Equation and its roots: definitions, examples. Which equation has no roots? Examples of equations When an equation has infinitely many roots

Having received a general idea of \u200b\u200bequalities, and having become acquainted with one of their types - numerical equalities, one can start talking about another very important form of equalities from a practical point of view - about equations. In this article we will analyze what is the equation, and what is called the root of the equation. Here we will give the corresponding definitions, as well as give various examples of equations and their roots.

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What is an equation?

A focused introduction to equations usually begins in 2nd grade mathematics. At this time the following is given equation definition:

Definition.

The equation Is an equality containing an unknown number to be found.

Unknown numbers in equations are usually denoted using small Latin letters, for example, p, t, u, etc., but the most commonly used letters are x, y and z.

Thus, the equation is defined in terms of the notation form. In other words, equality is an equation when it obeys the specified notation rules - it contains the letter whose value you want to find.

Here are some examples of the very first and simplest equations. Let's start with equations like x \u003d 8, y \u003d 3, etc. Equations that contain, together with numbers and letters, the signs of arithmetic operations, look a little more complicated, for example, x + 2 \u003d 3, z − 2 \u003d 5, 3 · t \u003d 9, 8: x \u003d 2.

The variety of equations grows after acquaintance with - equations with brackets begin to appear, for example, 2 (x − 1) \u003d 18 and x + 3 (x + 2 (x − 2)) \u003d 3. An unknown letter in the equation can appear several times, for example, x + 3 + 3 x − 2 − x \u003d 9, letters can also be on the left side of the equation, on its right side, or in both sides of the equation, for example, x (3 + 1) −4 \u003d 8, 7−3 \u003d z + 1, or 3x − 4 \u003d 2 (x + 12).

Further, after studying natural numbers, acquaintance with integers, rational, real numbers occurs, new mathematical objects are studied: degrees, roots, logarithms, etc., while more and more new types of equations appear that contain these things. Their examples can be found in the article main types of equationsstudying at school.

In the 7th grade, along with the letters, by which they mean some specific numbers, they begin to consider letters that can take on different meanings, they are called variables (see the article). In this case, the word "variable" is introduced into the definition of the equation, and it becomes like this:

Definition.

Equation is an equality containing a variable whose value you want to find.

For example, the equation x + 3 \u003d 6 x + 7 is an equation with variable x, and 3 · z − 1 + z \u003d 0 is an equation with variable z.

In algebra lessons in the same 7th grade, there is a meeting with equations that contain not one, but two different unknown variables in their record. They are called equations in two variables. In the future, the presence of three or more variables in the equations is allowed.

Definition.

Equations with one, two, three, etc. variables - these are equations containing one, two, three, ... unknown variables, respectively.

For example, the equation 3.2 x + 0.5 \u003d 1 is an equation with one variable x, while an equation of the form x − y \u003d 3 is an equation with two variables x and y. And one more example: x 2 + (y − 1) 2 + (z + 0.5) 2 \u003d 27. It is clear that such an equation is an equation with three unknown variables x, y and z.

What is the root of an equation?

The definition of the equation is directly related to the definition of the root of this equation. Let's do some reasoning that will help us understand what the root of the equation is.

Let's say we have an equation with one letter (variable). If, instead of the letter included in the record of this equation, a number is substituted, then the equation will turn into a numerical equality. Moreover, the resulting equality can be both true and false. For example, if you substitute the number 2 instead of the letter a in the equation a + 1 \u003d 5, you get an incorrect numerical equality 2 + 1 \u003d 5. If we substitute the number 4 for a in this equation, then we get the correct equality 4 + 1 \u003d 5.

In practice, in the overwhelming majority of cases, such values \u200b\u200bof the variable are of interest, substitution of which into the equation gives the correct equality, these values \u200b\u200bare called roots or solutions of this equation.

Definition.

Root of the equation Is the value of a letter (variable), when substituted, the equation turns into a true numerical equality.

Note that the root of an equation in one variable is also called a solution to the equation. In other words, the solution to the equation and the root of the equation are the same thing.

