The equation of the form AX4 + BX2 + C \u003d 0 is called a biquette equation. Absolutely any equation of this type can be solved using the introduction of a new variable and the subsequent solution of the equation relative to it. After spending a replacement and find the desired x.
Let's look at how to apply this method when solving rational equations.
The equation is given: x4 - 4x2 + 4 \u003d 0.
Decision
To solve this equation, you must enter a new variable, which has the form y \u003d x2. Also rightly the following equality: x4 \u003d (x2) 2 \u003d y2. The initial equation is rewriting as follows: Y2 - 4Y + 4 \u003d 0. This is a common square equation, deciding which, you get the roots y1 \u003d y2 \u003d 2. Since y \u003d x2, then the solution of this task is reduced to solving another equation, namely: x2 \u003d 2. Find the answer: + -√2.
In this situation, the method of introducing the variable was "adequate the situation", that is, it was clearly visible, which expression is replaced with a new variable, but it does not always happen. Basically, the expression that can be replaced is manifested only in the process of transformation and simplify the initial expression. You can see the analysis of this example in the video tutorial.
The properties of the function y \u003d k / x, when k\u003e 0
In the video tutorial, you will get acquainted with the basic properties of hyperboles, based on its geometric model.
1. D (f) \u003d (-∞; 0) ∪ (0; ∞) - the function of determining the function consists of all numbers, except 0.
2. At x\u003e 0 \u003d\u003e y\u003e 0, and with x< 0 => Y.< 0.
3. For k\u003e 0, the function decreases on an open beam (-∞; 0) and on the open beam (0; ∞).
4. The function y \u003d k / x does not have restrictions on top and bottom.
5. The function y \u003d k / x does not have the greatest and smallest values.
6. Continuous on the interval (-∞; 0) and (0; ∞), undergoing the gap at x \u003d 0.
With the method of introducing a new variable in solving rational equations from one variable, you learned to know the 8th grade algebras. The essence of this method in solving systems of equations is the same, but from a technical point of view there are some features that we discuss in the following examples.
Example 3. Solve the system of equations
Decision. We introduce a new variable then the first equation of the system can be rewritten in more simple sight: 
This is the equation relative to the variable T:
Both of these values \u200b\u200bsatisfy the condition, and therefore are the roots of the rational equation with the variable T. But it means, or from where we find that x \u003d 2, or
Thus, using the introduction method of the new variable, we managed to "bundle" the first system equation, quite complicated, on two simpler equations:
x \u003d 2 y; y - 2x.
What's next? And then each of the two obtained ordinary equations should be alternately considered in the system with equation x 2 - in 2 \u003d 3, which we have not yet remembered. In other words, the task is reduced to solving two systems of equations:
It is necessary to find solutions of the first system, the second system and all received vapors enable in response. Perform the first system of equations:
We use the substitution method, especially since everything is ready for him: we will substitute the expression 2y instead of x in the second equation of the system. Receive
Since x \u003d 2u, we are found respectively x 1 \u003d 2, x 2 \u003d 2. Thereby, two solutions of a given system were obtained: (2; 1) and (-2; -1). We will solve the second system of equations:
We will use the method of substitution again: we will substitute the expression 2x instead of in the second equation of the system. Receive
This equation does not have roots, it means that the equation system has no solutions. Thus, in response, only solutions of the first system must be included.
Answer: (2; 1); (-2; -1).
The method of introducing new variables when solving systems of two equations with two variables is used in two versions. First option: One new variable is introduced and is used in only one system equation. This is how the case was in the example 3.The order: two new variables are introduced and are used simultaneously in both equations of the system. So will be the case in example 4.
Example 4. Solve the system of equations
Lesson on the topic: solving equations
Amounted to: Volkova Vera Viktorovna - Mathematics Teacher
The subject of the lesson: solving the equations by introducing a new variable.
Objectives lesson: 1. Introduce students with a new method of solving equations;
2. Create Solutions Skills square equations and choosing the methods of their solutions;
3. Will the primary fixation of a new topic;
4. Develop the ability to defend its point of view, argued to conduct a dialogue with classmates;
Develop attention, memory and logical thinking, Observation
Check out the skills of communicability and culture of communication
Put the skills of independent work
During the classes
1. Argmoment
Message Topics lesson and setting goals.
2. Repetition
In previous lessons, we learned to solve square equations different ways and equations. Which can be brought to square.
What equation is called square.
What ways to solve them you know
What equations can be brought to square
a) (x + 3) 2 + (x - 2) 2 + (x + 5) (x -5) \u003d 11x +20
b) x 2 (x + 1) - (x + 4) x \u003d 12 (x - 1) 2
c) x 2 + x + 9 \u003d 3x-7,
d) x + 1 + x \u003d 2.5
X x + 1
e) x 2 + 2x + 2 + x 2 + 2x + 3 \u003d 9
X 2 + 2x + 5 x 2 + 2x + 6 10?
