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Some schoolchildren really don’t like equations and problems in which the root sign appears. But solving an example from the root is not so difficult; it is important to know from which side to approach the problem. The icon itself, which indicates the extraction of the root, is called a radical. How to solve roots? To extract the square root of a number means to select a number that, when squared, will give the same value under the radical sign.
So how to solve square roots
Decide square roots not difficult. For example, you need to figure out what the root of 16 is. In order to solve this simple example, you need to remember how much 2 squared is - 2 2, then 3 2, and finally 4 2. Only now we will see that the result (16) matches the request. That is, in order to extract the root, we had to select possible values. It turns out that there is no exact and proven algorithm for solving roots. To make the work of the “solver” easier, mathematicians recommend memorizing (precisely by heart, like a multiplication table) the values of the squares of numbers up to twenty. Then it will be possible to easily extract the root of numbers that are more than a hundred. And, on the contrary, you can immediately see that the root cannot be extracted from this number, that is, the answer will not be an integer.
We figured out how to solve square roots. Now let's figure out which square roots have no solution. For example, negative numbers. It is clear here that if two negative numbers multiply - the answer will be with a plus sign. Here's what you should know: The root can be extracted from any number (except negative, as mentioned above). The answer may simply turn out to be a decimal fraction. That is, contain a certain number of digits after the decimal point. For example, the root of two has the value 1.41421 and this is not all the numbers after the decimal point. Such values are rounded to facilitate calculations, sometimes to the second decimal place, sometimes to the third or fourth. In addition, it is often practiced to leave the number under the root as an answer if it looks good and compact. After all, it’s already clear what it means.
How to solve equations with roots?
To solve equations with roots, you need to use one of the methods not invented by us. For example, square both sides of such an equation. For example:
Root of X+3=5
Let's square the left and right sides of the equation:
Now you can see how to solve this equation. First, let's find out what X 2 is equal to (and it is equal to 16), and then take the root of it. Answer: 4. However, it is worth saying here that this equation actually has two solutions, two roots: 4 and -4. After all, -4 squared also gives 16.
In addition to this method, sometimes it is more attractive and convenient to replace the variable that is under the root with another variable in order to get rid of this root.
Y = root of X.
Subsequently, having solved the equation, we return to the substitution and finish the calculations with the root.
That is, we get X = Y 2. And this will be the solution.
It should be said that there are several more techniques for solving equations with roots.
How to solve roots in powers?
A radical, which does not have a power in its base, means that you need to take the square root of an expression or number, that is, the square power in reverse. It's simple and clear. For example: root of 9 = 3, (and 3 2 = 9), root of 16 = 4 (4 2 = 16) and everything in the same spirit. But what does it mean if the root has a degree? This means that it is necessary, again, to perform the action opposite to raising it to this very power. For example, you need to find out the value of the cube root of 27.
To do this, you need to choose a number that, when cubed, will give 27. This is 3 (3*3*3=27).
root 3 of 27 = 3
Similar actions need to be performed if the degree of the root is 4, 5. Only in this case it is necessary to select a number that, when raised to a power n will give the value under the root n-th degree.
Here it must be said that the degrees of roots and the degrees of radical expressions can be reduced. However, according to the rules. If the number or variable under the root has a degree that is a multiple of the degree of the root, they can be reduced. For example:
root 3 of X 6 = X 2
These rules for dealing with roots and powers are simple; you need to know them clearly, and then the calculation will be simple. We figured out how to solve roots to a degree, now we move on.
How to solve the root under the root?
This terrible expression is root by root and at first glance cannot be solved. But in order to correctly calculate the value of such an expression, you need to know the properties of the roots. In this case, you just need to replace two roots with one. To do this, the degrees of these radicals need to be simply multiplied. For example:
root 3 of root 729 = (root 3 * root 2) of 729
That is, here we multiplied the cube root by the square root. As a result, we got the sixth root:
root 6 of 729 = 3
Other similar roots under the root need to be addressed in the same way.
Having considered all the proposed examples, it is easy to agree that solving the roots is not such a difficult task. Of course, when it comes down to simple, banal arithmetic, sometimes it’s easier to use a familiar calculator. However, before making calculations, you need to do everything possible to simplify the task for yourself, reducing the number and complexity of arithmetic calculations as much as possible. Then the solution will become simple and, most importantly, interesting.
