Does the equation have roots and how many x. Which equation has no roots? Examples of equations. Examples of determining the roots of a quadratic equation

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 basic definitions and give various examples of equations and finding their roots.

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Concept of equation

Typically, the concept of an equation is taught at the very beginning of a school algebra course. Then it is defined like this:

Definition 1

Equation called an equality with an unknown number that needs to be found.

It is customary to denote unknowns in small Latin letters, for example, t, r, m, etc., but x, y, z are most often used. In other words, the equation is determined by 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.

Let us give some examples of the simplest equations. These can be equalities of the form x = 5, y = 6, etc., as well as those that include arithmetic operations, for example, x + 7 = 38, z − 4 = 2, 8 t = 4, 6: x = 3.

After the concept of brackets is learned, the concept of equations with brackets appears. These include 7 · (x − 1) = 19, x + 6 · (x + 6 · (x − 8)) = 3, etc. The letter that needs to be found can appear more than once, but several times, like, for example, in the equation x + 2 + 4 · x − 2 − x = 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 = 8, 3 − 3 = z + 3 or 8 x − 9 = 2 (x + 17) .

Further, after students become familiar with the concepts of integers, reals, rationals, 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 7th grade curriculum, 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, letter and variable expressions). Based on this concept, we can redefine the equation:

Definition 2

The equation is an equality involving a variable whose value needs to be calculated.

That is, for example, the expression x + 3 = 6 x + 7 is an equation with the variable x, and 3 y − 1 + y = 0 is an equation with the variable y.

One equation can have more than one variable, but two or more. They are called, respectively, equations with two, three variables, etc. Let us write down the definition:

Definition 3

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

For example, an equality of the form 3, 7 · x + 0, 6 = 1 is an equation with one variable x, and x − z = 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 = 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 a certain equation that includes one variable. If we substitute a number for the unknown letter, the equation becomes a numerical equality - true or false. So, if in the equation a + 1 = 5 we replace the letter with the number 2, then the equality will become false, and if 4, then the correct equality will be 4 + 1 = 5.

We are more interested in precisely those values with which the variable will turn into a true equality. They are called roots or solutions. Let's write down the definition.

Definition 4

Root of the equation They call the 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 = 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 = 5.

How many roots can one equation have? Does every 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 = 5. We can substitute an infinite number of different numbers into it, but none of them will turn it into a true equality, since multiplying by 0 always gives 0.

There are also equations that have several roots. They can have either a finite or an infinite number of roots.

Example 3

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

Now let us explain how to correctly write the roots of the equation. If there are none, then we write: “the equation has no roots.” In this case, you 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 simple 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 = 2 and y = 7. Sometimes subscripts are added to letters, for example, x 1 = 3, x 2 = 5. In this way we point to the numbers of the roots. If the equation has an infinite number of solutions, then we write the answer as a numerical interval or use generally accepted notation: the set of natural numbers is denoted N, integers - Z, real numbers - R. Let's say, if we need to write that the solution to the equation will be any integer, then we write that x ∈ Z, and if any real number from one to nine, then y ∈ 1, 9.

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

Definition 5

The solution to an equation with two, three or more variables is two, three or more values of the variables that turn the given equation into a correct numerical equality.

Let us explain the definition with examples.

Example 4

Let's say we have the expression x + y = 7, which is an equation with 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 will not be a solution to this equation. If we take the pair 3 and 4, then the equality becomes true, which means we have found a solution.

Such equations may also have no roots or an infinite number of them. If we need to write down 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, you most often have 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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Having received a general idea of equalities, and having become acquainted with one of their types - numerical equalities, you can start talking about another type of equalities that is very important from a practical point of view - equations. In this article we will look at what is an equation, and what is called the root of the equation. Here we will give the corresponding definitions, as well as provide various examples of equations and their roots.

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What is an equation?

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

Definition.

The equation is an equality containing an unknown number that needs to be found.

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

Thus, the equation is determined from the point of view of the form of writing. In other words, equality is an equation when it obeys the specified writing rules - it contains a letter whose value needs to be found.

Let us give examples of the very first and simplest equations. Let's start with equations of the form x=8, y=3, etc. Equations that contain arithmetic signs along with numbers and letters look a little more complicated, for example, x+2=3, z−2=5, 3 t=9, 8:x=2.

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

Further, after studying natural numbers, one gets acquainted with integer, rational, real numbers, new mathematical objects are studied: powers, roots, logarithms, etc., while more and more new types of equations containing these things appear. Examples of them can be seen in the article basic types of equations studying at school.

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

Definition.

Equation called an equality containing a variable whose value needs to be found.

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

During algebra lessons in the same 7th grade, we encounter equations containing not one, but two different unknown variables. 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 in their writing one, two, three, ... unknown variables, respectively.

