Solving complex irrational systems of equations. Basic methods for solving irrational equations

Your privacy is important to us. For this reason, we have developed a Privacy Policy that describes how we use and store your information. Please read our privacy policy and let us know if you have any questions.

Collection and use of personal information

Personal information refers to data that can be used to identify a specific person or contact him.

You may be asked to provide your personal information at any time when you contact us.

Below are some examples of the types of personal information we may collect and how we may use such information.

What personal information we collect:

• When you submit a request on the site, we may collect various information, including your name, phone number, email address, etc.

How we use your personal information:

• Collected by us personal information allows us to contact you and inform you about unique offers, promotions and other events and upcoming events.
• From time to time, we may use your personal information to send important notifications and messages.
• We may also use personal information for internal purposes, such as conducting audits, data analysis and various research in order to improve the services we provide and provide you with recommendations regarding our services.
• If you participate in a prize draw, competition or similar promotional event, we may use the information you provide to administer such programs.

Disclosure of information to third parties

We do not disclose information received from you to third parties.

Exceptions:

• If it is necessary - in accordance with the law, court order, in court proceedings, and / or on the basis of public requests or requests from state authorities in the Russian Federation - to disclose your personal information We may also disclose information about you if we determine that such disclosure is necessary or appropriate for security, law enforcement, or other socially important reasons.
• In the event of a reorganization, merger or sale, we may transfer the personal information we collect to an appropriate third party - the legal successor.

Protection of personal information

We take precautions - including administrative, technical and physical - to protect your personal information from loss, theft, and misuse, as well as from unauthorized access, disclosure, alteration and destruction.

Respect for your privacy at the company level

In order to make sure that your personal information is safe, we bring the rules of confidentiality and security to our employees, and strictly monitor the implementation of confidentiality measures.

Methodological developments for the elective course

"Methods for solving irrational equations" "

INTRODUCTION

The proposed elective course "Methods for solving irrational equations" is intended for grade 11 students comprehensive school and is subject-oriented, aimed at expanding the theoretical and practical knowledge of students. The elective course is built on the basis of the knowledge and skills gained by students in the study of mathematics in high school.

The specificity of this course lies in the fact that it is intended primarily for students who want to expand, deepen, systematize, generalize their mathematical knowledge, study common methods and techniques for solving irrational equations. The program includes questions that partly go beyond the current programs in mathematics and non-standard methods that allow you to more effectively solve various problems.

Most of the USE assignments require graduates to master various methods of solving various kinds of equations and their systems.The material related to equations and systems of equations makes up a significant part of the school mathematics course. The relevance of the choice of the topic of the elective course is determined by the importance of the topic "Irrational equations" in school course mathematics and, at the same time, the lack of time to consider non-standard methods and approaches to solving irrational equations, which are found in the tasks of the group "C" of the exam.

Along with the basis of the task of teaching mathematics - ensuring a lasting and conscious mastery by students of the system of mathematical knowledge and skills - this elective course provides for the formation of a stable interest in the subject, development math skills, increasing the level of mathematical culture of students, creates the basis for the successful delivery of the exam and the continuation of studies in universities.

Purpose of the course:

Raise the level of understanding and practical training in solving irrational equations;

Study the techniques and methods for solving irrational equations;

To form the ability to analyze, highlight the main thing, form elements of creative search based on generalization techniques;

To expand the knowledge of students on this topic, to improve the skills and abilities of solving various problems for the successful passing of the exam.

Course objectives:

Expansion of knowledge about methods and ways of solving algebraic equations;

Generalization and systematization of knowledge when teaching in grades 10-11 and preparing for the exam;

Development of the ability to independently acquire and apply knowledge;

Introducing students to work with mathematical literature;

Development logical thinking students, their algorithmic culture and mathematical intuition;

Improving the student's mathematical culture.

The elective course program involves the study of various methods and approaches in solving irrational equations, the development of practical skills on the issues under consideration. The course is designed for 17 hours.

The program is complicated, surpasses the usual course of study, promotes the development of abstract thinking, expands the field of knowledge of the student. At the same time, it maintains continuity with the existing programs, being their logical continuation.