Let us explain this definition with an example. To do this, we return to the above equation a + 1 \u003d 5. According to the sounded definition of the root of the equation, the number 4 is the root of this equation, since when substituting this number instead of the letter a, we obtain the correct equality 4 + 1 \u003d 5, and the number 2 is not its root, since it corresponds to an incorrect equality of the form 2 + 1 \u003d five .

At this point, a number of natural questions arise: "Does any equation have a root, and how many roots does a given equation have?" We will answer them.

There are both equations with roots and equations without roots. For example, the equation x + 1 \u003d 5 has a root of 4, and the equation 0 x \u003d 5 has no roots, since no matter what number we substitute in this equation instead of the variable x, we get the wrong equality 0 \u003d 5.

As for the number of roots of the equation, there are both equations that have a certain finite number of roots (one, two, three, etc.), and equations that have infinitely many roots. For example, the equation x − 2 \u003d 4 has a unique root 6, the roots of the equation x 2 \u003d 9 are two numbers −3 and 3, the equation x (x − 1) (x − 2) \u003d 0 has three roots 0, 1, and 2, and the solution to the equation x \u003d x is any number, that is, it has an infinite set of roots.

A few words should be said about the accepted writing of the roots of the equation. If the equation has no roots, then usually they write “the equation has no roots”, or use the empty set sign ∅. If the equation has roots, then they are written separated by commas, or written as elements of the set in curly braces. For example, if the roots of the equation are the numbers −1, 2, and 4, then they write −1, 2, 4 or (−1, 2, 4). It is also permissible to write the roots of the equation in the form of the simplest equalities. For example, if the letter x enters the equation, and the roots of this equation are the numbers 3 and 5, then you can write x \u003d 3, x \u003d 5, also the variable is often added with subscripts x 1 \u003d 3, x 2 \u003d 5, as if indicating numbers roots of the equation. The infinite set of roots of the equation is usually written in the form, also, if possible, use the notation of the sets of natural numbers N, integers Z, real numbers R. For example, if the root of an equation with variable x is any integer, then write, and if the roots of an equation with variable y are any real number from 1 to 9, inclusive, then write.

For equations with two, three and more variables, as a rule, the term "equation root" is not used, in these cases they say "equation solution". What is called solving equations in several variables? Let us give an appropriate definition.

Definition.

Solving an equation with two, three, etc. variables call a couple, three, etc. values \u200b\u200bof the variables, which turns this equation into a true numerical equality.

Let us show some illustrative examples. Consider an equation in two variables x + y \u003d 7. Substitute in it instead of x the number 1, and instead of y the number 2, and we have the equality 1 + 2 \u003d 7. Obviously, it is wrong, therefore, the pair of values \u200b\u200bx \u003d 1, y \u003d 2 is not a solution to the written equation. If we take a pair of values \u200b\u200bx \u003d 4, y \u003d 3, then after substitution into the equation we will arrive at the correct equality 4 + 3 \u003d 7, therefore, this pair of values \u200b\u200bof variables is by definition a solution to the equation x + y \u003d 7.

Equations with several variables, like equations with one variable, may not have roots, may have a finite number of roots, or may have infinitely many roots.

Pairs, threes, fours, etc. variable values \u200b\u200bare often written concisely, listing their values \u200b\u200bseparated by commas in parentheses. In this case, the written numbers in brackets correspond to the variables in alphabetical order. Let's clarify this point by returning to the previous equation x + y \u003d 7. The solution to this equation x \u003d 4, y \u003d 3 can be briefly written as (4, 3).

The greatest attention in the school course of mathematics, algebra and the beginnings of analysis is paid to finding the roots of equations with one variable. We will analyze the rules of this process in great detail in the article. solving equations.

List of references.

