Development of an algebra lesson in the 11th profile class

The lesson was conducted by the teacher of mathematics MBOU secondary school No. 6 Tupitsyna O.V.

Topic and lesson number in the topic:“Application of several transformations leading to an equation-consequence”, lesson No. 7, 8 in the topic: “Equation-consequence”

Academic subject:Algebra and the beginnings of mathematical analysis - grade 11 (profile training according to the textbook by S.M. Nikolsky)

Type of lesson: "systematization and generalization of knowledge and skills"

Lesson type: workshop

The role of the teacher: direct the cognitive activity of students to develop the ability to independently apply knowledge in a complex to select the desired method or methods of transformation, leading to an equation - a consequence and application of the method in solving the equation, in new conditions.

Required technical equipment:multimedia equipment, webcam.

The lesson used:

1. didactic learning model- creating a problematic situation,
2. pedagogical means- sheets indicating training modules, a selection of tasks for solving equations,
3. type of student activity- group (groups are formed in the lessons - "discoveries" of new knowledge, lessons No. 1 and 2 from students with different degrees of learning and learning), joint or individual problem solving,
4. personality-oriented educational technologies: modular training, problem-based learning, search and research methods, collective dialogue, activity method, work with a textbook and various sources,
5. health-saving technologies- to relieve stress, physical education is carried out,
6. competencies:

- educational and cognitive at the basic level- students know the concept of an equation - a consequence, the root of an equation and methods of transformation leading to an equation - a consequence, they are able to find the roots of equations and perform their verification at a productive level;

- at an advanced level- students can solve equations using well-known methods of transformations, check the roots of equations using the area of ​​\u200b\u200badmissible values ​​​​of equations; calculate logarithms using exploration-based properties; informational - students independently search, extract and select the information necessary for solving educational problems in sources of various types.

Didactic goal:

creating conditions for:

Formation of ideas about equations - consequences, roots and methods of transformation;

Formation of the experience of meaning creation on the basis of a logical consequence of the previously studied methods of transforming equations: raising an equation to an even power, potentiating logarithmic equations, freeing an equation from denominators, bringing like terms;

Consolidation of skills in determining the choice of the transformation method, further solving the equation and choosing the roots of the equation;

Mastering the skills of setting a problem based on known and learned information, forming requests to find out what is not yet known;

Formation of cognitive interests, intellectual and creative abilities of students;

Development of logical thinking, creative activity of students, project skills, the ability to express their thoughts;

Formation of a sense of tolerance, mutual assistance when working in a group;

Awakening interest in independent solution of equations;

Tasks:

Organize the repetition and systematization of knowledge about how to transform equations;

- to ensure mastery of methods for solving equations and checking their roots;

- to promote the development of analytical and critical thinking of students; compare and choose optimal methods for solving equations;

- create conditions for the development of research skills, group work skills;

Motivate students to use the studied material to prepare for the exam;

Analyze and evaluate your work and the work of your comrades in the performance of this work.

Planned results:

*personal:

Skills of setting a task based on known and learned information, generating requests to find out what is not yet known;

The ability to choose the sources of information necessary to solve the problem; development of cognitive interests, intellectual and creative abilities of students;

The development of logical thinking, creative activity, the ability to express one's thoughts, the ability to build arguments;

Self-assessment of performance results;

Teamwork skills;

*metasubject:

The ability to highlight the main thing, compare, generalize, draw an analogy, apply inductive methods of reasoning, put forward hypotheses when solving equations,

Ability to interpret and apply the acquired knowledge in preparation for the exam;

*subject:

Knowledge of how to transform equations,

The ability to establish a pattern associated with various types of equations and use it in solving and selecting roots,

Integrating lesson objectives:

1. (for the teacher) Formation in students of a holistic view of the ways of transforming equations and methods for solving them;
2. (for students) Development of the ability to observe, compare, generalize, analyze mathematical situations associated with types of equations containing various functions. Preparation for the exam.

Stage I of the lesson:

Updating knowledge to increase motivation in the field of application of various methods of transforming equations (input diagnostics)

The stage of updating knowledgecarried out in the form of a test work with self-test. Developmental tasks are proposed, based on the knowledge acquired in previous lessons, requiring active mental activity from students and necessary to complete the task in this lesson.

Verification work

1. Choose equations that require the restriction of unknowns on the set of all real numbers:

a) = X-2; b) 3 \u003d X-2; c) =1;

d) ( = (; e) = ; e) +6 =5;

g) = ; h) = .

(2) Specify the range of valid values ​​of each equation, where there are restrictions.