After we have studied the concept of equalities, namely one of their types - numerical equalities, we can move on to another important view– equations. Within the framework of this material, we will explain what an equation is and its root, formulate the basic definitions and give various examples equations and finding their roots.
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Concept of equation
Usually the concept of an equation is studied at the very beginning school course algebra. 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 designate unknowns as small with Latin letters, for example, t, r, m etc., but most often x, y, z are 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 are familiar 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 7th grade curriculum, the concept of variables appears for the first time. These are letters that can take different meanings(For more information, 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 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 be either finite or infinite a large 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 this pair of values will not be a solution given 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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While studying algebra, schoolchildren are faced with many types of equations. Among those that are the simplest are linear ones, containing one unknown. If a variable in a mathematical expression is raised to a certain power, then the equation is called quadratic, cubic, biquadratic, and so on. These expressions may contain rational numbers. But there are also irrational equations. They differ from others by the presence of a function where the unknown is under the radical sign (that is, purely externally, the variable here can be seen written under the square root). Solving irrational equations has its own characteristics. When calculating the value of a variable to obtain the correct answer, they must be taken into account.
"Unspeakable in Words"
It is no secret that ancient mathematicians operated mainly rational numbers. These include, as is known, integers expressed through ordinary and decimal periodic fractions, representatives of a given community. However, scientists of the Middle and Near East, as well as India, developing trigonometry, astronomy and algebra, also learned to solve irrational equations. For example, the Greeks knew similar quantities, but putting them into verbal form, they used the concept “alogos”, which meant “inexpressible”. Somewhat later, Europeans, imitating them, called such numbers “deaf.” They differ from all others in that they can only be represented in the form of an infinite non-periodic fraction, the final numerical expression of which is simply impossible to obtain. Therefore, more often such representatives of the kingdom of numbers are written in the form of numbers and signs as some expression located under the root of the second or higher degree.
Based on the above, let's try to define an irrational equation. Such expressions contain so-called "unexpressible numbers", written using the square root sign. They can represent all sorts of rather complex options, but in their in its simplest form They look like the photo below.
When starting to solve irrational equations, first of all it is necessary to calculate the range of permissible values of the variable.
Does the expression make sense?
The need to check the obtained values follows from the properties. As is known, such an expression is acceptable and has any meaning only under certain conditions. In cases of roots of even degrees, all radical expressions must be positive or equal to zero. If this condition is not met, then the presented mathematical notation cannot be considered meaningful.
Let's give a specific example of how to solve irrational equations (pictured below).
In this case, it is obvious that the specified conditions cannot be satisfied for any values accepted by the desired value, since it turns out that 11 ≤ x ≤ 4. This means that only Ø can be the solution.
Method of analysis
From the above, it becomes clear how to solve some types of irrational equations. Here in an effective way may be a simple analysis.
Let us give a number of examples that will again clearly demonstrate this (pictured below).
In the first case, upon careful examination of the expression, it immediately turns out to be extremely clear that it cannot be true. Indeed, on the left side of the equality we should get positive number, which cannot possibly be equal to -1.
In the second case, the sum of two positive expressions can be considered equal to zero only when x - 3 = 0 and x + 3 = 0 at the same time. And this is again impossible. And that means the answer should again be written Ø.
The third example is very similar to the one already discussed earlier. Indeed, here the conditions of the ODZ require that the following absurd inequality be satisfied: 5 ≤ x ≤ 2. And such an equation in the same way cannot have sensible solutions.
Unlimited zoom
The nature of the irrational can most clearly and completely be explained and known only through an endless series of numbers decimal. And specific, a shining example one of the members of this family is πi. It is not without reason that this mathematical constant has been known since ancient times, being used in calculating the circumference and area of a circle. But among Europeans it was first put into practice by the Englishman William Jones and the Swiss Leonard Euler.
This constant arises as follows. If we compare circles of different circumferences, then the ratio of their lengths and diameters is necessarily equal to the same number. This is pi. If we express it through common fraction, then we get approximately 22/7. This was first done by the great Archimedes, whose portrait is shown in the figure above. That is why similar number received his name. But this is not an explicit, but an approximate value of perhaps the most amazing of numbers. A brilliant scientist found the desired value with an accuracy of 0.02, but, in fact, this constant has no real meaning, but is expressed as 3.1415926535... It is an endless series of numbers, indefinitely approaching some mythical value.