Academic-thematic plan

№p / p

Lesson topic

Number of hours

Solution of equations taking into account the range of admissible values

Solving irrational equations by raising to a natural power

Solving equations by introducing auxiliary variables (replacement method)

Solution of an equation with a radical of the third degree.

Identical transformations when solving irrational equations

Unconventional tasks. Tasks of the group "C" of the exam

Control forms:home tests, independent work, essays and research work.

As a result of teaching this elective course, students should be able to solve various irrational equations using standard and non-standard methods and techniques;

master the algorithm for solving standard irrational equations;

be able to use the properties of equations to solve non-standard tasks;

be able to perform identical transformations when solving equations;

have a clear understanding of the topics of a single state examination, about the main methods of their solution;

gain experience in choosing methods for solving non-standard problems.

MAIN PART.

Equations in which the unknown quantity is under the radical sign are called irrational.

The simplest irrational equations include equations of the form:

The main idea of \u200b\u200bthe solutionan irrational equation consists in reducing it to a rational algebraic equation, which is either equivalent to the original irrational equation, or is its consequence. When solving irrational equations, we are always talking about finding real roots.

Let's consider some ways to solve irrational equations.

1.Solution of irrational equations taking into account the range of permissible values \u200b\u200b(ODZ).

The range of admissible values \u200b\u200bof an irrational equation consists of those values \u200b\u200bof the unknowns for which all expressions under the sign of an even degree radical are non-negative.

Sometimes knowledge of the ODZ allows you to prove that the equation has no solutions, and sometimes it allows you to find solutions to the equation by direct substitution of numbers from the ODZ.

Example 1 . Solve the equation.

Decision . Having found the ODV of this equation, we come to the conclusion that the ODV of the original equation is a one-element set ... Substitutingx \u003d 2 in given equation, we come to the conclusion thatx \u003d 2 Is the root of the original equation.

Answer : 2 .

Example 2.

The equation has no solutions, because for every valid value of the variable, the sum of two non-negative numbers cannot be negative.

Example 3.
+ 3 =
.

ODZ:

ODZ equation is an empty set.

Answer: the equation has no roots.

Example 4. 3
−4
−
=−(2+
).

ODZ:

ODZ:
... By checking, we make sure that x \u003d 1 is the root of the equation.

Answer: 1.

Prove that the equation does not have

roots.

1.
= 0.

2.
=1.

3. 5
.

4.
+
=2.

5.
=
.

Solve the equation.

1. .

2. = 0.

3.
= 92.

4. = 0.

5.
+
+ (x + 3) (2005 − x) \u003d 0.

2.In raising both sides of the equation to the natural power , that is, the transition from the equation

(1)

to the equation

. (2)

The following statements are true:

1) for any equation (2) is a consequence of equation (1);

2) if ( n – not even number), then equations (1) and (2 ) are equivalent;

3) if ( n Is an even number), then equation (2) is equivalent to the equation

, (3)

and equation (3) is equivalent to the set of equations

. (4)

In particular, the equation

(5)

is equivalent to a set of equations (4).

Example 1... Solve the equation

.

The equation is equivalent to the system

whence it follows that x \u003d 1, and the root does not satisfy the second inequality. At the same time, a competent decision does not require verification.

Answer:x \u003d 1.

Example 2... Solve the equation.

Solving the first equation of this system, which is equivalent to the equation , we get the roots and. However, at these values x inequality does not hold, and therefore the given equation has no roots.

Answer: no roots.

Example 3... Solve the equation

Removing the first radical, we obtain the equation

equivalent to the original.

Squaring both sides of this equation, since they are both positive, we obtain the equation

,

which is a consequence of the original equation. Squaring both sides of this equation under the condition that, we arrive at the equation

.

This equation has roots,. The first root satisfies the initial condition, and the second does not.

Answer: x \u003d 2.

If the equation contains two or more radicals, then they are first secluded and then squared.

Example 1.

By removing the first radical, we obtain an equation equivalent to this one. Let's square both sides of the equation:

After performing the necessary transformations, we will square the resulting equation

After checking, we notice that

out of range.

Answer: 8.

Answer: 2

Answer: 3; 1.4.

3. Many irrational equations are solved by introducing auxiliary variables.

A convenient means of solving irrational equations is sometimes the method of introducing a new variable, or "Replacement method". The method is usually used if in the equation some expression occurs repeatedly, depending on the unknown quantity. Then it makes sense to designate this expression with some new letter and try to solve the equation first with respect to the introduced unknown, and then find the original unknown.