(3) Choose an example of such an equation, where the transformation may cause the loss of the root (use the materials of the previous lessons on this topic).

Everyone checks the answers independently according to the ready-made ones highlighted on the screen. The most difficult tasks are analyzed and students pay special attention to examples a, c, g, h, where restrictions exist.

It is concluded that when solving equations, it is necessary to determine the range of values ​​allowed by the equation or to check the roots in order to avoid extraneous values. The previously studied methods of transforming equations leading to an equation - a consequence are repeated. That is, the students are thus motivated to find the right way to solve the equation proposed by them in further work.

II stage of the lesson:

Practical application of their knowledge, skills and abilities in solving equations.

The groups are given sheets with a module compiled on the issues of this topic. The module includes five learning elements, each of which is aimed at performing certain tasks. Students with different degrees of learning and learning independently determine the scope of their activities in the lesson, but since everyone works in groups, there is a continuous process of adjusting knowledge and skills, pulling those who are lagging behind to compulsory, others to advanced and creative levels.

In the middle of the lesson, a mandatory physical minute is held.

No. of educational element	Educational element with assignments	Guide to the development of educational material
UE-1	Purpose: To determine and justify the main methods for solving equations based on the properties of functions. Exercise: Specify the transformation method for solving the following equations: A) )= -8); b) = c) (=( d) ctg + x 2 -2x = ctg +24; e) = ; f) = sinx. 2) Task: Solve at least two of the proposed equations. Describe what methods were used in the solved equations.	Clause 7.3 p.212 Clause 7.4 p.214 Clause 7.5 p.217 Clause 7.2 p. 210
UE-2	Purpose: To master rational techniques and methods of solving Exercise: Give examples from the above or self-selected (use materials from previous lessons) equations that can be solved using rational methods of solution, what are they? (emphasis on the way to check the roots of the equation)
UE-3	Purpose: Using the acquired knowledge in solving equations of a high level of complexity Exercise: = (or ( = (	Clause 7.5
UE-4	Set the level of mastery of the topic: low - solution of no more than 2 equations; Medium - solution of no more than 4 equations; high - solution of no more than 5 equations
UE-5	Output control: Make a table in which to present all the methods of transforming equations you use and for each method write down examples of the equations you solved, starting from lesson 1 of the topic: “Equations - consequences”	Abstracts in notebooks

III stage of the lesson:

Output diagnostic work, representing the reflection of students, which will show readiness not only to write a test, but also readiness for the exam in this section.

At the end of the lesson, all students, without exception, evaluate themselves, then comes the teacher's assessment. If disagreements arise between the teacher and the student, the teacher can offer an additional task to the student in order to objectively be able to evaluate it. Homeworkaimed at reviewing the material before the control work.

School lecture

“Equivalent Equations. Corollary equation»

methodological comments. The concepts of equivalent equations, corollary equations, theorems on the equivalence of equations are important issues related to the theory of solving equations.

By 10th grade, students have gained some experience in solving equations. In grades 7-8, linear and quadratic equations are solved, there are no unequal transformations here. Further, in the 8th and 9th grades, rational and simplest irrational equations are solved, it turns out that in connection with the release from the denominator and squaring both parts of the equation, extraneous roots may appear. Thus, there is a need for the introduction of new concepts: the equivalence of equations, equivalent and non-equivalent transformations of the equation, extraneous roots and verification of roots. Based on the experience accumulated by students in solving the above classes of equations, it is possible to determine a new relation of equivalence of equations and “discover” theorems on the equivalence of equations together with students.

The lesson, the summary of which is presented below, precedes the consideration of topics related to the solution of irrational, exponential, logarithmic and trigonometric equations. The theoretical material of this lesson serves as a support for solving all classes of equations. In this lesson, it is necessary to define the concept of equivalent equations, corollary equations, to consider transformation theorems that lead to such types of equations. The material under consideration, as noted above, is a kind of systematization of students' knowledge about the transformations of equations, it is characterized by a certain complexity, therefore the most acceptable type of lesson is a school lecture. The peculiarity of this lesson is that the educational task (goals) set on it is solved over the course of many subsequent lessons (identifying transformations over equations leading to the acquisition of extraneous roots and the loss of roots).

Each stage of the lesson occupies an important place in its structure.

On the update stage students remember the main theoretical provisions related to the equation: what is an equation, the root of the equation, what does it mean to solve the equation, the range of acceptable values ​​(ODV) of the equation. They find the ODZ of specific equations that will serve as a support for the “discovery” of theorems in the lesson.

Target stage of motivation- to create a problem situation, which consists in finding the correct solution of the proposed equation.