For example, the equation 3.2 x+0.5=1 is an equation with one variable x, in turn, an equation of the form x−y=3 is an equation with two variables x and y. And one more example: x 2 +(y−1) 2 +(z+0.5) 2 =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 an equation is directly related to the definition of the root of this equation. Let's carry out 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 a letter included in the entry of this equation, a certain number is substituted, then the equation turns into a numerical equality. Moreover, the resulting equality can be either true or false. For example, if you substitute the number 2 instead of the letter a in the equation a+1=5, you will get the incorrect numerical equality 2+1=5. If we substitute the number 4 instead of a in this equation, we get the correct equality 4+1=5.

In practice, in the overwhelming majority of cases, the interest is in those values of the variable whose substitution into the equation gives the correct equality; these values are called roots or solutions of this equation.

Definition.

Root of the equation- this is the value of the letter (variable), upon substitution of which the equation turns into a correct numerical equality.

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

Let us explain this definition with an example. To do this, let's return to the equation written above a+1=5. According to the stated definition of the root of an equation, the number 4 is the root of this equation, since when substituting this number instead of the letter a we get the correct equality 4+1=5, and the number 2 is not its root, since it corresponds to an incorrect equality of the form 2+1= 5 .

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 that have roots and equations that do not have roots. For example, the equation x+1=5 has root 4, but the equation 0 x=5 has no roots, since no matter what number we substitute in this equation instead of the variable x, we will get the incorrect equality 0=5.

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

A few words should be said about the accepted notation for the roots of the equation. If an equation has no roots, then they usually 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 brackets. For example, if the roots of the equation are the numbers −1, 2 and 4, then write −1, 2, 4 or (−1, 2, 4). It is also permissible to write down the roots of the equation in the form of simple equalities. For example, if the equation includes the letter x, and the roots of this equation are the numbers 3 and 5, then you can write x=3, x=5, and subscripts x 1 =3, x 2 =5 are often added to the variable, as if indicating the numbers roots of the equation. An infinite set of roots of an equation is usually written in the form; if possible, the notation for sets of natural numbers N, integers Z, and real numbers R is also used. 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 or more variables, as a rule, the term “root of the equation” is not used; in these cases they say “solution of the equation”. What is called solving equations with several variables? Let us give the corresponding definition.

Definition.

Solving an equation with two, three, etc. variables called a pair, three, etc. values of the variables, turning this equation into a correct numerical equality.

Let us show explanatory examples. Consider an equation with two variables x+y=7. Let's substitute the number 1 instead of x, and the number 2 instead of y, and we have the equality 1+2=7. Obviously, it is incorrect, therefore, the pair of values x=1, y=2 is not a solution to the written equation. If we take a pair of values x=4, y=3, then after substitution into the equation we will arrive at the correct equality 4+3=7, therefore, this pair of variable values, by definition, is a solution to the equation x+y=7.

Equations with several variables, like equations with one variable, may have no roots, may have a finite number of roots, or may have an infinite number of roots.

Pairs, triplets, quadruples, etc. The values of variables are often written briefly, listing their values separated 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=7. The solution to this equation x=4, y=3 can be briefly written as (4, 3).

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

Bibliography.

Mathematics. 2 classes Textbook for general education institutions with adj. per electron carrier. At 2 p.m. Part 1 / [M. I. Moro, M. A. Bantova, G. V. Beltyukova, etc.] - 3rd ed. - M.: Education, 2012. - 96 p.: ill. - (School of Russia). - ISBN 978-5-09-028297-0.
Algebra: textbook for 7th grade general education institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; edited by S. A. Telyakovsky. - 17th ed. - M.: Education, 2008. - 240 p. : ill. - ISBN 978-5-09-019315-3.
Algebra: 9th grade: educational. for general education institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; edited by S. A. Telyakovsky. - 16th ed. - M.: Education, 2009. - 271 p. : ill. - ISBN 978-5-09-021134-5.

Solving equations in mathematics occupies a special place. This process is preceded by many hours of studying theory, during which the student learns how to solve equations, determine their type, and brings the skill to complete automation. However, searching 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 domains of definition, as well as cases when 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 non-specialist, 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 = -53 has no solution, since there is no number x whose product with zero would give something other than zero.

Now we will 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 = cx + d or in generalized form kx + b = 0. Where a, b, c, d are known numbers, and x is an unknown quantity . Which equation has no roots? Examples of linear equations are presented in the illustration below.

Basically, linear equations are solved by simply transferring the number part to one part and the contents of x to another. The result is an equation of the form mx = n, where m and n are numbers, and x is an unknown. To find x, just divide both sides by m. Then x = n/m. Most linear equations have only one root, but there are cases when there are either infinitely many roots or no roots at all. When m = 0 and n = 0, the equation takes the form 0 * x = 0. The solution to such an equation will be absolutely any number.

However, what equation has no roots?

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

2. Quadratic equation

A quadratic equation is an equation of the form ax 2 + bx + c = 0 for a = 0. The most common solution is through the discriminant. The formula for finding the discriminant of a quadratic equation is: D = b 2 - 4 * a * c. Next there are two roots x 1.2 = (-b ± √D) / 2 * a.