A good choice of a new variable makes the structure of the equation more transparent. The new variable is sometimes obvious, sometimes somewhat veiled, but “felt”, and sometimes “manifested” only in the process of transformations.

Example 1.

Let be
t\u003e 0, then

t \u003d
,

t 2 + 5t-14 \u003d 0,

t 1 \u003d -7, t 2 \u003d 2. t \u003d -7 does not satisfy the condition t\u003e 0, then

,

x 2 -2x-5 \u003d 0,

x 1 \u003d 1-
, x 2 \u003d 1 +
.

Answer: 1-
; 1+
.

Example 2. Solve an irrational equation

Replacement:

Reverse replacement: /

Answer:

Example 3. Solve the equation .

Let's make replacements:,. The original equation will be rewritten in the form, whence we find that and = 4b and. Further, raising both sides of the equation squared, we get: Hence x \u003d 15. It remains to check:

- right!

Answer: 15.

Example 4... Solve the equation

Putting, we obtain a substantially simpler irrational equation. Let's square both sides of the equation:.

; ;

; ; , .

Checking the found values, their substitution into the equation shows that is the root of the equation, and is an extraneous root.

Returning to the original variable x , we get an equation, that is, a quadratic equation, having solved which we find two roots:,. Both roots satisfy the original equation.

Answer: , .

Substitution is especially useful if a new quality is achieved as a result, for example, an irrational equation turns into a rational one.

Example 6... Solve the equation.

Let's rewrite the equation as follows:.

It can be seen that if we introduce a new variable , then the equation takes the form , whence is an extraneous root and.

From the equation we obtain,.

Answer: , .

Example 7... Solve the equation .

Let's introduce a new variable,.

As a result, the original irrational equation takes the form of a square

,

whence, taking into account the restriction, we obtain. Solving the equation, we get the root. Answer: 2,5.

Tasks for independent solution.

1.
+
=
.

2.
+
=.

3.
.

5.
.

4. Method of introducing two auxiliary variables.

Equations of the form (here a , b , c , d – some numbers, m , n – natural numbers) and a number of other equations can often be solved by introducing two auxiliary unknowns: and, where and the subsequent transition to equivalent system of rational equations.

Example 1... Solve the equation.

Raising both sides of this equation to the fourth power doesn't bode well. If we put,, then the original equation is rewritten as follows:. Since we introduced two new unknowns, we need to find one more equation connecting y and z ... To do this, we raise the equalities to the fourth power and note that. So, we need to solve the system of equations

By squaring we get:

After substitution we have: or. Then the system has two solutions:,; ,, and the system has no solutions.

It remains to solve the system of two equations with one unknown

and the system The first of them gives, the second gives.

Answer: , .

Example 2.

Let be

Answer:

5. Equations with a radical of the third degree.
When solving equations containing radicals of the 3rd degree, it can be useful to use the addition of identities:

Example 1. .
Let's raise both sides of this equation to the 3rd power and use the above identity:

Note that the expression in parentheses is 1, which follows from the original equation. Taking this into account and bringing similar terms, we get:
Let's open the brackets, give similar terms and solve the quadratic equation. Its roots and... If we assume (by definition) that the root of an odd degree can also be extracted from negative numbers, then both numbers obtained are solutions of the original equation.
Answer:.

6. Multiplication of both sides of the equation by the conjugate of one of them expression.

Sometimes an irrational equation can be solved quite quickly if both sides of it are multiplied by a well-chosen function. Of course, when both sides of the equation are multiplied by some function, extraneous solutions may appear, they may be the zeros of this function itself. Therefore, the proposed method requires a mandatory study of the resulting values.

Example 1. Solve the equation

Decision: Let's choose the function

We multiply both sides of the equation by the selected function:

Let us present similar terms and obtain the equivalent equation

We add the original equation and the last one, we get

Answer: .

7. Identical transformations when solving irrational equations

When solving irrational equations, it is often necessary to apply identical transformations associated with the use of well-known formulas. Unfortunately, these actions are sometimes just as unsafe, just like raising to an even power - solutions can be gained or lost.