Squaring
But let's return to irrational equations. To find the unknown, in this case they very often resort to simple method: square both sides of the existing equality. This method usually gives good results. But one should take into account the insidiousness of irrational quantities. All roots obtained as a result of this must be checked, because they may not be suitable.
But let's continue looking at the examples and try to find the variables using the newly proposed method.
It is not at all difficult, using Vieta’s theorem, to find the desired values of quantities after, as a result of certain operations, we have formed quadratic equation. Here it turns out that among the roots there will be 2 and -19. However, when checking, substituting the resulting values into the original expression, you can make sure that none of these roots are suitable. This is a common occurrence in irrational equations. This means that our dilemma again has no solutions, and the answer should indicate an empty set.
More complex examples
In some cases, it is necessary to square both sides of an expression not once, but several times. Let's look at examples where this is required. They can be seen below.
Having received the roots, do not forget to check them, because extra ones may appear. It should be explained why this is possible. When applying this method, the equation is somewhat rationalized. But by getting rid of roots we don’t like, which prevent us from performing arithmetic operations, we seem to expand the existing range of meanings, which is fraught (as one can understand) with consequences. Anticipating this, we carry out a check. In this case, there is a chance to make sure that only one of the roots is suitable: x = 0.
Systems
What should we do in cases where we need to solve systems of irrational equations, and we have not one, but two unknowns? Here we act in the same way as in ordinary cases, but taking into account the above properties of these mathematical expressions. And in every new task, of course, you should use a creative approach. But, again, it is better to consider everything specific example presented below. Here you not only need to find the variables x and y, but also indicate their sum in the answer. So, there is a system containing irrational quantities (see photo below).
As you can see, such a task does not represent anything supernaturally difficult. You just need to be smart and figure out what left side The first equation is the square of the sum. Similar tasks are found in the Unified State Exam.
Irrational in mathematics
Each time, the need to create new types of numbers arose among humanity when it did not have enough “space” to solve some equations. Irrational numbers are no exception. As facts from history testify, the great sages first paid attention to this even before our era, in the 7th century. This was done by a mathematician from India known as Manava. He clearly understood that it was impossible to extract a root from some natural numbers. For example, these include 2; 17 or 61, as well as many others.
One of the Pythagoreans, a thinker named Hippasus, came to the same conclusion by trying to make calculations using numerical expressions of the sides of the pentagram. Discovering mathematical elements that cannot be expressed digital values and do not have properties ordinary numbers, he angered his colleagues so much that he was thrown overboard the ship into the sea. The fact is that other Pythagoreans considered his reasoning a rebellion against the laws of the universe.
Sign of the Radical: Evolution
The root sign for expressing the numerical value of “deaf” numbers did not immediately begin to be used in solving irrational inequalities and equations. European, in particular Italian, mathematicians first began to think about the radical around the 13th century. At the same time, they came up with the idea of using the Latin R for designation. But German mathematicians acted differently in their works. They liked the letter V better. In Germany, the designation V(2), V(3) soon spread, which was intended to express the square root of 2, 3, and so on. Later, the Dutch intervened and modified the sign of the radical. And Rene Descartes completed the evolution, bringing the square root sign to modern perfection.
Getting rid of the irrational
Irrational equations and inequalities can include a variable not only under the square root sign. It can be of any degree. The most common way to get rid of it is to raise both sides of the equation to the appropriate power. This is the main action that helps in operations with the irrational. The actions in even-numbered cases are not particularly different from those that we have already discussed earlier. Here the conditions for the non-negativity of the radical expression must be taken into account, and at the end of the solution it is necessary to filter out extraneous values of the variables in the same way as was shown in the examples already considered.
Among the additional transformations that help find the correct answer, multiplication of the expression by its conjugate is often used, and it is also often necessary to introduce a new variable, which makes the solution easier. In some cases, it is advisable to use graphs to find the value of unknowns.