• Maths... 2 cl. Textbook. for general education. institutions with adj. to the electron. carrier. At 2 pm Part 1 / [M. I. Moro, MA Bantova, GV Beltyukova and others] - 3rd ed. - M .: Prosveshenie, 2012 .-- 96 p .: ill. - (School of Russia). - ISBN 978-5-09-028297-0.
• Algebra: study. for 7 cl. general education. institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; ed. S. A. Telyakovsky. - 17th ed. - M.: Education, 2008 .-- 240 p. : ill. - ISBN 978-5-09-019315-3.
• Algebra: Grade 9: textbook. for general education. institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; ed. S. A. Telyakovsky. - 16th ed. - M.: Education, 2009 .-- 271 p. : ill. - ISBN 978-5-09-021134-5.

After we have studied the concept of equalities, namely one of their types - numerical equalities, we can move on to another important type - equations. Within the framework of this material, we will explain what an equation and its root are, formulate the basic definitions and give various examples of equations and finding their roots.

Equation concept

Usually, the concept of an equation is studied at the very beginning of the school algebra course. Then it is defined as follows:

Definition 1

Equation called equality with an unknown number to be found.

It is customary to denote unknowns with small Latin letters, for example, t, r, m, etc., but most often x, y, z are used. In other words, the equation determines the form of its recording, that is, equality will be an equation only when it is reduced to a certain form - it must contain a letter, the value that must be found.

Here are some examples of the simplest equations. These can be equalities of the form x \u003d 5, y \u003d 6, etc., as well as those that include arithmetic operations, for example, x + 7 \u003d 38, z - 4 \u003d 2, 8 t \u003d 4, 6: x \u003d 3.

After the concept of brackets is studied, the concept of equations with brackets appears. These include 7 (x - 1) \u003d 19, x + 6 (x + 6 (x - 8)) \u003d 3, etc. The letter to be found can occur not once, but several times, like, for example, in the equation x + 2 + 4 x - 2 - x \u003d 10. Also, unknowns can be located not only on the left, but also on the right, or in both parts at the same time, for example, x (8 + 1) - 7 \u003d 8, 3 - 3 \u003d z + 3 or 8 x - 9 \u003d 2 (x + 17).

Further, after the students get acquainted with the concept of integers, real, rational, natural numbers, as well as logarithms, roots and powers, new equations appear that include all these objects. We have devoted a separate article to examples of such expressions.

In the program for the 7th grade, the concept of variables appears for the first time. These are letters that can take on different meanings (for more details, see the article on numeric, literal and variable expressions). Based on this concept, we can redefine the equation:

Definition 2

The equation Is an equality that includes the variable whose value you want to evaluate.

That is, for example, the expression x + 3 \u003d 6 x + 7 is an equation with a variable x, and 3 y - 1 + y \u003d 0 is an equation with a variable y.

One equation may contain not one variable, but two or more. They are called, respectively, equations with two, three variables, etc. Let us write the definition:

Definition 3

Equations with two (three, four or more) variables are equations that include the corresponding number of unknowns.

For example, an equality of the form 3, 7 x + 0, 6 \u003d 1 is an equation with one variable x, and x - z \u003d 5 is an equation with two variables x and z. An example of an equation with three variables would be x 2 + (y - 6) 2 + (z + 0, 6) 2 \u003d 26.

Root of the equation

When we talk about an equation, the need immediately arises to define the concept of its root. Let's try to explain what it means.

Example 1

We are given some kind of equation that includes one variable. If we substitute a number for the unknown letter, then the equation becomes a numerical equality - true or false. So, if in the equation a + 1 \u003d 5 we replace the letter with the number 2, then the equality will become incorrect, and if 4, then we will get the correct equality 4 + 1 \u003d 5.

We are more interested in exactly those values \u200b\u200bwith which the variable will turn into the correct equality. They are called roots or solutions. Let's write down the definition.

Definition 4

The root of the equation is called a value of a variable that turns a given equation into a true equality.

The root can also be called a solution, or vice versa - both of these concepts mean the same thing.

Example 2

Let's take an example to clarify this definition. Above we gave the equation a + 1 \u003d 5. According to the definition, the root in this case will be 4, because when substituted instead of a letter, it gives the correct numerical equality, and two will not be a solution, since it corresponds to the incorrect equality 2 + 1 \u003d 5.

