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”

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: to direct the cognitive activity of students to the development of skills 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 the 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) = sin x. 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 ways you use to transform equations and for each way 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.

This presentation can be used when conducting an algebra lesson and starting analysis in grade 11 when studying the topic "Equations - consequences" according to the teaching materials of the authors S.M. Nikolsky, M.K. Potapov, N.N. Reshetnikov, A.V. Shevkin

View document content
“Equations of consequence. Other transformations leading to the equation corollary"

EQUATIONS - CONSEQUENCES

ORAL WORK

• What equations are called corollary equations?
• What is called the transition to the consequence equation
• What transformations lead to the corollary equation?

ORAL WORK

• √ x= 6
• √ x-2 = 3
• 3 √x= 4;
• √ x 2 \u003d 9
• √ x+4=-2
• √x+1+√x+2=-2

No solutions

No solutions

ORAL WORK

No solutions

Transformations leading to the corollary equation

transformation

Influence on the roots of the equation

Raising an Equation to an EVEN Power

f(x)=g(x) (f(x)) n =(g(x)) n

Potentiation of logarithmic equations, i.e. replacement:

log a f(x)=log a g(x) f(x)= g(x)

Can lead to extraneous roots

Releasing the equation from denominators:

Can lead to the appearance of extraneous roots, i.e. those numbers x i for which or

Replacing the difference f(x)-f(x) by zero, i.e. reduction of similar members

Can lead to the appearance of extraneous roots, i.e. those numbers for each of which the function f(x) is not defined.

If, when solving this equation, a transition to the consequence equation is made, then it is necessary to check whether all the roots of the consequence equation are the roots of the original equation.

Checking the obtained roots is a mandatory part of solving the equation.

№ 8.2 2 (a) Solve the Equation :

2) No. 8.23(a)

№ 8.24 (a, c) Solve the Equation :

№ 8.25 (a, c) Solve the Equation :

№ 8.28 (a, c) Solve the Equation :

№ 8.29 (a, c) Solve the Equation :

HOMEWORK

• Run No. 8.24 (b, d), p. 236
• No. 8.25(b, d)
• 8.28 (b, d)
• 8.29 (b, d)

Class: 11

Duration: 2 lessons.

The purpose of the lesson:

• (for teacher) the formation of a holistic view of the methods for solving irrational equations among students.
• (for students) Development of the ability to observe, compare, generalize, analyze mathematical situations (slide 2). Preparation for the exam.

First lesson plan(slide 3)

1. Knowledge update
2. Analysis of the theory: Raising an equation to an even power
3. Workshop on solving equations

Plan of the second lesson

1. Differentiated independent work on groups "Irrational equations on the exam"
2. Summary of lessons
3. Homework

Course of lessons

I. Updating knowledge

Target: repeat the concepts necessary for the successful development of the topic of the lesson.

front poll.

What two equations are said to be equivalent?

What transformations of the equation are called equivalent?

- Replace this equation with an equivalent one with an explanation of the applied transformation: (slide 4)

a) x + 2x +1; b) 5 = 5; c) 12x = -3; d) x = 32; e) = -4.

What equation is called the equation-consequence of the original equation?

– Can the consequence equation have a root that is not the root of the original equation? What are these roots called?

– What transformations of the equation lead to the equation-consequences?

What is an arithmetic square root?

Let's dwell today in more detail on the transformation "Raising an equation to an even power".

II. Analysis of the theory: Raising an equation to an even power

Explanation by the teacher with the active participation of students:

Let 2m(mN) is a fixed even natural number. Then the consequence of the equationf(x) =g(x) is the equation (f(x)) = (g(x)).

Very often this statement is used in solving irrational equations.

Definition. An equation containing the unknown under the sign of the root is called irrational.

When solving irrational equations, the following methods are used: (slide 5)

Attention! Methods 2 and 3 require mandatory checks.

ODZ does not always help to eliminate extraneous roots.