Let's consider several situations in which these problems occur, and learn how to recognize and prevent them.

I. Example 1... Solve the equation.

Decision.The formula applies here .

You just need to think about the safety of its use. It is easy to see that its left and right sides have different domains of definition and that this equality is true only under the condition. Therefore, the original equation is equivalent to the system

Solving the equation of this system, we get the roots and. The second root does not satisfy the set of inequalities of the system and, therefore, is an extraneous root of the original equation.

Answer: -1 .

II . The next dangerous transformation when solving irrational equations is determined by the formula.

If you use this formula from left to right, the DHS expands and outside solutions can be purchased. Indeed, both functions on the left and must be non-negative; and their product must be non-negative on the right.

Let's look at an example where a problem is implemented using a formula.

Example 2... Solve the equation.

Decision. Let's try to solve this equation by factoring

Note that this action turned out to be a lost solution, since it fits the original equation and no longer fits the obtained one: it makes no sense for. Therefore, this equation is better solved by the usual squaring

Solving the equation of this system, we get the roots and. Both roots satisfy the system inequality.

Answer: , .

III There is an even more dangerous action - reduction by a common factor.

Example 3... Solve the equation .

Incorrect reasoning: Reduce both sides of the equation by, we get .

There is nothing more dangerous and wrong than doing this. First, a suitable solution to the original equation was lost; secondly, two third-party solutions were acquired. It turns out that the new equation has nothing to do with the original! Here is the correct solution.

Decision... Move all the terms to the left side of the equation and factor it into factors

.

This equation is equivalent to the system

which has the only solution.

Answer: 3 .

CONCLUSION.

As part of the study of the elective course, non-standard techniques for solving complex problems are shown that successfully develop logical thinking, the ability to find among the many ways of solving the one that is comfortable and rational for the student. This course requires students to do a lot of independent work, contributes to the preparation of students for continuing education, raising the level of mathematical culture.

The work considered the main methods for solving irrational equations, some approaches to solving equations of higher degrees, the use of which is supposed to solve the USE tasks, as well as when entering universities and continuing mathematics education. Also, the content of the basic concepts and statements related to the theory of solving irrational equations was disclosed. Having identified the most common method for solving equations, we identified its application in standard and non-standard situations. In addition, the typical mistakes when performing identical transformations and ways to overcome them.

During the course, students will have the opportunity to master various methods and techniques for solving equations, while learning to systematize and generalize theoretical information, independently search for solutions to some problems and, in this regard, compose a number of tasks and exercises on these topics. The choice of complex material will help students to express themselves in research activities.

The positive side of the course is the possibility of further application by students of the studied material when passing the exam, admission to universities.

The negative side is that not every student is able to master all the techniques of this course, even with a desire, due to the difficulty of most of the problems being solved.

LITERATURE:

Sharygin I.F. "Mathematics for university applicants." - 3rd ed., - M.: Bustard, 2000.

Equations and inequalities. Reference book. / Vavilov V.V., Melnikov I.I., Olekhnik S.N., Pasichenko P.I. –M .: Exam, 1998.

Cherkasov O.Yu., Yakushev A.G. "Mathematics: Intensive Exam Preparation Course". - 8th ed., Rev. and add. - M.: Iris, 2003. - (Home Tutor)

Balayan E.N. Complex exercises and options training tasks to the exam in mathematics. Rostov-on-Don: Phoenix Publishing House, 2004.

Skanavi M.I. "Collection of problems in mathematics for applicants to universities." - M., "High School", 1998.

Igusman O.S. "Mathematics on the oral exam". - M., Iris, 1999.

Examination materials for preparing for the exam - 2008 - 2012.

VV Kochagin, MN Kochagina "Unified State Exam - 2010. Mathematics. Tutor "Moscow" Education "2010.

V. A. Gusev, A. G. Mordkovich “Mathematics. Reference materials "Moscow" Education "1988.

Lesson summary

"Methods for solving irrational equations"

11th grade of physical and mathematical profile.

Zelenodolsk municipal district of the Republic of Tatarstan "

Valieva S.Z.

Lesson topic: Methods for solving irrational equations

The purpose of the lesson: 1.Explore different ways to solve irrational equations.