How many roots can one equation have? Does any equation have a root? Let's answer these questions.

Equations that do not have a single root also exist. An example would be 0 x \u003d 5. We can substitute infinitely many different numbers into it, but none of them will turn it into a true equality, since multiplication by 0 always gives 0.

There are also equations that have multiple roots. They can have both a finite and an infinitely large number of roots.

Example 3

So, in the equation x - 2 \u003d 4 there is only one root - six, in x 2 \u003d 9 there are two roots - three and minus three, in x (x - 1) (x - 2) \u003d 0 there are three roots - zero, one and two, in the equation x \u003d x there are infinitely many roots.

Now let's explain how to write the roots of the equation correctly. If they do not exist, then we write like this: "the equation has no roots." In this case, one can also indicate the sign of the empty set ∅. If there are roots, then we write them separated by commas or indicate them as elements of a set, enclosing them in curly braces. So, if any equation has three roots - 2, 1 and 5, then we write - 2, 1, 5 or (- 2, 1, 5).

It is allowed to write roots in the form of the simplest equalities. So, if the unknown in the equation is denoted by the letter y, and the roots are 2 and 7, then we write y \u003d 2 and y \u003d 7. Sometimes subscripts are added to the letters, for example, x 1 \u003d 3, x 2 \u003d 5. Thus, we indicate the numbers of the roots. If the equation has infinitely many solutions, then we write the answer as a numerical interval or use the generally accepted notation: the set of natural numbers is denoted by N, integers - Z, real - R. For example, if we need to write down that the solution to the equation will be any integer, then we write that x ∈ Z, and if any real from one to nine, then y ∈ 1, 9.

When an equation has two, three or more roots, then, as a rule, one speaks not of roots, but of solutions to the equation. Let us formulate the definition of a solution to an equation in several variables.

Definition 5

The solution to an equation with two, three or more variables is two, three or more values \u200b\u200bof the variables that turn this equation into a true numerical equality.

Let us explain the definition with examples.

Example 4

Let's say we have an expression x + y \u003d 7, which is an equation in two variables. Let's substitute one instead of the first, and two instead of the second. We will get an incorrect equality, which means that this pair of values \u200b\u200bwill not be a solution to this equation. If we take a pair of 3 and 4, then the equality becomes true, which means that we have found a solution.

Such equations may also have no roots or have an infinite number of them. If we need to write two, three, four or more values, then we write them separated by commas in parentheses. That is, in the example above, the answer will look like (3, 4).

In practice, most often one has to deal with equations containing one variable. We will consider the algorithm for solving them in detail in the article devoted to solving equations.

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The solution of equations in mathematics has a special place. This process is preceded by many hours of theory study, during which the student learns ways to solve equations, determine their type and bring the skill to complete automatism. However, the search for roots does not always make sense, since they may simply not exist. There are special techniques for finding roots. In this article, we will analyze the main functions, their areas of definition, as well as cases where their roots are missing.

Which equation has no roots?

An equation has no roots if there are no real arguments x for which the equation is identically true. For a layman, this formulation, like most mathematical theorems and formulas, looks very vague and abstract, but this is in theory. In practice, everything becomes extremely simple. For example: the equation 0 * x \u003d -53 has no solution, since there is no such number x, the product of which with zero would give something other than zero.

We will now look at the most basic types of equations.

1. Linear equation

An equation is called linear if its right and left sides are represented as linear functions: ax + b \u003d cx + d or in generalized form kx + b \u003d 0. Where a, b, c, d are known numbers and x is an unknown value ... Which equation has no roots? Examples of linear equations are shown in the illustration below.

Basically, linear equations are solved by simply transferring the numeric part to one part, and the content with x to the other. An equation of the form mx \u003d n is obtained, where m and n are numbers, and x is an unknown. To find x, it is enough to divide both parts by m. Then x \u003d n / m. Basically, linear equations have only one root, but there are cases when there are either infinitely many roots or no roots at all. For m \u003d 0 and n \u003d 0, the equation takes the form 0 * x \u003d 0. The solution to such an equation will be absolutely any number.

