Presentation for the lesson "solving quadratic equations". Solving quadratic equations. The purpose of the lesson: To generalize and put into the system all the knowledge and skills that we possess. Form skills, solve square ones. Download presentation solution of quadratic equations

Key to the assignment Evaluation criterion No errors - 5 points 1-2 errors - 4 points 3-4 errors - 3 points 5-6 errors - 2 points More than 6 errors - 0 points

The first quadratic equations appeared a long time ago. They were solved in Babylon around 2000 BC, and Europe seven years ago celebrated the 800th anniversary of the quadratic equations, because it was in 1202 that the Italian scientist Leonard Fibonacci outlined the quadratic equations. And only in the 17th century, thanks to Newton, Descartes and other scientists, these formulas took a modern form.

0, then the equation has two roots 4. If D \u003d 0, then the equation has one root 5. If D "title \u003d" (! LANG: Algorithm for solving a quadratic equation 1. Find the coefficients of the equation 2. Calculate the discriminant by the formula D \u003d в² - 4ac 3. If D\u003e 0, then the equation has two roots 4. If D \u003d 0, then the equation has one root 5. If D" class="link_thumb"> 7 !} Algorithm for solving a quadratic equation 1. Find the coefficients of the equation 2. Calculate the discriminant by the formula D \u003d в² - 4ac 3. If D\u003e 0, then the equation has two roots 4. If D \u003d 0, then the equation has one root 5. If D 0, then the equation has two roots 4. If D \u003d 0, then the equation has one root 5. If D "\u003e 0, then the equation has two roots 4. If D \u003d 0, then the equation has one root 5. If D"\u003e 0, then the equation has two roots 4. If D \u003d 0, then the equation has one root 5. If D "title \u003d" (! LANG: Algorithm for solving a quadratic equation 1. Find the coefficients of the equation 2. Calculate the discriminant by the formula D \u003d в² - 4ac 3. If D\u003e 0, then the equation has two roots 4. If D \u003d 0, then the equation has one root 5. If D"> title="Algorithm for solving a quadratic equation 1. Find the coefficients of the equation 2. Calculate the discriminant by the formula D \u003d в² - 4ac 3. If D\u003e 0, then the equation has two roots 4. If D \u003d 0, then the equation has one root 5. If D"> !}

"Hurry, don't be mistaken!" Key to the test Evaluation criterion 1-B 2-B No errors - 5 points 1 error - 4 points 3 errors - 2 points 2 errors - 1 point 4-5 errors - 0 points

Performance map F.I.Warm-upSlightly - think a little Theory Questions Solving equations Catch the errorTestTotal Evaluation criteria: points - "5" 9-14 points - "4" 5-8 points - "3"

20.01.2017 18:27

The presentation reflects the main stages of the consolidation lesson. There is musical accompaniment.

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"lesson 1"

Lesson topic: Solving quadratic equations using a formula.

The purpose of the lesson:

Educational

1. To form the ability to solve quadratic equations in different ways.

2. To form an idea of \u200b\u200bthe methods of mathematics as a science (general cultural competence).

Developing

Develop

1.the ability to compare, analyze, build analogies (educational and cognitive competence);

2. the ability to set a goal and plan activities, implement the plan (educational and cognitive competence);

3. the ability to listen, work in pairs, in a group (communicative competence).

Educational

1. To develop skills of control and self-control (competence of personal self-improvement).

2. To foster responsibility (social and labor competence).

During the classes:

1.Org. moment

Hello, my name is Aigul Anapievna Yarboldieva, today I will give you an algebra lesson.

Let the words of the great Goethe be the motto of our lesson today:

What are the keywords that reflect our activities in today's lesson? ... (Know. Be able to use)

So, in today's lesson we will find out what we know, what we can, and how you can use it in a variety of tasks.

I propose to start our work by deciphering the words that will help us determine the topic of the lesson.

- What words are encrypted?

Taiimdkisrnn (discriminant)

Nivarene (equation)

Fecocinetyf (coefficient)

Erokn (root)

Ormfual (formula)

So, what is the topic of the lesson? (Today in the lesson we will continue to deal with solving quadratic equations using the formula.)

Let's write down the topic of our lesson and the date.

Today not only I will evaluate you, but you yourself. The scorecard is on the tables, sign it. For each correct answer or decision, you will put 1 point

To earn a good grade, you must earn as many points as possible.

Last name, first name

Activities				EVALUATION
Oral exercises		Solving Equations

2.Oral work.