Conclusion: when solving irrational equations, it is important to go through three stages: technical, solution analysis, verification (slide 6).

III. Workshop on solving equations

Solve the equation:

After discussing how to solve the equation by squaring, solve by passing to an equivalent system.

Conclusion: the solution of the simplest equations with integer roots can be carried out by any familiar method.

b) \u003d x - 2

Solving by raising both parts of the equation to the same power, students get the roots x = 0, x = 3 -, x = 3 +, which are difficult and time-consuming to check by substitution. (Slide 7). Transition to an equivalent system

allows you to quickly get rid of extraneous roots. The condition x ≥ 2 is satisfied only by x.

Answer: 3+

Conclusion: It is better to check irrational roots by passing to an equivalent system.

c) \u003d x - 3

In the process of solving this equation, we obtain two roots: 1 and 4. Both roots satisfy the left side of the equation, but for x \u003d 1, the definition of the arithmetic square root is violated. The ODZ equation does not help eliminate extraneous roots. The transition to an equivalent system gives the correct answer.

Conclusion:a good knowledge and understanding of all the conditions for determining the arithmetic square root helps to move on toperforming equivalent transformations.

By squaring both sides of the equation, we get the equation

x + 13 - 8 + 16 \u003d 3 + 2x - x, separating the radical to the right side, we get

26 - x + x \u003d 8. Applying further steps to squaring both parts of the equation, will lead to an equation of the 4th degree. The transition to the ODZ equation gives a good result:

find the ODZ equation:

x = 3.

Check: - 4 = , 0 = 0 is correct.

Conclusion:sometimes it is possible to carry out a solution using the definition of the ODZ equation, but be sure to check.

Solution: ODZ equation: -2 - x ≥ 0 x ≤ -2.

For x ≤ -2,< 0, а ≥ 0.

Therefore, the left side of the equation is negative, and the right side is non-negative; so the original equation has no roots.

Answer: no roots.

Conclusion:having made the correct reasoning on the restriction in the condition of the equation, you can easily find the roots of the equation, or establish that they do not exist.

Using the example of solving this equation, show the double squaring of the equation, explain the meaning of the phrase "solitude of radicals" and the need to check the found roots.

h) + = 1.

The solution of these equations is carried out by the method of changing the variable until the return to the original variable. Finish the decision to offer those who will cope with the tasks of the next stage earlier.

test questions

• How to solve the simplest irrational equations?
• What should be remembered when raising an equation to an even power? ( extraneous roots may appear)
• What is the best way to check irrational roots? ( using the ODZ and the conditions for the coincidence of the signs of both parts of the equation)
• Why is it necessary to be able to analyze mathematical situations when solving irrational equations? ( For the correct and quick choice of a method for solving an equation).

IV. Differentiated independent work on groups "Irrational equations on the exam"

The class is divided into groups (2-3 people each) according to the levels of training, each group chooses an option with a task, discusses and solves the selected tasks. When necessary, contact the teacher for advice. After completing all the tasks of their version and checking the answers by the teacher, the group members individually complete the solution of equations g) and h) of the previous stage of the lesson. For options 4 and 5 (after checking the answers and the teacher's decision), additional tasks are written on the board, which are performed individually.

All individual solutions at the end of the lessons are handed over to the teacher for verification.

Option 1

Solve the equations:

a) = 6;
b) = 2;
c) \u003d 2 - x;
d) (x + 1) (5 - x) (+ 2 = 4.

Option 5

1. Solve the equation:

a) = ;
b) = 3 - 2x;

2. Solve the system of equations:

Additional tasks:

v. Summary of lessons

What difficulties did you experience in completing the exam assignments? What is needed to overcome these difficulties?

VI. Homework

Repeat the theory of solving irrational equations, read paragraph 8.2 in the textbook (pay attention to example 3).

Solve No. 8.8 (a, c), No. 8.9 (a, c), No. 8.10 (a).

