Studying algebra, students are faced with equations of many kinds. Among those that are the simplest, one can name linear ones containing one unknown. If a variable in a mathematical expression is raised to a certain degree, 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 sign of the radical (that is, purely outwardly the variable here can be seen written under the square root). The solution of irrational equations has its own characteristic features. When calculating the value of a variable to obtain the correct answer, they should be taken into account.
Inexpressible words
It is no secret that ancient mathematicians operated mainly on rational numbers. These include, as you know, whole, expressed through ordinary and decimal periodic fractions representatives of this community. However, scientists from the Middle and Middle East, as well as India, developing trigonometry, astronomy and algebra, also learned to solve irrational equations. For example, the Greeks knew such quantities, but clothed them in verbal form, they used the concept of “alogos”, which meant “inexpressible”. A little later, Europeans, imitating them, called such numbers "deaf." They differ from all the 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 realm of numbers are recorded in the form of numbers and signs as some expression that is under the root of the second or greater degree.
Based on the foregoing, let us try to define an irrational equation. Similar expressions contain the so-called “unspeakable numbers” written using the square root sign. They can be all kinds of rather complicated options, but in their simplest form they look like the one in the photo below.
Coming to the solution of 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 verify the obtained values follows from the properties of the arithmetic square root. As you know, such an expression is acceptable and has any meaning only under certain conditions. In cases of an even degree root, all radical expressions must be positive or equal to zero. If this condition is not fulfilled, then the presented mathematical record cannot be considered meaningful.
We give a concrete example of how to solve irrational equations (in the photo below).
In this case, it is obvious that the indicated conditions cannot be satisfied for any values accepted by the desired quantity, since it turns out that 11 ≤ x ≤ 4. Therefore, the solution can only be Ø.
Analysis method
From the above it becomes clear how to solve the irrational equation of certain types. Here, a simple analysis can be an effective way.
Here are a few examples that will again demonstrate this clearly (in the photo below).
In the first case, a careful examination of the expression immediately makes it extremely clear that it cannot be true. Indeed, the left side of the equality should yield a positive number, which in no way can turn out to be -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. But this again is impossible. And therefore, the answer should again write Ø.
The third example is very similar to the one considered earlier. Indeed, after all, here the SDL conditions require that the following absurd inequality holds: 5 ≤ x ≤ 2. And a similar equation in the same way can in no way have sound solutions.
Unlimited zoom
The nature of the irrational can be most clearly and fully explained and known only through an endless series of decimal numbers. A concrete, striking example of the members of this family is πi. Not without reason, it is assumed that this mathematical constant has been known since ancient times, used in calculating the circumference and circle area. But among Europeans it was first applied in practice by the Englishman William Jones and the Swiss Leonard Euler.
This constant arises as follows. If we compare the most different in circumference, then the ratio of their lengths and diameters is necessarily equal to the same number. This is pi. If to express it through an ordinary fraction, then approximately we will receive 22/7. For the first time this was done by the great Archimedes, whose portrait is presented in the figure above. That is why a similar number received his name. But this is not an obvious, but an approximate value of perhaps the most surprising of the numbers. An ingenious scientist, up to 0.02, found the desired value, but, in fact, this constant has no real value, but is expressed as 3.1415926535 ... It represents an infinite series of numbers, unlimitedly approaching some mythical value.
Squaring
But back to the irrational equations. In order to find the unknown, in this case they very often resort to a simple method: they square both sides of the existing equality. A similar method usually gives good results. But the insidiousness of irrational values should be taken into account. All the roots obtained as a result of this need to be checked, because they may not be suitable.
But we continue the consideration of examples and try to find the variables in the newly proposed way.
Using the Vieta theorem, it is not difficult to find the desired values of the quantities after we have formed a quadratic equation as a result of certain operations. It turns out that among the roots there will be 2 and -19. However, when checking, substituting the obtained value in the original expression, we can make sure that none of these roots is 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 complicated examples
In some cases, it is required to square both parts of the expression not one, but several times. Let's consider examples where the specified is required. You can see them below.
Having received the roots, do not forget to check them, because extra ones may arise. It should be explained why this is possible. When applying such a method, the equation is somewhat rationalized. But getting rid of unwanted roots that prevent us from performing arithmetic, we kind of expand the existing range of values, which is fraught (as you can understand) with consequences. Anticipating this, we are checking. In this case, there is a chance to make sure that only one of the roots is suitable: x = 0.
Systems
What to do in cases when it is required to carry out the solution of systems of irrational equations, and we have not one, but two whole unknowns? Here we do the same as in ordinary cases, but taking into account the above properties of these mathematical expressions. And in each new task, of course, a creative approach should be applied. But, again, it is better to consider everything with a specific example, presented below. Here, it is not only required to find the variables x and y, but also indicate their sum in the answer. So, there is a system containing irrational values (see photo below).
As you can see, such a task is not supernaturally complicated. All you need is to be smart and guess that the left side of the first equation is the square of the sum. Similar tasks are found in the exam.
Irrational in mathematics
Each time, the need for the creation of new types of numbers arose for mankind when he lacked the "space" to solve some equations. Irrational numbers are no exception. As the facts from history testify, for the first time the great sages drew attention to this before our era, the century in VII. A mathematician from India, known as Manawa, did this. 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 Gippas, came to the same conclusion, trying to make calculations with numerical expressions of the sides of the pentagram. Having discovered mathematical elements that cannot be expressed in numerical values and do not have the properties of ordinary numbers, he angered his colleagues so much that he was thrown overboard, 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 began to be used to solve irrational inequalities and equations far from immediately. For the first time, European, in particular Italian, mathematicians began to think about the radical around the 13th century. Then, for designation, they came up with the use of the Latin R. But German mathematicians acted differently in their works. They liked the letter V more. 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 an appropriate degree. This is the main action that helps with operations with the irrational. Actions in even cases are not particularly different from those that we have already analyzed. Here, the conditions for the non-negativity of the radical expression must be taken into account, and also at the end of the solution, it is necessary to screen out the extraneous values of the variables in the same way as was shown in the examples already considered.
Of the additional transformations that help to find the right answer, the multiplication of the expression by the conjugate is often used, and the introduction of a new variable is often required, which makes the solution easier. In some cases, in order to find the meaning of unknowns, it is advisable to use graphs.