There are 10 equations on the screen:

1.x 2 + 9x - 12 \u003d 0;

2.4x 2 - 1 \u003d 0;

3.x 2 - 2x + 5 \u003d 0;

4.2z 2 - 5z +2 \u003d 0;

5.4y 2 \u003d 1;

6. -2x 2 - x + 1 \u003d 0;

7.x 2 + 8x \u003d 0;

9.x 2 - 8x \u003d 1;

10.2x + x 2 - 1 \u003d 0

Answer the questions:

Give the definition of a quadratic equation.	An equation of the form ax 2 + bx + c \u003d 0, where a ≠ 0, is called square.
2. What are the types of quadratic equations	Complete; -incomplete; - given
3. What are the numbers of the given quadratic equations written on the board
4. What are the numbers of incomplete equations written on the board
5. What are the numbers of the complete equations written on the board	1, 3, 4, 6, 9, 10
6. What are the coefficients of the quadratic equation called?	a - first coefficient, b- second coefficient, c - free term
7. What are the coefficients of the quadratic equation number 7	a \u003d 1, b \u003d 8, c \u003d 0
8. What are the coefficients of the quadratic equation number 2	a \u003d 4, b \u003d 0, c \u003d -1

3. Work in a notebook. (scores in the table)

4. Recall the algorithm for solving square. equations by the formula

5. Let's solve the square using the formula.

In the next. year you have the OGE. Quadratic equations are present in the first and second parts of the examination paper. Let's solve the task from the open FIPI task bank.

5x 2 -18x + 16 \u003d 0

Answer: 2; 1.6

What number is the coefficient in? (even)

What other formula can you use to solve this equation?

We solve by the formula

5. FIZMINUTE for eyes

Let's rest our eyes. Set aside pens and pencils. Straighten up. Close your eyes. With your eyes closed, look to the right, left, up, down. Close your eyes tightly, relax. Make circular movements with your eyes, first in one direction, then in the other. Close your eyes again, relax. Sit a little with your eyes closed. Okay.

We open our eyes smoothly. We restore the sharpness of the image.

6. Game« black box"

Pascal spoke

Let's make math more fun.

You need to guess what is in the black box.

I give three definitions to this subject:

And now you will have to determine which plant this root is by solving the following equations in pairs, and from the key, choose the letter corresponding to the correct answer and write it in the form.

At the blackboard

5x 2 -4x - 1=0 x 2 -6x + 9 \u003d 0 2x 2 + 2x + 3 \u003d 0 –x 2 + 3x + 10 \u003d 0

		No roots

- What is this plant? (Rose flower)

- This means that in the black box lay the root of a rose, about which the people say: "The flowers are angelic, and the devil's claws." There is an interesting legend about the rose: according to Anacreon, a rose was born from the snow-white foam that covered the body of Aphrodite when the goddess of love came out of the sea. At first, the rose was white, but from a drop of the goddess's blood that pricked on a thorn, it became scarlet.

- You see, guys, everything in this world is interconnected: mathematics, Russian language and literature, biology.

7. INTERESTING EQUATIONS

Slide: "This is interesting!"

x 2 - 1999x + 1998 \u003d 0

- I can name the roots of this equation orally. This is 1 and 1998

- Would you like to learn how to do it?

Orally. 2x 2 + 3x + 1 \u003d 0 -No.533 (a) -s.121 account.

x 2 + 5x-6 \u003d 0- # 533 (g) -s.121 account.

HOW CAN I FIND ITS ROOTS WITHOUT SOLVING THESE EQUATIONS?

x 2 + 2000x - 2001 \u003d 0-reserve

8. Application in life

Studying the topic of quadratic equations, we somehow did not think about the fact that quadratic equations have wide practical application.

Let's think about where quadratic equations are now used, if we do not take into account their study in schools and various universities.

You cannot do without quadratic equations for various calculations. They can be used in construction to figure out the trajectory of the planets, in aircraft construction. Arithmetic calculations are also important in sports.

9. Lesson summary:

“Just knowing is not enough, you need to be able to use knowledge”.

What did you do?

What have you learned?

What new things have we learned today?

Have we achieved our goals?

Summing up the results of the evaluation table.

10. Homework

No. 534 (a, b) No. 533 (c)

No. 541 (b) No. 543 (a)

Any three equations.

Additionally. ON THE TRAIL. LESSON YOU LEARN TO SOLVE PROBLEMS USING SQUARE EQUATIONS. TRY AT HOME.

Task:

Thank you for the lesson!

I am glad that each of you contributed, you know the formulas, you know how to apply them.

Were active in the lesson, worked with interest on various tasks, at each

stage tracked their results, you know how to evaluate yourself and your friend, are attentive and

friendly to each other.