Literature:

1. Nikolsky S.M., Potapov M.K., N.N. Reshetnikov N.N., Shevkin A.V. Algebra and beginning of mathematical analysis , textbook for the 11th grade of educational institutions, M .: Education, 2009.
2. Mordkovich A.G. On some methodological issues related to the solution of equations. Mathematics at school. -2006. -Number 3.
3. M. Shabunin. Equations. Lectures for high school students and entrants. Moscow, "Chistye Prudy", 2005. (library "First of September")
4. E.N. Balayan. Workshop on problem solving. Irrational equations, inequalities and systems. Rostov-on-Don, "Phoenix", 2006.
5. Maths. Preparation for the exam-2011. Edited by F.F. Lysenko, S.Yu. Kulabukhov Legion-M, Rostov-on-Don, 2010.

Some transformations allow us to move from the equation being solved to equivalent equations, as well as to consequence equations, which simplifies the solution of the original equation. In this material, we will tell you what these equations are, formulate the main definitions, illustrate them with illustrative examples and explain exactly how the roots of the original equation are calculated from the roots of the consequence equation or an equivalent equation.

Yandex.RTB R-A-339285-1

The concept of equivalent equations

Definition 1

Equivalent called such equations that have the same roots, or those in which there are no roots.

Definitions of this type are often found in various textbooks. Let's give some examples.

Definition 2

The equation f (x) = g (x) is considered equivalent to the equation r (x) = s (x) if they have the same roots or they both have no roots.

Definition 3

Equations with the same roots are considered equivalent. Also, they are considered two equations that equally do not have roots.

Definition 4

If the equation f (x) \u003d g (x) has the same set of roots as the equation p (x) \u003d h (x), then they are considered equivalent with respect to each other.

When we talk about a coinciding set of roots, we mean that if a certain number is the root of one equation, then it will fit as a solution to another equation. None of the equations that are equivalent can have a root that is not suitable for the other.

We give several examples of such equations.

Example 1

For example, 4 x \u003d 8, 2 x \u003d 4 and x \u003d 2 will be equivalent, since each of them has only one root - two. Also, x · 0 = 0 and 2 + x = x + 2 will be equivalent, since their roots can be any numbers, that is, the sets of their solutions are the same. The equations x = x + 5 and x 4 = − 1 will also be equivalent, each of which has no solution.

For clarity, consider several examples of non-equivalent equations.

Example 2

For example, x = 2 and x 2 = 4 will be, because their roots are different. The same applies to the equations x x \u003d 1 and x 2 + 5 x 2 + 5, because in the second the solution can be any number, and in the second the root cannot be 0.

The definitions given above are also suitable for equations with several variables, however, in the case when we are talking about two, three or more roots, the expression "solution of the equation" is more appropriate. Thus, to summarize: equivalent equations are those equations that have the same solutions or none at all.

Let's take examples of equations that contain several variables and are equivalent to each other. So, x 2 + y 2 + z 2 = 0 and 5 x 2 + x 2 y 4 z 8 = 0 include three variables each and have only one solution equal to 0 in all three cases. And the pair of equations x + y = 5 and x y = 1 will not be equivalent to each other, since, for example, the values ​​5 and 3 are suitable for the first, but will not be a solution to the second: when substituting them into the first equation, we get the correct equality, and in the second - false.

The concept of corollary equations

Let us quote several examples of definitions of corollary equations taken from textbooks.

Definition 5

The consequence of the equation f (x) = g (x) will be the equation p (x) = h (x), provided that each root of the first equation is at the same time the root of the second.

Definition 6

If the first equation has the same roots as the second, then the second will be a consequence of the first.

Let's take a few examples of such equations.

Example 3

So, x 2 = 32 will be a consequence of x - 3 = 0, since the first has only one root equal to three, and it will also be the root of the second equation, so in the context of this definition, one equation will be a consequence of another. Another example: the equation (x − 2) (x − 3) (x − 4) = 0 will be a consequence of x - 2 x - 3 x - 4 2 x - 4 because the second equation has two roots, equal to 2 and 3, which at the same time will be the roots of the first.