Decision learning task (operational-cognitive stage) in the presented lesson lies in the "discovery" of theorems on the equivalence of equations and their proof. The main attention in the presentation of the material is given to the definition of equivalent equations, equations-consequences, "finding" theorems on the equivalence of equations.

The notes that the teacher makes during the lesson are presented directly in the abstract. Registration of notes by students in notebooks is given at the end of the lesson summary.

Lesson summary

Subject. Equivalent equations. Equation-consequence.

(Algebra and the beginning of analysis: A textbook for grades 10-11 of educational institutions / Sh.A. Alimov, Yu.M. Kolyagin, Yu.V. Sidorov and others - M .: Education, 2003).

Lesson goals. In joint activities with students, identify the equivalence relation on a set of equations, “discover” theorems on the equivalence of equations.

As a result, the student

knows

Definition of equivalent equations,

Definitions of the corollary equation,

Statements of the main theorems;

can

From the proposed equations, choose equivalent equations and equations-consequences,

Apply definitions of equivalent equations and corollary equations in standard situations;

understands

What transformations lead to equivalent equations or to equations-consequences,

That there are transformations, as a result of which the equation can acquire extraneous roots,

That as a result of some transformations, the loss of roots may occur.

Lesson type. School lecture (2 hours).

Lesson structure.

I. Motivational and orienting part:

Knowledge update,

Motivation, setting a learning task.

II. Operational-cognitive part:

The solution of the educational and research problem (the purpose of the lesson).

III. Reflective-evaluative part:

Summing up the lesson

Issuance of homework.

During the classes

I. Motivational and orienting part.

Today in the lesson we will talk about the equation, but we will not write down the topic yet. Recall the basic concepts associated with the equation. First of all, what is an equation?

(An equation is an analytical record of the problem of finding the values ​​of the arguments for which the values ​​of one function are equal to the values ​​of another function).

What other concepts are related to the equation?

(The root of the equation and what it means to solve the equation. The root of the equation is a number, when substituting into the equation, the correct numerical equality is obtained. Solve the equation - find all its roots or establish that they do not exist).

What is the ODZ equation?

(The set of all numbers for which the functions on the left and right sides of the equation make sense at the same time).

Find the ODZ of the following equations.

5)

6)
.

The solution to the equation is written on the blackboard.

What is the process of solving an equation?

(Performing transformations that bring this equation to an equation of a simpler form, i.e. such an equation, finding the roots of which is not difficult).

True, i.e. there is a sequence of simplifications from equation to equation
etc. to
. Let's see what happens to the roots of the equation at each stage of transformations. In the presented solution, two roots of the equation are obtained
. Check if they are numbers and numbers
and
roots of the original equation.

(numbers , and are the roots of the original equation, and
- No).

So, in the process of solving these roots were lost. In general, the performed transformations led to the loss of two roots
and the acquisition of an extraneous root.

How can you get rid of extraneous roots?

(Make a check).

Is it possible to lose roots? Why?

(No, because solving an equation means finding all its roots).

How to avoid losing roots?

(Probably, when solving the equation, do not perform transformations that lead to the loss of roots).

So, in order for the process of solving an equation to lead to correct results, what is important to know when performing transformations on equations?

(Probably, to know which transformations over the equations preserve the roots, which lead to the loss of roots or the acquisition of extraneous roots. Know what transformations they can be replaced so that there is no loss or acquisition of roots).

That's what we're going to do in this lesson. How would you formulate the goal of the upcoming activity in today's lesson?

(To identify transformations over equations that preserve roots, lead to the loss of roots or the acquisition of extraneous roots. Know what transformations can be replaced so that there is no loss or acquisition of roots).

II . Operational-cognitive part.

Let's go back to the equation written on the blackboard. Let's trace at what stage and as a result of what transformations, two roots were lost and an outsider appeared. (The teacher to the right of each equation puts down the numbers).

Name the equations that have the same set (set) of roots.

(Equations , , ,
and ,).

Such equations are called equivalent. Try to formulate a definition of equivalent equations.

(Equations that have the same set of roots are called equivalent).

Let's write down the definition.

Definition 1. Equations
and
are said to be equivalent if the sets of their roots are the same.

It should be noted that equations without horses are also equivalent.

To denote equivalent equations, you can use the symbol "
». The process of solving the equation using the new concept can be reflected as follows:

Thus, the transition from a given equation to an equivalent one does not affect the set of roots of the resulting equation.

And what are the main transformations performed when solving linear equations?

(Opening brackets; transferring terms from one part of the equation to another, changing the sign to the opposite; adding an expression containing an unknown to both parts of the equation).