1. 
   Develop the ability to generalize, choose the right ways to solve irrational equations.
2. 
   Develop independence, educate speech literacy

Lesson type:seminar.
Lesson plan:

1. 
   Organizing time
2. 
   Learning new material
3. 
   Anchoring
4. 
   Homework
5. 
   Lesson summary

During the classes
I... Organizing time: message of the topic of the lesson, the purpose of the lesson.

In the previous lesson, we looked at solving irrational equations containing square roots by squaring them. In this case, we get an equation-consequence, which sometimes leads to the appearance of extraneous roots. And then a mandatory part of solving the equation is checking the roots. Also considered solving equations using the definition square root... In this case, the check can be omitted. However, when solving equations, it is not always necessary to immediately start "blind" application of algorithms for solving the equation. In the tasks of the Unified State Exam, there are quite a few equations, when solving which it is necessary to choose a solution method that allows solving the equations easier and faster. Therefore, it is necessary to know other methods for solving irrational equations, with which we will get acquainted today. Previously, the class was divided into 8 creative groups, and they were given specific examples to reveal the essence of one method or another. We give them the floor.

II. Learning new material.

From each group, 1 student explains to the children how to solve irrational equations. The whole class listens and notes their story.

1 way. Introducing a new variable.

Solve the equation: (2x + 3) 2 - 3

4x 2 + 12x + 9 - 3

4x 2 - 8x - 51 - 3

, t ≥0

x 2 - 2x - 6 \u003d t 2;

4t 2 - 3t - 27 \u003d 0

x 2 - 2x - 15 \u003d 0

x 2 - 2x - 6 \u003d 9;

Answer: -3; five.

Method 2. LDZ study.

Solve the equation

ODZ:


x \u003d 2. By checking we make sure that x \u003d 2 is the root of the equation.

Method 3. Multiplication of both sides of the equation by the conjugate factor.

+
(multiply both sides by -
)

x + 3 - x - 8 \u003d 5 (-)

2 \u003d 4, hence x \u003d 1. By checking, we make sure that x \u003d 1 is the root of this equation.

Method 4. Reduction of an equation to a system by introducing a variable.

Solve the equation

Let \u003d u,
\u003d v.

We get the system:

Let's solve by substitution method. We get u \u003d 2, v \u003d 2. Hence,

we get x \u003d 1.

Answer: x \u003d 1.

Method 5. Selecting a complete square.

Solve the equation

Let's open the modules. Because -1≤сos0.5x≤1, then -4≤сos0.5x-3≤-2, which means. Similarly,

Then we get the equation

x \u003d 4πn, nZ.

Answer: 4πn, nZ.

Method 6. Assessment method

Solve the equation

ODZ: x 3 - 2x 2 - 4x + 8 ≥ 0, by definition the right side -x 3 + 2x 2 + 4x - 8 ≥ 0

get
those. x 3 - 2x 2 - 4x + 8 \u003d 0. Solving the equation by factoring, we get x \u003d 2, x \u003d -2

Method 7: Using the properties of monotonicity of functions.

Solve the equation. The functions are strictly increasing. The sum of increasing functions is increasing and this equation has no more than one root. By selection, we find x \u003d 1.

Method 8. Using vectors.

Solve the equation. ODZ: -1≤x≤3.

Let the vector
. Scalar product vectors - there is a left side. Let's find the product of their lengths. This is the right side. Got
, i.e. vectors a and b are collinear. From here
... Let's square both sides. Solving the equation, we get x \u003d 1 and x \u003d
.

1. 
   Anchoring.(sheets with assignments are distributed to each student)

Frontal oral work

Find the idea of \u200b\u200bsolving equations (1-10)

1.
(ODZ - )

2.
x \u003d 2

3.x 2 - 3x +
(replacement)

4. (selection of a full square)

5.
(Reduction of an equation to a system by introducing a variable.)

6.
(multiplication by the conjugate expression)

7.
since
... Then the given equation has no roots.

8. Because each term is non-negative, equate them to zero and solve the system.

9. 3

10. Find the root of the equation (or the product of roots, if there are several) of the equation.

Written independent work with subsequent verification

solve equations numbered 11,13,17,19

Solve equations:

12. (x + 6) 2 -

14.

Assessment method

Using the monotonicity properties of functions.

Using vectors.