For D > 0 the equation has two roots, for D = 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 by graphing the function, which is a parabola. For a > 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 in which direction the branches are directed. The x coordinate of the vertex can be determined using the formula: x 0 = -b / 2a. In this case, the y coordinate of the vertex is found by simply substituting the x 0 value into the original equation.

The quadratic equation x 2 - 8x + 72 = 0 has no roots, since it has a negative discriminant D = (-8) 2 - 4 * 1 * 72 = -224. This means that the parabola does not touch the x-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 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 radius 1, |sinx| and |cosx| cannot be greater than 1. So, which sinx equation 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 = 5 will not have roots, since its absolute value 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 used the wrong formula and need to start all over again. Sometimes, even if you find the roots correctly, you may forget to take into account the restrictions on OD, which is why an extra root or interval appears in the answer, and the entire answer turns into an error. Therefore, strictly follow all the restrictions, because not all roots fit into the scope of the task.

4. Systems of equations

A system of equations is a set of equations joined by curly or square brackets. The curly brackets indicate that all equations are run together. That is, if at least one of the equations does not have roots or contradicts another, the entire system has no solution. Square brackets indicate the word "or". This means that if at least one of the equations of the system has a solution, then the entire system has a solution.

The answer of the system c is the set of all the roots of the individual equations. And systems with curly braces have only common roots. Systems of equations can include completely different functions, so such complexity does not allow us to immediately say which equation does not have 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 they are not there at all. Perhaps you made a mistake somewhere, then you just need to carefully double-check your decision.

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

So, the equation has no roots if:

in the linear equation mx = n the value is m = 0 and n = 0;
in a quadratic equation, if the discriminant is less than zero;
in a trigonometric equation of the form cosx = m / sinx = n, if |m| > 0, |n| > 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.

Consider the quadratic equation:
(1) .
Roots of a quadratic equation(1) are determined by the formulas:
; .
These formulas can be combined like this:
.
When the roots of a quadratic equation are known, then a polynomial of the second degree can be represented as a product of factors (factored):
.

Next we assume that are real numbers.
Let's consider discriminant of a quadratic equation:
.
If the discriminant is positive, then the quadratic equation (1) has two different real roots:
; .
Then the factorization of the quadratic trinomial has the form:
.
If the discriminant is equal to zero, then the quadratic equation (1) has two multiple (equal) real roots:
.
Factorization:
.
If the discriminant is negative, then the quadratic equation (1) has two complex conjugate roots:
;
.
Here is the imaginary unit, ;
and are the real and imaginary parts of the roots:
; .
Then

.

Graphic interpretation

If you plot the function
,
which is a parabola, then the points of intersection of the graph with the axis will be the roots of the equation
.
At , the graph intersects the x-axis (axis) at two points.
When , the graph touches the x-axis at one point.
When , the graph does not cross the x-axis.

Below are examples of such graphs.

Useful formulas related to quadratic equation

(f.1) ;
(f.2) ;
(f.3) .

Derivation of the formula for the roots of a quadratic equation

We carry out transformations and apply formulas (f.1) and (f.3):

,
Where
; .

So, we got the formula for a polynomial of the second degree in the form:
.
This shows that the equation

performed at
And .
That is, and are the roots of the quadratic equation
.

Examples of determining the roots of a quadratic equation

Example 1

(1.1) .

.
Comparing with our equation (1.1), we find the values of the coefficients:
.
We find the discriminant:
.
Since the discriminant is positive, the equation has two real roots:
;
;
.

From here we obtain the factorization of the quadratic trinomial:

.

Graph of the function y = 2 x 2 + 7 x + 3 intersects the x-axis at two points.

Let's plot the function
.
The graph of this function is a parabola. It crosses the abscissa axis (axis) at two points:
And .
These points are the roots of the original equation (1.1).

;
;
.

Example 2

Find the roots of a quadratic equation:
(2.1) .

Let's write the quadratic equation in general form:
.
Comparing with the original equation (2.1), we find the values of the coefficients:
.
We find the discriminant:
.
Since the discriminant is zero, the equation has two multiple (equal) roots:
;
.

Then the factorization of the trinomial has the form:
.

Graph of the function y = x 2 - 4 x + 4 touches the x-axis at one point.

Let's plot the function
.
The graph of this function is a parabola. It touches the x-axis (axis) at one point:
.
This point is the root of the original equation (2.1). Because this root is factored twice:
,
then such a root is usually called a multiple. That is, they believe that there are two equal roots:
.

;
.

Example 3

Find the roots of a quadratic equation:
(3.1) .

Let's write the quadratic equation in general form:
(1) .
Let's rewrite the original equation (3.1):
.
Comparing with (1), we find the values of the coefficients:
.
We find the discriminant:
.
The discriminant is negative, . Therefore there are no real roots.

You can find complex roots:
;
;

Let's plot the function
.
The graph of this function is a parabola. It does not intersect the x-axis (axis). Therefore there are no real roots.

There are no real roots. Complex roots:
;
;
.