3. Studying a new material.
Now we will work in groups (remind about the procedure for work and rules of behavior when working in groups). Your task is to solve the proposed equations (cards are distributed with a task, a poster is hung on the board).
but) x + 1 + x \u003d 2.5
X x + 1
b) x 2 + 2x + 2 + x 2 + 2x + 3 \u003d 9
X 2 + 2x + 5 x 2 + 2x + 6 10
The teacher watches the work under work and selects the form of checking the first equation:
Orally or on the board depending on the success of the class.
Let's check that you did.
The first equation is reduced to a square equation x 2 + x -2 \u003d 0.
The solution of which is numbers -2 and 1.
And now we turn to solve the second equation. In all groups it turned out the fourth degree equation, which you do not know how to decide.
Let's try to figure it out yet.
As be the solution of any problem, the solution of the equation consists of a number of stages:
- Analysis of the equation
- Drawing up a solution plan.
- Implementation of this plan.
- Solutions check.
- Analysis of the decision method to systematize experience.
- - How is the equation analysis is usually carried out?
First of all, we answer the question, did we meet with the equations of this species earlier?
Yes, they met, this is a fractional equation.
You can try to solve this "heavy" equation, and you can return to
the initial equation and once again analyze it.
For this:
- We highlight some elements of the equation,
- We will establish their general properties,
- We study the link between different elements of the equation,
- We use this information.
We will work for 5 minutes in groups on this plan.
Most single elements included in the numerators and denominators of fractions in the equation. So that the equation becomes easier, replace this expression by one letter, for example Z:
X 2 + 2x \u003d z
Z +2 + z +3 \u003d 9
Z +5 z +6 10
It can be considered as a new equation for a relatively new unknown Z. In it, the variable x is not present in a clear form.
They say that the variable is replaced.
Is such a replacement appropriate? To answer this question, it is enough to figure out:
Is it possible to solve a new equation and find Z values,
Is it possible to find the value of the variable x for the source equation.
Try working in groups to answer the first part of the question.
The teacher watches the work of work. Then the results of the search for values \u200b\u200bof the variable Z are checked.
So, we found the values \u200b\u200bof the variable z: z 1 \u003d 0, z 2 \u003d - 61 | eleven
But we are interested in all values \u200b\u200bof variable x, satisfying the initial equation. We find these values. The relationship between the roots of the original and the new equation is contained in the formula x 2 + 2x \u003d z. The values \u200b\u200bof the Z variable we already found. Consequently, any root of the original fractional - rational equation is the root of one of the equations: x 2 + 2x \u003d z 1 or x 2 + 2x \u003d z 2
Decide these equations yourself according to options.
Check the results: the first equation has roots x 1 \u003d 0, x 2 \u003d -2, and the second equation does not have roots.
It remains to check the results obtained for the source equation and write down the answer.
Answer: x 1 \u003d 0, x 2 \u003d -2.
So, we decided the initial equation with a new method called the introduction of a new variable.
Make an algorithm for solving our equation the introduction of a new variable.(Work in groups)
- We allocate the expression x 2 + 2x;
- We indicate the expression of one letter x 2 + 2x \u003d z;
- We carry out the substitution and get a new equation;
- We give it to the square and decide;
- By values \u200b\u200bof the variable z, we find the values \u200b\u200bof the variable x;
- We make a check of the results obtained and write the answer.
3. Reflowing material.
What do you think it was possible to carry out another replacement of variables? (For example, x 2 + 2x
2 \u003d z or x 2 + 2x +6 \u003d Z.) What kind of will then have a new equation? How to solve them? Can the first home equation solve the introduction of a new variable? What expression can be replaced with a new variable? What is the equation? How to solve it? What are the values \u200b\u200bof the variable z? What are the values \u200b\u200bof the variable x?
4. Approach the outcome.
- What have we studied today at the lesson?
- What new way solutions equations did you find out?
- What is the method of introducing a new variable?
- What is the algorithm of this method?
- Did you find this method difficult, inconvenient?
- Is it possible to apply it for everyone?
5. Maximum task.
- Write down and learn the algorithm for the application of the introduction of a new variable;
- To solve this method No. 2.43 (1; 2) GIA p.117.
Compliance with 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 inform us if you have any questions.
Collection and use of personal information
Under personal information is subject to data that can be used to identify a certain person or communicating with it.
You can be requested to provide your personal information at any time when you connect with us.
Below are some examples of the types of personal information that we can collect, and how we can use such information.