1. 
   Which of these methods are used to solve other types of equations?
2. 
   Which of these methods did you like best and why?

1. 
   Homework: Solve the remaining equations.

List of references:

1. 
   Algebra and the beginning of mathematical analysis: textbook. for 11 cl. general education. institutions / S.M. Nikolsky, M.K. Potapov, N.N. Reshetnikov, A.V. Shevkin. M: Education, 2009

1. 
   Didactic materials on algebra and the principles of analysis for grade 11 / B.M. Ivlev, S.M. Sahakyan, S.I. Schwarzburd. - M .: Education, 2003.
2. 
   Mordkovich A.G. Algebra and the beginning of analysis. 10 - 11 grades: Problem book for general education. institutions. - M .: Mnemosina, 2000.
3. 
   Ershova A.P., Goloborodko V.V. Independent and test papers on algebra and the principles of analysis for 10 - 11 grades. - M .: Ileksa, 2004
4. 
   KIMs Unified State Examination 2002 - 2010

6. Algebraic trainer. A.G. Merzlyak, V.B. Polonsky, M.S. Yakir. A guide for schoolchildren and applicants. Moscow .: "Ileksa" 2001.
7. Equations and inequalities. Non-standard solution methods. Educational - toolkit... 10-11 grades. S. N. Oleinik, M. K. Potapov, P.I. Pasichenko. Moscow. "Bustard". 2001

Lesson workshop "Methods for solving irrational equations"

Kornyushina Tatiana Anatolievna

Purpose:

Systematize ways to solve irrational equations.

Promote the formation of the ability to choose the most rational ways to solve irrational equations.

To consolidate the basic methods for solving irrational equations:

The method of raising both sides of the equation to the same power;

Method for introducing a new variable.

Recall non-standard ways of solving irrational equations.

On the desk:

List of equations (on the central part of the board III and IV)

Independent work (closed board I and VI)

At each desks: a notebook-lecture, a notebook-workshop, a sheet for independent work;

On the desks by the group: A4 sheet and a marker.

During the classes

1. Lesson stage

W. Guys, we are studying the topic "Generalization of the concept of degree." We have already systematized and generalized knowledge on the topics “Root of the n-th degree and its properties”, “Degree with rational indicator”.

And today, our goals are: to generalize knowledge on the topic "Irrational equations", to repeat the methods of solving them and to learn how to choose the most rational for a particular group of irrational equations.

Q. Guys, let's remember: what equations are called irrational?

A. Irrational are equations in which the variable is contained under the radical sign or the variable is raised to a fractional power.

C. Formulate a basic algorithm for solving irrational equations.

A. (The guys formulate the algorithm, and the teacher posts it on board V).

Algorithm

Find ODZ

Raise both sides of the equation to the same power

Solve the resulting equation

Make a check

W. But it is possible to solve irrational equations not only by an algorithm. There are also ways to solve them.

C. What are the ways you know about solving irrational equations?

A. (The students name, and the teacher hangs the nameplates on board II in the order given, but as the students receive answers)

Ways to solve irrational equations

Solitude of the radical (raising to the same power)

Introducing a new variable

Multiplication by a conjugate expression

Gaze method

Equations Containing Cubic Radicals

Equations Reducible to Equations with Modules

Exploring Domain and Domain

Method of equivalent transitions (transition to the system)

2. Lesson stage

W. Let's talk about one of the main methods of solving irrational equations - the method of seclusion of the root.

Guys, today in the lesson we work in pairs, and we unite the pairs into groups according to the following principle: the first desks of the rows (3 pairs) - 1 group; second desks in rows (3 pairs) - group 2; the third desks of the rows (3 pairs) - group 3; fourth desks of the rows (3 pairs) - group 4.

Note! In case of difficulty in working in pairs, she can get help-advice in her group. To do this, just get up and walk up to any pair. All clear? Any questions?

W. So, let us consider the first method for solving irrational equations and characterize some of its features.

Let's start by solving the equations (board IV).

Z
writing to each group on the board. You have 5 minutes.

W. We discuss the solution.

B. 1 step. What did group 1 do?

A. Moved the term to the right side.

Q. What did the second group do?

Q. What did the third group do?

Q. Fourth group, did you follow this step?

ABOUT ... Not. (As answers are received, the teacher fills out the table on the board: “+” - yes; “-” - no).