I wish you creative success with your homework!

Bye! Looking forward to meeting you!

Grade 8 student (s) grade table ____________________________

Last name, first name

Activities				EVALUATION
Oral exercises	Make the equations for the coefficients.	Solving Equations

Grade 8 student (s) grade table ____________________________

Last name, first name

Activities				EVALUATION
Oral exercises	Make the equations for the coefficients.	Solving Equations

Grade 8 student (s) grade table ____________________________

Last name, first name

Activities				EVALUATION
Oral exercises	Make the equations for the coefficients.	Solving Equations

Grade 8 student (s) grade table ____________________________

Last name, first name

Activities				EVALUATION
Oral exercises	Make the equations for the coefficients.	Solving Equations

Grade 8 student (s) grade table ____________________________

Last name, first name

Activities				EVALUATION
Oral exercises	Make the equations for the coefficients.	Solving Equations

Grade 8 student (s) grade table ____________________________

Last name, first name

Activities				EVALUATION
Oral exercises	Make the equations for the coefficients.	Solving Equations

Homework

The task "to your taste and color."

No. 534 (a, b) No. 533 (c)

No. 541 (b) No. 543 (a)

Any three equations.

Quadratic equations are first encountered in the works of the Indian mathematician and astronomer Aryabhatta. Another Indian scientist Brahmagupta outlined a general rule for solving quadratic equations, which practically coincides with the modern one. Find information about an astronomer or scientist and prepare a report.

Additionally

At that time, public competition in solving difficult problems was widespread in ancient India. These tasks were often clothed in poetic form. Here is one such challenge. Solve it at home.

Task:

Frisky flock of monkeys, having eaten their fill, had fun.

The eighth part of them in the square amused themselves in the clearing.

And twelve began to jump over the vines, hanging.

How many monkeys were there, tell me, in this pack?

Homework

The task "to your taste and color."

No. 534 (a, b) No. 533 (c)

No. 541 (b) No. 543 (a)

Any three equations.

Quadratic equations are first encountered in the works of the Indian mathematician and astronomer Aryabhatta. Another Indian scientist Brahmagupta outlined a general rule for solving quadratic equations, which practically coincides with the modern one. Find information about an astronomer or scientist and prepare a report.

Additionally

At that time, public competition in solving difficult problems was widespread in ancient India. These tasks were often clothed in poetic form. Here is one such challenge. Solve it at home.

Task:

Frisky flock of monkeys, having eaten their fill, had fun.

The eighth part of them in the square amused themselves in the clearing.

And twelve began to jump over the vines, hanging.

How many monkeys were there, tell me, in this pack?

View presentation content
"lesson"

"Just - little knowledge

need to ».

Goethe.

know

be able to use

What words are encrypted?

Root

The equation

Coefficient

Discriminant Formula

• Yerokn Nivarene Fecocinetyf Taiimdkisrnn Ormfual
• Yerokn
• Nivarene
• Fecocinetyf
• Taiimdkisrnn
• Ormfual

Lesson topic:

"Solving quadratic equations by formula"

Oral work

1. x 2 + 9x - 12 \u003d 0;

2. 4x 2 – 1 = 0;

3. x 2 - 2x + 5 \u003d 0;

4. 2 z 2 – 5 z +2 = 0;

5. 4 y 2 = 1;

6. -2x 2 - x + 1 \u003d 0;

7. x 2 + 8x \u003d 0;

8. 2x 2 = 0;

9. x 2 - 8x \u003d 1;

10. 2x + x 2 – 1 = 0

Make and write down quadratic equations by coefficients:

The equation

0 D \u003d 0 Equation has no valid roots "width \u003d" 640 "

ax 2 + in + c \u003d 0

Write out the coefficients a, b, c

Discriminant

D \u003d b 2 -4ac

Equation has no real roots

Solve the equation using the formula

5X 2 –18X + 16 \u003d 0

Answer: 2; 1.6

• Answer: 2; 1.6
• Answer: 2; 1.6
• Answer: 2; 1.6
• Answer: 2; 1.6

"The subject of mathematics is so serious that it is helpful to not miss an opportunity to make it a little entertaining."

Pascal.

What's in the black box?