From the above definition, we can conclude that any equation that does not have roots will also be a consequence of any equation. Here are some other consequences of all the rules formulated in this article:

Definition 7

1. If one equation is equivalent to another, then each of them will be a consequence of the other.
2. If of two equations each is a consequence of the other, then these equations will be equivalent to each other.
3. Equations will be equivalent with respect to each other only if each of them is a consequence of the other.

How to find the roots of an equation from the roots of a consequence equation or an equivalent equation

Based on what we wrote in the definitions, then in the case when we know the roots of one equation, then we also know the roots of equivalent ones, since they will coincide.

If we know all the roots of the consequence equation, then we can determine the roots of the second equation, of which it is a consequence. To do this, you just need to weed out extraneous roots. We wrote a separate article about how this is done. We advise you to read it.

If you notice a mistake in the text, please highlight it and press Ctrl+Enter

To study today's topic, we need to repeat which equation is called the consequence equation, which theorems are "restless" and what steps the solution of any equation consists of.

Definition. If each root of the equation ef from x is equal to x (we denote it by the number one) is at the same time the root of the equation pe from x, equal to ash from x (we denote it by the number two), then equation two is called a consequence of equation one.

Theorem four. If both sides of the equation ef from x is equal to the same from x, multiply by the same expression ash from x, which is:

First, it makes sense everywhere in the domain of definition (in the range of admissible values) of the equation eff from x, which is equal to from x.

Secondly, nowhere in this region does it vanish, then we get the equation ef from x, multiplied by ash from x is equal to x, multiplied by ash from x, equivalent to the given in its ODZ.

Consequence theorem four is another "calm" statement: if both parts of the equation are multiplied or divided by the same non-zero number, then an equation is obtained that is equivalent to the given one.

Theorem five. If both sides of the equation

ef from x is equal to x is non-negative in the ODZ equation, then after raising both of its parts to the same even power n, we get the equation eff from x to the power of x is equal to x to the power of x, equivalent to this equation in its o de ze.

Theorem six. Let a be greater than zero, and not equal to one, and eff from x greater than zero,

zhe from x is greater than zero, the tolologarithmic equation is the logarithm of ef from x to the base a, equal to the logarithm of zhe from x to the base a,

is equivalent to the equation ef from x is the same as from x .

As we have already said, the solution of any equations occurs in three stages:

The first stage is technical. With the help of a chain of transformations from the original equation, we come to a fairly simple equation, which we solve and find the roots.

The second stage is the analysis of the solution. We analyze the transformations that we performed and find out if they are equivalent.

The third stage is verification. Checking all the roots found by substituting them into the original equation is mandatory when performing transformations that can lead to a consequence equation.

In this lesson, we will find out, when applying what transformations does this equation go into a consequence equation? Consider the following tasks.

Exercise 1

Which equation is a consequence of the equation x minus three equals two?

Solution

The equation x minus three equals two has a single root - x equals five. Multiply both sides of this equation by the expression x minus six, add like terms and get the quadratic equation x square minus eleven x plus thirty equals zero. Let's calculate its roots: x the first is equal to five; x second equals six. It already contains two roots. The equation x square minus eleven x plus thirty equals zero contains a single root - x equals five; of the equation x minus three equals two, so x squared minus eleven x plus thirty is a consequence of the equation x minus three equals two.

Task 2

What other equation is a consequence of the equation x-3=2?

Solution

In the equation x minus three is equal to two, we square both parts of it, apply the formula for the square of the difference, add like terms, we get the quadratic equation x squared minus six, x plus five is equal to zero.

Let's calculate its roots: x the first is equal to five, x the second is equal to one.

The root x equals one is extraneous to the equation x minus three equals two. This happened because both sides of the original equation were squared (an even power). But at the same time, its left side - x minus three - can be negative (conditions theorem five). So the equation x square minus six x plus five equals zero is a consequence of the equation x minus three equals two.