Q. What did group 1 do?

A. Squared both sides of the equation (Same for groups 2 and 3).

Q. What did the 4th group do?

A. In number 4, they raised it to a cube. No 5 was built.

Q. Why?

A. The root value does not meet the definition of a nth arithmetic root.

Conclusion: Equation 5 has no roots.

W. Attention! Here it is rational to pay attention to the 1st stage of the algorithm: Find the ODZ.

Q. What did you do?

A. Solved the resulting equations.

Q. How many roots did you get during the solution?

Group 1 - 2 roots;

Group 2 - 2 roots;

Group 3 - 2 roots;

Group 4 - 2 roots;

Q. What's the next step 4?

A. We do a check.

Q. During the check, it was established that the irrational equation has in 1 group ..?

A. 2 roots.

W. Have group 2?

A. 1 root left.

D: The third group?

O. 1 root

U. 4 group?

A. The check could not have been done, because when raising both sides of the equation to the same and that burns an odd power, we obtain an equation equivalent to the given one.

W. Checking the answers!

No. 5 - no roots

3. Lesson stage

W. At the beginning of the lesson, we found out that the solution of irrational equations in 1 way is not the only one.

Q. Try to guess: in what ways can you solve the equations written on the board?

Assignment: Rewrite them in a notebook and put the method number next to the condition. We discuss in pairs!

Q. What methods did you choose? 1 equation? 2? 3? 4? five? 6? 7? 8? (Look at board III), (I write down the methods under the dictation of the students).

W. Any irrational equation can be solved in different ways, so our task in the next lesson is to learn how to choose the most rational ways to solve irrational equations.

And today, for example, let's take equation # 4 and solve it in the first four ways, and at home try to solve it in other ways (if possible).

1 group decides in 1 way; Group 2 solves in 2 ways; Group 3 solves in 3 ways; Group 4 decides in 4 ways. (The decision is written by 1 person from the group on sheet A4 with a marker, the rest of the group decide and prepare to comment on the decision).

Ready-made solutions are hung out on the board using magnets.

Q. Which of the following solutions do you think is more rational? (Vote)

W. We did not receive a definite answer.

4. Lesson stage

U. I would like to recall the words of the 12th century Indian mathematician Bhaskara: “A spark of knowledge ignites in the one who achieves understanding on his own.”

Let this “spark of knowledge” help you cope with independent work.

W. Homework in a notebook. Thank you for the lesson.

P
appendix 1

P
verification:

1 \u003d 1, true, 0 is the root of the equation

0 \u003d 0, true, 1 is the root of the equation

Methods for solving irrational equations.

Preliminary preparation for the lesson: students should be able to solve irrational equations in a variety of ways.

Three weeks prior to this lesson, students receive homework # 1: Solve various irrational equations. (Students independently find 6 different irrational equations and solve them in pairs.)

One week before this lesson, students receive homework # 2, which they complete individually.

1. Solve the equation different ways.

2. Assess the advantages and disadvantages of each method.

3. Make a record of conclusions in the form of a table.

№ p / p	Way	Advantages	disadvantages

Lesson objectives:

Educational:generalization of students' knowledge on this topic, demonstration of various methods for solving irrational equations, the ability of students to approach the solution of equations from a research standpoint.

Educational: fostering independence, the ability to listen to others and communicate in groups, increasing interest in the subject.

Developing: development of logical thinking, algorithmic culture, skills of self-education, self-organization, work in pairs when doing homework, the ability to analyze, compare, generalize, draw conclusions.

Equipment: computer, projector, screen, table "Rules for solving irrational equations", a poster with a quote by M.V. Lomonosov "Mathematics should then be taught that it puts the mind in order", cards.

Rules for solving irrational equations.

Lesson type: lesson-seminar (work in groups of 5-6 people, each group must have strong students).

During the classes

I . Organizing time

(Communication of the topic and objectives of the lesson)

II ... Presentation research work "Methods for solving irrational equations"

(The work is presented by the student who conducted it.)

III . Analysis of homework solution methods

(One student from each group writes down their proposed solutions on the board. Each group analyzes one of the solutions, assesses the advantages and disadvantages, and draws conclusions. The students of the groups supplement, if necessary. The analysis and conclusions of the group are assessed. Answers should be clear and full.)