What personal information we collect:
- When you leave a bid on the site, we can collect various information, including your name, phone number, address email etc.
As we use your personal information:
- We collected personal information allows us to contact you and report on unique proposals, promotions and other events and nearest events.
- From time to time, we can use your personal information to send important notifications and messages.
- We can also use personalized information for internal purposes, such as auditing, data analysis and various studies in order to improve the services of our services and providing you with recommendations for our services.
- If you participate in the prizes, competition or similar stimulating event, we can use the information you provide to manage such programs.
Information disclosure to third parties
We do not reveal the information received from you to third parties.
Exceptions:
- If necessary, in accordance with the law, judicial order, in the trial, and / or on the basis of public queries or requests from government agencies On the territory of the Russian Federation - to disclose your personal information. We can also disclose information about you if we define that such disclosure is necessary or appropriate for the purpose of security, maintaining law and order, or other socially important cases.
- In the case of reorganization, mergers or sales, we can convey the personal information we collect the corresponding to the third party - a successor.
Protection of personal information
We are making precautions - including administrative, technical and physical - to protect your personal information from loss, theft, and unscrupulous use, as well as from unauthorized access, disclosure, changes and destruction.
Compliance with your privacy at the company level
In order to make sure that your personal information is safe, we bring the norm of confidentiality and security to our employees, and strictly follow the execution of confidentiality measures.
2.2.3. Method of introducing a new variable.
The powerful means of solving the irrational equations is the method of introducing a new variable, or "replacement method". The method is usually applied if a certain expression is repeatedly encountered in the equation, depending on an unknown value. Then it makes sense to designate this expression of some new letter and try to solve the equation first about the introduced unknown, and then find the original unknown. In some cases, successfully introduced new unknown persons sometimes make it possible to get a decision faster and easier; Sometimes without replacement, the task is generally impossible. .
Example 7. Solve equation.
Decision. Putting, we obtain a significantly simpler irrational equation. Erected both parts of the equation in the square :.
;
;
;
Checking the found values \u200b\u200bof their substitution to the equation shows that - the root of the equation, and is an extraneous root.
Returning to the original variable x, we obtain the equation, that is, a square equation 
Deciding which we find two roots: ,. Both roots, as the test shows, satisfy the source equation.
The replacement is especially useful if a result is achieved by a new quality, for example, the irrational equation turns into a square.
Example 8. Solve equation.
Decision. I rewrite the equation like this :.
It can be seen that if you enter a new variable 
then the equation will take the form 
Where,.
Now the task is reduced to solving the equation 
and equations 
. The first of these solutions does not have, but from the second we get ,. Both roots, as the test shows, satisfy the source equation.
It should be noted that "thoughtless" application in Example 8 of the "Radical Privacy" method and the construction of the square would lead to the fourth degree equation, the solution of which is generally extremely complex task.
Example 9. Solve equation 
.
We introduce a new variable
As a result, the initial irrational equation takes the view of the square
,
where taking into account the restriction, we get. Solving equation, we get the root. As the test shows, satisfies the initial equation.
Sometimes, by some substitution, it is possible to bring an irrational equation to rational form, as considered examples 8, 9. In this case, it is said that this substitution rationalizes the irrational equation in question, and is called its rationalizing. Based on the use of rationalizing substitutions is called the rationalization method.
With all students in the lesson, this method of solving irrational equations is not necessary, but it can be considered within the framework of optional or circle classes in mathematics with students showing increased interest in mathematics.
Based on knowledge of communication between the result and components arithmetic action (i.e. knowledge of the methods of finding unknown components). These requirements of the program define the work technique on the equations. 2. Methods of studying inequalities in high schools 2.1 The content and role of the line of equations and inequalities in the modern school course of mathematics due to the importance and extensity of the material, ...
On a qualitatively new stage of mastering the content of school mathematics. Chapter II. Methodico - pedagogical foundations of using independent work, as a means of learning to solve equations in 5 - 9 classes. § 1. Organization of independent work when learning to solve equations in 5 - 9 classes. For traditional method Teaching Teacher often puts a student to the position of the object ...
It can be concluded that insufficient lighting of the question under study in modern methodical literature. Object research work: the process of learning mathematics. Subject: Formation of the ability to solve square equations in students of the 8th grade. Contingent: 8th grade students. Chapter 1. Theoretical aspects learning to solve equations in grade 8 1.1. From the history of the occurrence of square ...
Numerical argument, so with this approach there is a certain redundancy in the formation of a function as a generalized concept. 2. The main directions of introducing the concept of function in the school course of mathematics in the modern school course of mathematics the leading approach is considered genetic with the addition of elements of logical. The formation of concepts and ideas, methods and techniques in the composition ...
