2 - 4)=0 x(x 2 -12x+36)=0

x=1 x=0, x=2, x= -2 x=0, x=6

1 root 3 roots 2 roots

• How many roots can an equation of degree I I I have?

No more than three!

• How many roots do you think the equation can have?

IV, V, VI, VII, n th degrees?

• No more than four, five, six, seven roots!

No more at all n roots!

ax²+bx+c=0

Quadratic equation

ax + b = 0

Linear equation

No roots

No roots

One root

Let's expand the left side of the equation

by multipliers:

x²(x-8)-(x-8)=0

Answer:=1, =-1.

• Third degree equation of the form: ax³+bx²+cx+d=0

By factorization

(8x-1)(2x-3)-(4x-1)²=38

Let's open the brackets and give

similar terms

16x²-24x-2x+3-16x²+8x-138=0

Answer: x=-2