1. Non-derivative stem of the word.

2. A number that, after putting it into an equation, turns the equation into identity.

3. One of the main organs of plants.

Solve equations using the formula

• 5x 2 -4x-1 \u003d 0 x 2 -6x + 9 \u003d 0 2x 2 + 2x + 3 \u003d 0 – x 2 + 3x + 10 \u003d 0
• 5x 2 -4x-1 \u003d 0
• x 2 -6x + 9 \u003d 0
• 2x 2 + 2x + 3 \u003d 0
• – x 2 + 3x + 10 \u003d 0

No roots

• According to Anacreon, the rose was born from the snow-white foam that covered the body of Aphrodite when the goddess of love came out of the sea. At first, the rose was white, but from a drop of the goddess's blood that pricked on a thorn, it became scarlet.

It is interesting!

x 2 - 1999x + 1998 \u003d 0

2x 2 + 3x + 1 \u003d 0 - No. 533 (a) -s.121 account.

Answer: -1; -0.5

x 2 + 5x-6 \u003d 0 - No. 533 (g) -s.121 account.

Answer: 1; -6

Takeoff

Takeoff is the main component of the flight. Here, the calculation is taken for small drag and accelerated takeoff.

“Just knowing is not enough, you need to be able to use knowledge”.

5-6 points- "3"

7-8 points - "4"

9 and more - "5"

Homework.

• Find historical background on the topic .

Quadratic equations are first encountered in the works of the Indian mathematician and astronomer Aryabhatta. Another Indian scientist Brahmagupta outlined a general rule for solving quadratic equations, which practically coincides with the modern one. Find information about an astronomer or scientist and prepare a report.

2. The task "to your taste and color."

No. 534 (a, b) No. 533 (d)

No. 541 (b) No. 543 (a)

Any three equations.

Homework.

3. In those days in ancient India, public competition in solving difficult problems was widespread. These tasks were often clothed in poetic form. Here is one such challenge. Solve it at home.

Task:

Frisky flock of monkeys, having eaten their fill, had fun.

The eighth part of them in the square amused themselves in the clearing.

And twelve began to jump over the vines, hanging.

How many monkeys were there, tell me, in this pack?

What determines the number of roots of a quadratic equation? Answer: From the sign D. D \u003d 0 D 0 1 root No roots two roots X \u003d -b / 2 aX \u003d (- b + D) / 2 a 0 1 root No roots two roots X \u003d -b / 2 aX \u003d (- b + D) / 2 a "\u003e 0 1 root No roots two roots X \u003d -b / 2 aX \u003d (- b + D) / 2 a "\u003e 0 1 root No roots two roots X \u003d -b / 2 aX \u003d (- in + D) / 2 a" title \u003d "(! LANG: What determines the number of roots of a quadratic equation? Answer: From the sign D. D \u003d 0 D 0 1 root No roots two roots X \u003d -b / 2 aX \u003d (- b + D) / 2 a"> title="What determines the number of roots of a quadratic equation? Answer: From the sign D. D \u003d 0 D 0 1 root No roots two roots X \u003d -b / 2 aX \u003d (- b + D) / 2 a"> !}

The task. The flasks are filled with liquids in which quadratic equations float. If D\u003e 0, then a vapor is released from the flask, in which the roots of the equation are located. If D 0, then a pair is released from the flask, in which the roots of the equation are located. If D "\u003e 0, then a vapor is released from the flask, in which the roots of the equation are located. If D"\u003e 0, then a pair is released from the flask, in which the roots of the equation are located. If D "title \u003d" (! LANG: Task. Flasks are filled with liquids in which quadratic equations float. If D\u003e 0, then steam is released from the flask, in which the roots of the equation are located. If D"> title="The task. The flasks are filled with liquids in which quadratic equations float. If D\u003e 0, then a vapor is released from the flask, in which the roots of the equation are located. If D"> !}

The treatise and its contents The first book that has come down to us, which describes the classification of quadratic equations and gives methods of their solution, as well as geometric proofs of these solutions, is the treatise "Kitab al-jabr wal-muqabala" by Muhammad al-Khwarizmi. Mathematician Muhammad al-Khwarizmi explains how to solve equations of the form ax 2 \u003d bx, ax 2 \u003d c, ax 2 + c \u003d bx, ax 2 + bx \u003d c, bx + c \u003d ax 2 (letters a, b, c denote only positive numbers) and only finds positive roots.

Problem “The square and number 21 is equal to 10 roots. Find the root (meaning the root of the equation is X 2 + 21 \u003d 10X). The author's decision sounds something like this: “Divide the number of roots in half - you get 5, multiply 5 by itself, subtract 21 from the product, there will be 4. Extract the root from 4 - you get 2. subtract 2 from 5 - you get 3, this will be the desired root ... Or add 5 to 7, which is also his root.