Have their roots changed?

On the basis of one of these transformations, namely: the transfer of terms from one part of the equation to another, while changing the sign to the opposite, in the 7th grade they formulated a property of equations. Formulate it using a new concept.

(If any term of the equation is transferred from one part of the equation to another with the opposite sign, then an equation equivalent to the given one will be obtained).

What other property of the equation do you know?

(Both sides of the equation can be multiplied by the same non-zero number.)

Applying this property also replaces the original equation with an equivalent one. Let's go back to the equation written on the blackboard. Compare the set of roots of equations and ?

(The root of the equation is the root of the equation).

That is, when passing from one equation to another, the set of roots, although expanded, did not lose the roots. In this case, the equation is called a consequence of the equation. Try to formulate a definition of an equation that is a consequence of this equation.

(If there is no loss of roots during the transition from one equation to another, then the second equation is called a consequence of the first equation).

Definition 2. An equation is called a consequence of an equation if each root of the equation is a root of the equation.

- As a result of what transformation did you get the equation from the equation?

(Squaring both sides of the equation).

This means that this transformation can lead to the appearance of extraneous roots, i.e. the original equation is transformed into a consequence equation. Are there any other corollary equations in the presented chain of equation transformations?

(Yes, for example, the equation is a consequence of the equation, and the equation is a consequence of the equation).

What are these equations?

(Equivalent).

Try, using the concept of a consequence equation, to formulate an equivalent definition of equivalent equations.

(Equations are said to be equivalent if each of them is a consequence of the other).

Are there any other corollary equations in the proposed solution of the equation?

(Yes, the equation is a consequence of the equation).

What happens to roots when going from to ?

(Two roots are lost).

What transformation resulted in this?

(Error in applying the identity
).

Applying the new concept of the equation-corollary, and using the symbol "
”, the process of solving the equation will look like this:

.

So, the resulting scheme shows us that if equivalent transitions are made, then the sets of roots of the resulting equations do not change. But it is not always possible to apply only equivalent transformations. If the transitions are not equivalent, then two cases are possible: and . In the first case, the equation is a consequence of the equation, the set of roots of the resulting equation includes the set of roots of the given equation, here extraneous roots are acquired, they can be cut off by performing a check. In the second case, an equation was obtained for which this equation is a consequence: , which means that there will be a loss of roots, such transitions should not be performed. Therefore, it is important to ensure that when transforming an equation, each subsequent equation is a consequence of the previous one. What do you need to know so that the transformations are only such? Let's try to install it. Let's write task 1 (it offers equations; their ODZ found at the update stage; the set of roots of each equation is recorded).

Task 1. Are the equations of each group (a, b) equivalent? Name the transformation, as a result of which the first equation of the group is replaced by the second.

a)
b)

Let us turn to the equations of the group a), are these equations equivalent?

(Yes, and they are equivalent).

(We used the identity).

That is, the expression in one part of the equation was replaced by an identically equal expression. Has the ODZ equation changed under this transformation?

Consider the group of equations b). Are these equations equivalent?

(No, the equation is a consequence of the equation).

As a result of what transformation did you get from ?

(We replaced the left side of the equation with an identically equal expression).

What happened to the odz equation?

(ODZ expanded).

As a result of the expansion of the ODZ, we obtained a consequence equation and an extraneous root
for the equation. This means that the expansion of the ODZ equation can lead to the appearance of extraneous roots. For both cases a) and b), formulate the statement in general form. (Students formulate, teacher corrects).

(Let in some equation
, expression
replaced by the identical expression
. If such a transformation does not change the ODZ equation, then we pass to the equivalent equation
. If the ODZ expands, then the equation is a consequence of the equation ).

This statement is a transformation theorem leading to equivalent equations or corollary equations.

Theorem 1.,
a) ODZdoes not change	b) ODZ is expanding

We accept this theorem without proof. Next task. Three equations and their roots are presented.

Task 2. Are the following equations equivalent? Name the transformation, as a result of which the first equation is replaced by the second equation, the third equation.

Which of the following equations are equivalent?

(Only equations and ).

What transformations were performed in order to pass from the equation to the equation , ?

(To both sides of the equation in the first case we added
, in the second case we added
).

That is, in each case, some function was added
. Compare the domain of the function in the equation with the ODZ equation.

(Function
defined on the ODZ equation ).

What equation was obtained by adding the function to both sides of the equation?

(We get an equivalent equation).

What happened to the ODZ equation compared to the ODZ equation?

(It has narrowed due to the function
).

What did you get in this case? Will the equation be equivalent to the equation or - the equation-corollary for the equation?