The first way: raising both sides of the equation to the same power, followed by verification.

Decision.

Squaring both sides of the equation again:

From here

Check:

1. Ifx \u003d42, thenso the number42 is not the root of the equation.

2. Ifx \u003d2, thenso the number2 is the root of the equation.

Answer:2.

№ p / p	Way	Advantages	disadvantages
	Raising both sides of the equation to the same power	1. Got it. 2. Available.	1. Verbal notation. 2. Complicated check.

Output. When solving irrational equations by raising both sides of the equation to the same power, it is necessary to keep a verbal record, which makes the solution clear and accessible. However, mandatory verification is sometimes difficult and time-consuming. This method can be used to solve simple irrational equations containing 1-2 radicals.

Second way: equivalent transformations.

Decision:Let's square both sides of the equation:

Answer:2.

№ p / p	Way	Advantages	disadvantages
	Equivalent transformations	1. Lack of verbal description. 2. No verification. 3. Clear logical record. 4. A sequence of equivalent transitions.	1. Cumbersome recording. 2. You can make a mistake when combining the signs of the system and the collection.

Output. When solving irrational equations by the method of equivalent transitions, you need to clearly know when to put the sign of the system, and when - to set. The cumbersomeness of the recording, various combinations of system and system symbols often lead to errors. However, a sequence of equivalent transitions, a clear logical record without a verbal description, which does not require verification, are the indisputable advantages of this method.

The third way: functional graphic.

Decision.

Consider the functions and.

1. Function power-law; is increasing, because exponent is a positive (not integer) number.

D (f).

Let's create a table of valuesxandf( x).

	1,5		3,5
f (x)

2. Function power-law; is decreasing.

Find the domain of the functionD( g).

Let's create a table of valuesxandg( x).

g (x)

Let's construct these graphs of functions in one coordinate system.

Function graphs intersect at a point with an abscissa Because functionf( x) increases, and the functiong( x) decreases, then there will be only one solution to the equation.

Answer: 2.

№p / p	Way	Advantages	disadvantages
	Functional-graphic	1. Visibility. 2. No need to do complex algebraic transformations and monitor the ODV. 3. Lets you find the number of solutions.	1. verbal recording. 2. It is not always possible to find the exact answer, and if the answer is accurate, then a check is needed.

Output. The functional-graphical method is visual, it allows you to find the number of solutions, but it is better to use it when you can easily plot graphs of the functions under consideration and get an accurate answer. If the answer is approximate, then it is better to use another method.

Fourth way: introducing a new variable.

Decision.We introduce new variables, denoting We obtain the first equation of the system

Let's compose the second equation of the system.

For a variable:

For a variable

therefore

We obtain a system of two rational equations for and

Back to the variable, we get

Introducing a new variable

Simplification - obtaining a system of equations that do not contain radicals

1. The need to track the DHS of new variables

2. The need to return to the original variable

Output. This method is best used for irrational equations containing radicals of different degrees, or the same polynomials under the root sign and behind the root sign, or reciprocal expressions under the root sign.

- So guys, for each irrational equation, you need to choose the most convenient solution: an understandable one. Accessible, logically and well-designed. Raise your hand, which of you would prefer when solving this equation:

1) the method of raising both sides of the equation to the same power with verification;

2) the method of equivalent transformations;

3) functional-graphic method;

4) the method of introducing a new variable.

IV ... Practical part

(Work in groups. Each group of students receives an equation card and solves it in their notebooks. During this time, one representative from the group is solving an example on the board. Students in each group solve the same example as a member of their group and make sure that they are completed correctly tasks on the blackboard. If the respondent at the blackboard makes mistakes, then the person who notices them raises his hand and helps to correct them. During the lesson, each student, in addition to the example solved by his group, should write down in a notebook and others suggested to the groups and solve them at home .)

Group 1.

Group 2.

Group 3.

V . Independent work

(In groups, there is a discussion first, and then the students begin the assignment. The correct solution, prepared by the teacher, is displayed.)

VI ... Lesson summary

Now you know that solving irrational equations requires you to have good theoretical knowledge, the ability to apply them in practice, attention, diligence, ingenuity.

Homework

Solve the equations given to the groups during the session.