Research: a) consider the given quadratic equation X 2 + 3X-10 \u003d 0; rewrite it as X 2 -10 \u003d -3X. Solution: 1) divide the number of roots in half: -3: 2 \u003d -1.5 2) multiply (-1.5) by itself: -1.5 * (- 1.5) \u003d 2.25 3) from the product subtract (-10): 2.25 - (- 10) \u003d 2.25 + 10 \u003d 12.25

4) extract the square root of 12.25: we get 3.5 5) subtract 3.5 from (-1.5): -1.5-3.5 \u003d -5- this will be the desired root first 6) add 3, 5 to (-1.5): -1.5 + 3.5 \u003d 2- this will be the desired root of the second. Let's check: With X 1 \u003d -5 With X 2 \u003d \u003d \u003d 0 0 \u003d 0 (true) Answer: X 1 \u003d -5, X 2 \u003d 2.

Conclusion: Indeed, the above method for solving the reduced quadratic equation in the treatise by mathematician Muhammad al-Khwarizmi only for positive numbers, is applicable for negative numbers too. Let's compose an algorithm for solving the reduced quadratic equations by the method of Muhammad al-Khwarizmi.

Solution algorithm 1) Write the equation in the form: X 2 + c \u003d bX 2) Divide by 2 the number of roots b 3) Square the result of item 2 4) Subtract the free term from the result of item 3 5) Extract the square root of the result item 4 6) From the result of item 2, subtract the result of item 5, we get the first root 7) Add the result of item 5 to the result of item 2, we get the second root

Lesson presentation

"Solving quadratic equations"

Updating basic knowledge

1. What kind of equation is called square?

A quadratic equation is an equation of the form oh 2 + in + s \u003d 0, where x is a variable, a, in and from - some numbers, and and not equal to 0.

2. Which expression is a quadratic equation?

7x - x 2 + 5 \u003d 0

3. Name the coefficients in the equations:

5x 2 + 4x + 1 \u003d 0 x 2 + 5 \u003d 0 - x 2 + x \u003d 0

and = 1; at = 0; from = 5

and = -1; at = 1; from = 0

and = - 5 ; at = 4; from = 1

4. Make a quadratic equation if

and = 5, at = -3, from = -2.

5x 2 - 3x - 2 \u003d 0

5. What quadratic equations are called incomplete quadratic equations?

If in a quadratic equation and x 2 + at x + from \u003d 0 at least one of the coefficients at or from is equal to zero,

then such an equation is called an incomplete quadratic equation.

6. Name the types of incomplete quadratic equations.

1) a x 2 + from = 0

2) a x 2 + at x \u003d 0

3) a x 2 \u003d 0

7 what is the expression called at 2 – 4 ace ?

Discriminant

0 two roots in 2 - 4 ac \u003d 0 one root in 2 - 4 ac has no roots 9. Write the formula for the roots of a general quadratic equation. "Width \u003d" 640 "

8. What does it mean?

at 2 – 4 ace 0

two roots

at 2 – 4 ace = 0

one root

at 2 – 4 ace

has no roots

9.Write the formula for the roots of a general quadratic equation.

1. Which expression is a quadratic equation?

Option 1. Option 2.

a) 3x + 1 \u003d 0 a) 5x 2 + x - 4 \u003d 0

b) 5x + 4x 2 \u003d 0 b) 4x - 3 \u003d 0

c) 4x 2 + x - 1 c) x 2 - x - 12

2. Which of the numbers are the roots of the equation?

Option 1. Option 2.

x 2 + 3x + 2 \u003d 0 x 2 - 6x + 8 \u003d 0

a) -1 and - 2 a) - 4 and 2

b) 2 and -1 b) 4 and -2

c) -2 and 1 c) 4 and 2

0 for 𝐃 \u003d 0 a) one a) one b) two b) two c) none c) none "width \u003d" 640 "

3. Determine the signs of the roots of the equation without solving it:

Option 1. Option 2.

x 2 -14x + 21 \u003d 0 x 2 - 2x - 35 \u003d 0

a) (- and +) a) (+ and +)

b) (- and -) b) (- and +)

c) (+ and +) c) (- and -)

4. How many roots does the equation have and x 2 + at x + from = 0

Option 1. Option 2.

for 𝐃 0 for 𝐃 \u003d 0

a) one a) one

b) two b) two

c) none c) none

5. Without solving the equation, determine how many roots it has:

Option 1. Option 2.

5x 2 - 6x + 2 \u003d 0 x 2 + 10x + 9 \u003d 0

a) one a) one

b) two b) two

c) none c) none

Mutual verification:

Option 1. Option 2.