(No, not both).

Having considered two cases of transformation of the equation, which are presented in task 2, try to draw a conclusion.

(If we add to both parts of the equation the function defined on the ODZ of this equation, then we get an equation equivalent to the given one).

Indeed, this statement is a theorem.

Theorem2. , - defined	on the odz equation

But we used a statement similar to the formulated theorem when solving equations. How does it sound?

(The same number can be added to both sides of the equation.)

This property is a particular case of Theorem 2 when
.

Task 3. Are the following equations equivalent? Name the transformation, as a result of which the first equation is replaced by the second equation, the third equation.

Which of the equations in task 3 are equivalent?

(Equations and ).

As a result of what transformation from the equation are the equations , ?

(Both sides of the equation are multiplied by
and get the equation. To get the equation, both sides of the equation are multiplied by
).

What condition must the function satisfy so that by multiplying both sides of the equation by , an equation equivalent to would be obtained?

(The function must be defined on the entire ODZ of the equation).

Have such transformations been performed on equations before?

(Performed, both parts of the equation were multiplied by a number other than zero).

This means that the condition imposed on the function must be supplemented.

(The function must not go to zero for any from the ODZ equation).

So, we write in symbolic form a statement that allows us to pass from a given equation to an equivalent one. (The teacher, under the dictation of the students, writes down Theorem 3).

Theorem 3. - defined throughout the ODZ for any of the ODZ

Let's prove the theorem. What does it mean that two equations are equivalent?

(It must be shown that all the roots of the first equation are the roots of the second equation and vice versa, i.e. the second equation is a consequence of the first and the first equation is a consequence of the second).

Let us prove that is a consequence of the equation . Let be - the root of the equation, what does it mean?

(When substituting in we get the correct numerical equality
).

At a point, the function is defined and does not vanish. What does this mean?

(Number
. Therefore, the numerical equality can be multiplied by
. We get the correct numerical equality ).

What does this equality mean?

( - the root of the equation. This showed that the equation is an equation-consequence for the equation).

Let us prove that is a consequence of the equation . (Students work independently, then after the discussion, the teacher writes the second part of the proof on the board).

Task 4. Are the equations of each group (a, b) equivalent? Name the transformation, as a result of which the first equation of the group is replaced by the second.

a)
b)

Are the equations and ?

(Equivalent).

As a result of what transformation from can be obtained?

(We raise both sides of the equation to a cube).

From the right and left sides of the equation, you can take the function
. On which set is the function defined?
?

(On the common part of the sets of function values
and
).

Describe the group of equations under the letter b)?

(They are not equivalent, is a consequence, the function was applied to the equation
and passed to the equation , the function is defined on the common part of the sets of function values
and
).

What is the difference between the properties of functions in the group a) and b)?

(In the first case, the function is monotonic, but not in the second).

Let us formulate the following assertion. (The teacher, under the dictation of the students, writes down the theorem).

Theorem 4. - is defined on the common part of the sets of function values ​​and
a) - monotonous	b) - not monotonous

Let's discuss how this theorem will "work" when solving the following equations.

Example. solve the equation

1)
; 2)
.

Which function is applicable to both sides of equation 1)?

(Let's raise both sides of the equation to a cube, i.e., apply the function).

(This function is defined on the common part of the sets of values ​​of functions on the left and right sides of the equation; it is monotonic).

So, by raising both sides of the original equation to a cube, what equation will we get?

(Equivalent to this).

Which function is applicable to both sides of equation 2)?

(Let's raise both sides of the equation to the fourth power, i.e. apply the function
).

List the properties of this function necessary to apply Theorem 4.

(This function is defined on the common part of the value sets of functions on the left and right sides of the equation; it is not monotonic).

What equation, relative to the original one, will we get by raising this equation to the fourth power?

(Consequence equation).

Will the set of roots of the original equation and the set of roots of the resulting equation differ?

(Extraneous roots may appear. So, a check is necessary).

Solve these equations at home.

III . Reflective-evaluative part.

Today we “discovered” four theorems together. Look at them again and say what equations they say.

(On equivalent equations and the equation-corollary).

Let's write the topic of the lesson. Let's return to the equation that was considered at the beginning of today's conversation. Which of Theorems 1-4 were applied when passing from one equation to another? (Students together with the teacher find out which theorem worked at each step, the teacher marks the number of the theorem on the diagram).

T.2 T.2 T.1 T.4 T.2 T.4

What new did you learn at the lesson today?

(The concepts of equivalent equations, corollary equations, theorems on the equivalence of equations).

What task did we set at the beginning of the lesson?