Task 3

Find the equation-corollary for the equation

the logarithm of x plus one to base three plus the logarithm of x plus three to base three equals one.

Solution

We represent unity as the logarithm of three to the base of three, potentiate the logarithmic equation, perform multiplication, add like terms and get the quadratic equation x squared plus four x equals zero. Let's calculate its roots: x the first is equal to zero, x the second is equal to minus four. The root x is equal to minus four is extraneous for the logarithmic equation, since when it is substituted into the logarithmic equation, the expressions x plus one and x plus three take negative values ​​- the conditions are violated theorem six.

So the equation x squared plus four x equals zero is a consequence of this equation.

Based on the solution of these examples, we can do conclusion:the consequence equation is obtained from the given equation by expanding the domain of the equation. And this is possible when performing such transformations as

1) getting rid of denominators containing a variable;

2) raising both parts of the equation to the same even power;

3) exemption from signs of logarithms.

Remember! If in the process of solving the equation the domain of definition of the equation has expanded, then it is necessary to check all the roots found.

Task 4

Solve the equation x minus three divided by x minus five plus one divided by x is equal to x plus five divided by x times x minus five.

Solution

The first stage is technical.

Let's perform a chain of transformations, get the simplest equation and solve it. To do this, we multiply both parts of the equation by a common denominator of fractions, that is, by the expression x multiplied by xminus five.

We get the quadratic equation x square minus three x minus ten equals zero. Let's calculate the roots: x the first is equal to five, x the second is equal to minus two.

The second stage is the analysis of the solution.

When solving the equation, we multiplied both parts of it by an expression containing a variable. This means that the domain of definition of the equation has expanded. Therefore, checking the roots is required.

The third stage is verification.

When x equals minus two, the common denominator does not vanish. So x equals minus two is the root of this equation.

When x equals five, the common denominator goes to zero. Therefore x is equal to five - an extraneous root.

Answer: minus two.

Task 5

Solve the equation square root of x minus six is ​​equal to square root of four minus x.

Solution

The first stage is technical .

In order to obtain a simple equation and solve it, we perform a chain of transformations.

Let's square (an even power) both parts of this equation, move the x's to the left side, and the numbers to the right side of the equation, give like terms, we get: two x equals ten. X is equal to five.

The second stage is the analysis of the solution.

Let's check the performed transformations for equivalence.

When solving an equation, we squared both sides of it. This means that the domain of definition of the equation has expanded. Therefore, checking the roots is required.

The third stage is verification.

We substitute the found roots into the original equation.

If x is equal to five, the expression square root of four minus x is undefined, so x equal to five is an extraneous root. So this equation has no roots.

Answer: The equation has no roots.

Task 6

Solve the equation The natural logarithm of x squared plus two x minus seven is equal to the natural logarithm of x minus one.

Solution

The first stage is technical .

Let's perform a chain of transformations, get the simplest equation and solve it. To do this, we potentiate

equation, we transfer all the terms to the left side of the equation, we bring similar terms, we get a quadratic equation x square plus x minus six is ​​equal to zero. Let's calculate the roots: x the first is equal to two, x the second is equal to minus three.

The second stage is the analysis of the solution.

Let's check the performed transformations for equivalence.

In the process of solving this equation, we got rid of the signs of logarithms. This means that the domain of definition of the equation has expanded. Therefore, checking the roots is required.

The third stage is verification.

We substitute the found roots into the original equation.

If x is equal to two, then we get the natural logarithm of unity is equal to the natural logarithm of unity -

correct equality.

Hence, x equal to two is the root of this equation.

If x is minus three, then the natural logarithm of x squared plus two x minus seven and the natural logarithm of x minus one are undefined. So x equal to minus three is an extraneous root.

Answer: two.

Is it always necessary to distinguish three stages when solving an equation? How else can you check?

We will get answers to these questions in the next lesson.