Every new action in mathematics instantly generates its opposite. Once upon a time, the ancient Greeks discovered that a square piece of land 2 meters long and 2 meters wide would have an area of 2*2 = 4 square meters(hereinafter will be denoted by m^2). Now, on the contrary, if a Greek knew that his plot of land was square and had an area of 4 m^2, how would he know what the length and width of his plot was? An operation was introduced that was the inverse of the squaring operation and became known as square root extraction. People began to understand that 2 squared (2^2) is equal to 4. Conversely, the square root of 4 (hereinafter referred to as √(4)) will be equal to two. Models became more complex, and records describing processes with roots also became more complex. The question has arisen many times: how to solve an equation with a root.
Let a certain value x, when multiplied by itself once, give 9. This can be written as x*x=9. Or through a degree: x^2=9. To find x, you need to take the root of 9, which to some extent is already an equation with a radical: x=√(9) . The root can be extracted orally or using a calculator. Next we should consider the inverse problem. A certain quantity, when the square root is taken from it, gives the value 7. If we write this in the form of an irrational equation, we get: √(x) = 7. To solve this problem, it is necessary to square both sides of the expression. Considering that √(x) *√(x) =x, it turns out x = 49. The root is immediately ready in its pure form. Next, we should look at more complex examples of equations with roots.
Let us subtract 5 from a certain quantity, then raise the expression to the power of 1/2. As a result, the number 3 was obtained. Now this condition must be written as an equation: √(x-5) =3. Next, you should multiply each part of the equation by itself: x-5 = 3. After raising to the second power, the expression was freed from radicals. Now it’s time to solve the simplest linear equation, moving the five to the right side and changing its sign. x = 5+3. x = 8. Unfortunately, not all life processes can be described by such simple equations. Very often you can find expressions with several radicals; sometimes the degree of the root can be higher than the second. There is no single solution algorithm for such identities. It is worth looking for a special approach to each equation. An example is given in which the equation with the root has the third degree.
The cube root will be denoted by 3√. Find the volume of a container shaped like a cube with a side of 5 meters. Let the volume be x m^3. Then the cube root of the volume will be equal to side cube and equal to five meters. The resulting equation is: 3√(x) =5. To solve it, you need to raise both parts to the third power, x = 125. Answer: 125 cubic meters. Below is an example of an equation with a sum of roots. √(x) +√(x-1) =5. First you need to square both parts. To do this, it is worth remembering the abbreviated multiplication formula for the square of the sum: (a+b) ^2=a^2+2*ab+b^2. Applying this to the equation, we get: x + 2*√(x) *√(x-1) + x-1 = 25. Next, the roots are left on the left side, and everything else is transferred to the right: 2*√(x) *√ (x-1) = 26 - 2x. It is convenient to divide both sides of the expression by 2: √((x) (x-1)) = 13 - x. A simpler irrational equation is obtained.
Next, both sides should be squared again: x*(x-1) = 169 - 26x + x^2. It is necessary to open the brackets and bring similar terms: x^2 - x = 169 - 26x + x^2. The second degree disappears, hence 25x = 169. x = 169/25 = 6.6. By checking, substituting the resulting root into the original equation: √(6.6) +√(6.6-1) = 2.6 + √(5.6) = 2.6 + 2.4 = 5, you can get a satisfactory answer. It is also very important to understand that an expression with a root of even degree cannot be negative. Indeed, multiplying any number by itself an even number of times, it is impossible to obtain the value less than zero. Therefore, equations such as √(x^2+7x-11) = -3 can be safely not solved, but written that the equation has no roots. As mentioned above, solving equations with radicals can take a variety of forms.
A simple example of an equation where it is necessary to change variables. √(y) - 5*4√(y) +6 = 0, where 4√(y) is the fourth root of y. The proposed replacement looks like this: x = 4√(y) . After doing this, we get: x^2 - 5x + 6 = 0. The resulting quadratic equation is obtained. Its discriminant: 25 - 4*6 = 25 - 24 = 1. The first root x1 will be equal to (5 + √1) /2 = 6/2 = 3. The second root x2 = (5 - √1) /2 = 4/ 2 = 2. You can also find the roots using a corollary of Vieta’s theorem. The roots have been found, a reverse replacement should be carried out. 4√(y) = 3, hence y1 = 1.6. Also 4√(y) = 2, taking the 4th root turns out that y2 = 1.9. Values calculated using a calculator. But you don’t have to do them, leaving the answer in the form of radicals.
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