(Select transformations that do not change the set of roots of the equation, transformations leading to the acquisition and loss of roots).

Have we solved it completely?

We solved the problem in part, we will continue its study in the next lessons when solving new types of equations.

Using the concept of equivalent equations, new for us, reformulate the first part of the task "to select transformations that do not change the set of roots of the equation."

(How to know if going from one equation to another is an equivalent transformation).

What will help answer this question?

(Theorems on the equivalence of equations).

And have any transformations been applied today that lead to the acquisition of extraneous roots?

(Applied, this is the squaring of both parts of the equation; the use of formulas, the left and right parts of which make sense for different values ​​of the letters included in them).

There are other "specific" reasons that lead to both the appearance and loss of the roots of the equation, we talked about some of them. But there are also those that, as a rule, are associated with a certain class of equations, and we will talk about this later.

Let's write homework:

know the definitions of equivalent equations, corollary equations;

know the formulations of theorems 1-4;

carry out, by analogy with the proof of Theorem 3, the proof of Theorems 1 and 2;

4) Nos. 139(4,6), 141(2) - find out if the equations are equivalent; solve equations; .

Notebook entries

Equivalent equations. Equation-consequence.

Definition 1. Equations and are said to be equivalent if the sets of their roots coincide.

Definition 2. An equation is called a consequence of an equation if each root of the equation is a root of the equation. replaced by an identical expression.

Example.solve the equation

Let two equations be given

If each root of equation (1) is also a root of equation (2), then equation (2) is called a consequence of equation (1). Note that the equivalence of the equations means that each of the equations is a consequence of the other.

In the process of solving an equation, it is often necessary to apply such transformations that lead to an equation that is a consequence of the original one. The consequence equation is satisfied by all the roots of the original equation, but, in addition to them, the consequence equation can also have solutions that are not the roots of the original equation, these are the so-called extraneous roots. To identify and weed out extraneous roots, they usually do this: all found roots of the consequence equation are checked by substitution into the original equation.

If, when solving an equation, we replaced it with a consequence equation, then the above verification is an integral part of solving the equation. Therefore, it is important to know under what transformations this equation goes into the corollary.

Consider the equation

and multiply both its parts by the same expression that makes sense for all values ​​of x. We get the equation

whose roots are both the roots of equation (3) and the roots of the equation . Hence, equation (4) is a consequence of equation (3). It is clear that equations (3) and (4) are equivalent if the "outside" equation has no roots.

So, if both parts of the equation are multiplied by an expression that makes sense for any values ​​of x, then we get an equation that is a consequence of the original one. The resulting equation will be equivalent to the original if the equation has no roots. Note that the reverse transformation, i.e., the transition from equation (4) to equation (3) by dividing both parts of equation (4) by the expression, as a rule, is unacceptable, since it can lead to loss of solutions (in this case, they can “lose” roots of the equation For example, the equation has two roots: 3 and 4. Dividing both parts of the equation by leads to an equation that has only one root 4, that is, the root has been lost.

Again, take equation (3) and square both sides. We get the equation

whose roots are both the roots of equation (3) and the roots of the "outside" equation, i.e. the equation is a consequence of equation (3).

Can lead to the appearance of so-called extraneous roots. In this article, we will firstly analyze in detail what is extraneous roots. Secondly, let's talk about the reasons for their occurrence. And thirdly, using examples, we will consider the main ways of sifting out extraneous roots, that is, checking the roots for the presence of extraneous ones among them in order to exclude them from the answer.

Extraneous roots of the equation, definition, examples

Algebra school textbooks do not define an extraneous root. There, the idea of ​​an extraneous root is formed by describing the following situation: with the help of some transformations of the equation, the transition from the original equation to the consequence equation is carried out, the roots of the obtained consequence equation are found, and the found roots are checked by substitution into the original equation, which shows that some of the found roots are not the roots of the original equation, these roots are called extraneous roots for the original equation.

Based on this base, you can take for yourself the following definition of an extraneous root:

Definition

extraneous roots are the roots of the equation-consequence obtained as a result of transformations, which are not the roots of the original equation.

Let's take an example. Consider the equation and the corollary of this equation x·(x−1)=0 , obtained by replacing the expression with the expression x·(x−1) which is identically equal to it. The original equation has a single root 1 . The equation obtained as a result of the transformation has two roots 0 and 1 . So 0 is an extraneous root for the original equation.

Causes of the possible appearance of extraneous roots

If no “exotic” transformations are used to obtain the consequence equation, but only basic transformations of equations are used, then extraneous roots can arise only for two reasons:

• due to the expansion of the ODZ and
• because both sides of the equation are raised to the same even power.