However, which equation has no roots?

For m \u003d 0 and n \u003d 0, the equation has no roots in the set of real numbers. 0 * x \u003d -1; 0 * x \u003d 200 - these equations have no roots.

2. Quadratic equation

A quadratic equation is an equation of the form ax 2 + bx + c \u003d 0 for a \u003d 0. The most common solution is through the discriminant. The formula for finding the discriminant of a quadratic equation: D \u003d b 2 - 4 * a * c. Next, there are two roots x 1,2 \u003d (-b ± √D) / 2 * a.

For D\u003e 0 the equation has two roots, for D \u003d 0 it has one root. But what quadratic equation has no roots? The easiest way to observe the number of roots of a quadratic equation is from the function graph, which is a parabola. For a\u003e 0, the branches are directed upward, for a< 0 ветви опущены вниз. Если дискриминант отрицателен, такое квадратное уравнение не имеет корней на множестве действительных чисел.

You can also visually determine the number of roots without calculating the discriminant. To do this, you need to find the vertex of the parabola and determine which direction the branches are directed. You can determine the x coordinate of the vertex using the formula: x 0 \u003d -b / 2a. In this case, the y-coordinate of the vertex is found by simply substituting x 0 into the original equation.

The quadratic equation x 2 - 8x + 72 \u003d 0 has no roots, since it has a negative discriminant D \u003d (-8) 2 - 4 * 1 * 72 \u003d -224. This means that the parabola does not touch the abscissa axis and the function never takes the value 0, therefore, the equation has no real roots.

3. Trigonometric equations

Trigonometric functions are considered on a trigonometric circle, but they can also be represented in a Cartesian coordinate system. In this article we will look at two basic trigonometric functions and their equations: sinx and cosx. Since these functions form a trigonometric circle with a radius of 1, | sinx | and | cosx | cannot be greater than 1. So which equation sinx has no roots? Consider the graph of the sinx function shown in the picture below.

We see that the function is symmetric and has a repetition period of 2pi. Based on this, we can say that the maximum value of this function can be 1, and the minimum -1. For example, the expression cosx \u003d 5 will not have roots, since the modulus is greater than one.

This is the simplest example of trigonometric equations. In fact, solving them can take many pages, at the end of which you realize that you have used the wrong formula and you need to start all over again. Sometimes, even with the correct finding of the roots, you can forget to take into account the constraints on the LDV, which is why an extra root or interval appears in the answer, and the whole answer turns into an error. Therefore, strictly follow all restrictions, because not all roots fit into the scope of the task.

4. Systems of equations

A system of equations is a collection of equations united by curly or square brackets. The curly brackets denote the joint execution of all equations. That is, if at least one of the equations has no roots or contradicts another, the entire system has no solution. Square brackets represent the word "or". This means that if at least one of the equations of the system has a solution, then the whole system has a solution.

The answer of system c is the set of all roots of individual equations. And curly brace systems have only common roots. Systems of equations can include absolutely diverse functions, so such complexity does not allow you to immediately tell which equation has no roots.

In problem books and textbooks, there are different types of equations: those that have roots, and those that do not. First of all, if you can't find the roots, don't think that there are none at all. Perhaps you have made a mistake somewhere, then it is enough just to carefully double-check your decision.

We have considered the most basic equations and their types. Now you can tell which equation has no roots. In most cases, this is not difficult at all. Success in solving equations requires only attention and concentration. Practice more, this will help you navigate the material much better and faster.

So, the equation has no roots if:

• in the linear equation mx \u003d n, the value m \u003d 0 and n \u003d 0;
• in a quadratic equation, if the discriminant is less than zero;
• in a trigonometric equation of the form cosx \u003d m / sinx \u003d n, if | m | \u003e 0, | n | \u003e 0;
• in a system of equations with curly brackets if at least one equation has no roots, and with square brackets if all equations have no roots.