Here it is worth recalling that the expansion of the ODZ as a result of the transformation of the equation mainly occurs

• When reducing fractions;
• When replacing a product with one or more zero factors by zero;
• When replacing zero with a fraction with a zero numerator;
• When using some properties of powers, roots, logarithms;
• When using some trigonometric formulas;
• When multiplying both parts of the equation by the same expression, which vanishes on the ODZ for this equation;
• When released in the process of solving the signs of logarithms.

The example from the previous paragraph of the article illustrates the appearance of an extraneous root due to the expansion of the ODZ, which takes place when passing from the equation to the corollary equation x·(x−1)=0 . The ODZ for the original equation is the set of all real numbers, except for zero, the ODZ for the resulting equation is the set R, that is, the ODZ is extended by the number zero. This number eventually turns out to be an extraneous root.

We will also give an example of the appearance of an extraneous root due to the raising of both parts of the equation to the same even power. The irrational equation has a single root 4, and the consequence of this equation, obtained from it by squaring both parts of the equation, that is, the equation , has two roots 1 and 4 . From this it can be seen that squaring both sides of the equation led to the appearance of an extraneous root for the original equation.

Note that the expansion of the ODZ and the raising of both parts of the equation to the same even power does not always lead to the appearance of extraneous roots. For example, when passing from the equation to the corollary equation x=2, the ODZ expands from the set of all non-negative numbers to the set of all real numbers, but extraneous roots do not appear. 2 is the only root of both the first and second equations. Also, there is no appearance of extraneous roots during the transition from the equation to the equation-consequence. The only root of both the first and second equations is x=16 . That is why we are not talking about the causes of the appearance of extraneous roots, but about the reasons for the possible appearance of extraneous roots.

What is weeding out extraneous roots?

The term "eliminating extraneous roots" can only be called a well-established term, it is not found in all algebra textbooks, but it is intuitive, which is why it is usually used. What is meant by sifting out extraneous roots becomes clear from the following phrase: “... verification is an obligatory step in solving the equation, which will help to detect extraneous roots, if any, and discard them (usually they say “weed out”)” .

Thus,

Definition

Weeding out extraneous roots is the detection and rejection of extraneous roots.

Now you can move on to ways to weed out extraneous roots.

Methods for weeding out extraneous roots

Substitution check

The main way to weed out extraneous roots is a substitution check. It allows you to weed out extraneous roots that could arise both due to the expansion of the ODZ, and due to the raising of both parts of the equation to the same even power.

The substitution check is as follows: the found roots of the consequence equation are substituted in turn into the original equation or into any equation equivalent to it, those that give the correct numerical equality are the roots of the original equation, and those that give an incorrect numerical equality or expression, meaningless are extraneous roots for the original equation.

Let's use an example to show how extraneous roots are screened out through substitution into the original equation.

In some cases, weeding out extraneous roots is more appropriate to carry out in other ways. This applies mainly to those cases where the substitution check is associated with significant computational difficulties or when the standard way of solving equations of a certain type involves a different check (for example, sifting out extraneous roots when solving fractional-rational equations is carried out according to the condition that the denominator of the fraction is not equal to zero ). Let's analyze alternative ways of sifting out extraneous roots.

According to ODZ

In contrast to the substitution check, screening out extraneous roots by ODZ is not always appropriate. The fact is that this method allows you to filter out only extraneous roots that arise due to the expansion of the ODZ, and it does not guarantee the elimination of extraneous roots that could arise for other reasons, for example, due to raising both parts of the equation to the same even power . Moreover, it is not always easy to find the ODZ for the equation being solved. Nevertheless, the method of sifting out extraneous roots by ODZ should be kept in service, since its use often requires less computational work than the use of other methods.

The sifting of extraneous roots according to the ODZ is carried out as follows: all found roots of the consequence equation are checked for belonging to the region of permissible values ​​of the variable for the original equation or any equation equivalent to it, those that belong to the ODZ are the roots of the original equation, and those of them which do not belong to the ODZ are extraneous roots for the original equation.

An analysis of the information provided leads to the conclusion that it is advisable to screen out extraneous roots according to the ODZ if at the same time:

• it is easy to find the ODZ for the original equation,
• extraneous roots could arise only due to the expansion of the ODZ,
• substitution verification is associated with significant computational difficulties.

We will show how weeding out extraneous roots is carried out in practice.

Under the terms of the ODZ

As we said in the previous paragraph, if extraneous roots could arise only due to the expansion of the ODZ, then they can be filtered out according to the ODZ for the original equation. But it is not always easy to find ODZ in the form of a numerical set. In such cases, it is possible to screen out extraneous roots not according to the ODZ, but according to the conditions that determine the ODZ. Let us explain how the screening of extraneous roots is carried out according to the conditions of the ODZ.

The found roots are substituted in turn into the conditions that determine the ODZ for the original equation or any equation equivalent to it. Those of them that satisfy all conditions are the roots of the equation. And those of them that do not satisfy at least one condition or give an expression that does not make sense are extraneous roots for the original equation.

Let us give an example of screening out extraneous roots according to the conditions of the ODZ.

Screening out extraneous roots arising from raising both sides of the equation to an even power

It is clear that weeding out extraneous roots arising from raising both parts of the equation to the same even power can be done by substituting into the original equation or into any equation equivalent to it. But such verification can be associated with significant computational difficulties. In this case, it is worth knowing an alternative way to weed out extraneous roots, which we will talk about now.

Screening out extraneous roots that may arise when both parts of irrational equations of the form are raised to the same even power , where n is some even number, can be carried out according to the condition g(x)≥0 . This follows from the definition of an even root: an even root n is a non-negative number whose nth power is equal to the root number, whence . Thus, the voiced approach is a kind of symbiosis of the method of raising both parts of the equation to the same degree and the method of solving irrational equations by determining the root. That is, the equation , where n is an even number, is solved by raising both parts of the equation to the same even power, and sifting out extraneous roots is performed according to the condition g(x)≥0 taken from the method for solving irrational equations to determine the root.

In the presentation, we will continue to consider equivalent equations, theorems, and dwell in more detail on the stages of solving such equations.

First, let's recall the condition under which one of the equations is a consequence of the other (slide 1). The author once again cites some theorems on equivalent equations that were considered earlier: on the multiplication of parts of an equation by the same value h (x); raising the parts of the equation to the same even power; obtaining an equivalent equation from the equation log a f (x) = log a g (x).

On the 5th slide of the presentation, the main stages are highlighted, with the help of which it is convenient to solve equivalent equations:

Find solutions to an equivalent equation;

Analyze solutions;

Check.

Consider example 1. It is necessary to find a consequence of the equation x - 3 = 2. Find the root of the equation x = 5. Write the equivalent equation (x - 3)(x - 6) = 2(x - 6), applying the method of multiplying the parts of the equation by (x - 6). Simplifying the expression to the form x 2 - 11x +30 = 0, we find the roots x 1 = 5, x 2 = 6. each root of the equation x - 3 \u003d 2 is also a solution to the equation x 2 - 11x +30 \u003d 0, then x 2 - 11x +30 \u003d 0 is a consequence equation.

Example 2. Find another consequence of the equation x - 3 = 2. To obtain an equivalent equation, we use the method of raising to an even power. Simplifying the resulting expression, we write x 2 - 6x +5 = 0. Find the roots of the equation x 1 = 5, x 2 = 1. x \u003d 5 (the root of the equation x - 3 \u003d 2) is also a solution of the equation x 2 - 6x +5 \u003d 0, then the equation x 2 - 6x +5 \u003d 0 is also a consequence equation.

Example 3. It is necessary to find a consequence of the equation log 3 (x + 1) + log 3 (x + 3) = 1.

Let us replace 1 = log 3 3 in the equation. Then, applying the statement from Theorem 6, we write the equivalent equation (x + 1)(x +3) = 3. Simplifying the expression, we obtain x 2 + 4x = 0, where the roots are x 1 = 0, x 2 = - 4. So the equation x 2 + 4x = 0 is a consequence for the given equation log 3 (x + 1) + log 3 (x + 3) = 1.

So, we can conclude: if the domain of definition of the equation is expanded, then an equation-consequence is obtained. We single out the standard actions in finding the equation-consequence:

Getting rid of the denominators that contain the variable;

Raising the parts of the equation to the same even power;

Exemption from logarithmic signs.

But it is important to remember: when the domain of definition of the equation is expanded during the solution, it is necessary to check all the roots found - whether they will fall into the ODZ.

Example 4. Solve the equation presented on slide 12. First, find the roots of the equivalent equation x 1 \u003d 5, x 2 \u003d - 2 (first stage). It is imperative to check the roots (second stage). Checking the roots (third stage): x 1 \u003d 5 does not belong to the range of permissible values ​​\u200b\u200bof the given equation, therefore the equation has one solution only x \u003d - 2.

In example 5, the found root of the equivalent equation is not included in the ODZ of the given equation. In example 6, the value of one of the two found roots is not defined, so this root is not a solution to the original equation.