This article contains material covering the topic " solution of square inequalities". First, it is shown what quadratic inequalities with one variable are, their general form is given. And then it is analyzed in detail how to solve quadratic inequalities. The main approaches to the solution are shown: the graphical method, the method of intervals, and by highlighting the square of the binomial on the left side of the inequality. Solutions of typical examples are given.

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What is a quadratic inequality?

Naturally, before talking about solving quadratic inequalities, one must clearly understand what a quadratic inequality is. In other words, you need to be able to distinguish square inequalities from inequalities of other types by the type of record.

Definition.

Square inequality is an inequality of the form a x 2 +b x+c<0 (вместо знака >there can be any other inequality sign ≤, >, ≥), where a, b and c are some numbers, and a≠0, and x is a variable (the variable can be denoted by any other letter).

Let's immediately give another name for quadratic inequalities - inequality of the second degree. This name is explained by the fact that on the left side of the inequalities a x 2 +b x+c<0 находится второй степени - квадратный трехчлен. Термин «неравенства второй степени» используется в учебниках алгебры Ю. Н. Макарычева, а Мордкович А. Г. придерживается названия «квадратные неравенства».

You can also sometimes hear that quadratic inequalities are called quadratic inequalities. This is not entirely correct: the definition of "quadratic" refers to functions given by equations of the form y=a x 2 +b x+c . So there are quadratic inequalities and quadratic functions, but not quadratic inequalities.

Let's show some examples of square inequalities: 5 x 2 −3 x+1>0 , here a=5 , b=−3 and c=1 ; −2.2 z 2 −0.5 z−11≤0, the coefficients of this quadratic inequality are a=−2.2 , b=−0.5 and c=−11 ; , in this case .

Note that in the definition of the quadratic inequality, the coefficient a at x 2 is considered non-zero. This is understandable, the equality of the coefficient a to zero will actually “remove” the square, and we will be dealing with a linear inequality of the form b x + c>0 without the square of the variable. But the coefficients b and c can be equal to zero, both separately and simultaneously. Here are examples of such square inequalities: x 2 −5≥0 , here the coefficient b for the variable x is equal to zero; −3 x 2 −0.6 x<0 , здесь c=0 ; наконец, в квадратном неравенстве вида 5·z 2 >0 and b and c are zero.

How to solve quadratic inequalities?

Now you can be puzzled by the question of how to solve quadratic inequalities. Basically, three main methods are used to solve:

• graphical method (or, as in A.G. Mordkovich, functional-graphical),
• interval method,
• and solving quadratic inequalities through highlighting the square of the binomial on the left side.

Graphically

Let us make a reservation right away that the method of solving quadratic inequalities, which we are starting to consider, is not called graphical in algebra school textbooks. However, in essence, this is what he is. Moreover, the first acquaintance with graphical way of solving inequalities usually begins when the question arises of how to solve quadratic inequalities.

Graphical way to solve quadratic inequalities a x 2 +b x+c<0 (≤, >, ≥) is to analyze the graph of the quadratic function y=a x 2 +b x+c to find the intervals in which the specified function takes negative, positive, non-positive or non-negative values. These intervals constitute the solutions of the quadratic inequalities a x 2 +b x+c<0 , a·x 2 +b·x+c>0 , a x 2 +b x+c≤0 and a x 2 +b x+c≥0 respectively.

interval method

To solve square inequalities with one variable, in addition to the graphical method, the interval method is quite convenient, which in itself is very versatile, and is suitable for solving various inequalities, not just square ones. Its theoretical side lies outside the algebra course of grades 8, 9, when they learn to solve quadratic inequalities. Therefore, here we will not go into the theoretical justification of the interval method, but will focus on how quadratic inequalities are solved with its help.

The essence of the interval method, in relation to the solution of square inequalities a x 2 +b x + c<0 (≤, >, ≥), consists in determining the signs that have the values ​​of the square trinomial a x 2 + b x + c on the intervals into which the coordinate axis is divided by the zeros of this trinomial (if any). The gaps with minus signs make up the solutions of the quadratic inequality a x 2 +b x+c<0 , со знаками плюс – неравенства a·x 2 +b·x+c>0 , and when solving non-strict inequalities, points corresponding to the zeros of the trinomial are added to the indicated intervals.

You can get acquainted with all the details of this method, its algorithm, the rules for placing signs on the intervals and consider ready-made solutions for typical examples with the illustrations given by referring to the material of the article solving quadratic inequalities by the interval method.

By isolating the square of the binomial

In addition to the graphical method and the interval method, there are other approaches that allow solving quadratic inequalities. And we come to one of them, which is based on squaring a binomial on the left side of the quadratic inequality.

The principle of this method of solving quadratic inequalities is to perform equivalent transformations of the inequality , allowing one to go to the solution of an equivalent inequality of the form (x−p) 2 , ≥), where p and q are some numbers.

And how is the transition to the inequality (x−p) 2 , ≥) and how to solve it, the material of the article explains the solution of quadratic inequalities by highlighting the square of the binomial. There are also examples of solving quadratic inequalities in this way and the necessary graphic illustrations are given.

Quadratic inequalities

In practice, very often one has to deal with inequalities that can be reduced with the help of equivalent transformations to quadratic inequalities of the form a x 2 +b x + c<0 (знаки, естественно, могут быть и другими). Их можно назвать неравенствами, сводящимися к квадратным неравенствам.

Let's start with examples of the simplest inequalities that can be reduced to square ones. Sometimes, in order to pass to a quadratic inequality, it is enough to rearrange the terms in this inequality or transfer them from one part to another. For example, if we transfer all the terms from the right side of the inequality 5≤2 x−3 x 2 to the left side, then we get a quadratic inequality in the form specified above 3 x 2 −2 x+5≤0 . Another example: rearranging the inequality 5+0.6 x 2 −x on the left side<0 слагаемые по убыванию степени переменной, придем к равносильному квадратному неравенству в привычной форме 0,6·x 2 −x+5<0 .

At school, in algebra lessons, when they learn to solve quadratic inequalities, they simultaneously deal with solution of rational inequalities, reducing to square. Their solution involves the transfer of all terms to the left side with the subsequent transformation of the expression formed there to the form a x 2 +b x + c by executing . Consider an example.

Example.

Find a set of solutions to the inequality 3 (x−1) (x+1)<(x−2) 2 +x 2 +5 .irrational inequality is equivalent to the quadratic inequality x 2 −6 x−9<0 , а logarithmic inequality – inequality x 2 +x−2≥0 .

Bibliography.

• Algebra: textbook for 8 cells. general education institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; ed. S. A. Telyakovsky. - 16th ed. - M. : Education, 2008. - 271 p. : ill. - ISBN 978-5-09-019243-9.
• 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.
• Mordkovich A. G. Algebra. 8th grade. At 2 pm Part 1. A textbook for students of educational institutions / A. G. Mordkovich. - 11th ed., erased. - M.: Mnemozina, 2009. - 215 p.: ill. ISBN 978-5-346-01155-2.
• Mordkovich A. G. Algebra. Grade 9 At 2 pm Part 1. Textbook for students of educational institutions / A. G. Mordkovich, P. V. Semenov. - 13th ed., Sr. - M.: Mnemosyne, 2011. - 222 p.: ill. ISBN 978-5-346-01752-3.
• Mordkovich A. G. Algebra and beginning of mathematical analysis. Grade 11. At 2 pm Part 1. Textbook for students of educational institutions (profile level) / A. G. Mordkovich, P. V. Semenov. - 2nd ed., erased. - M.: Mnemosyne, 2008. - 287 p.: ill. ISBN 978-5-346-01027-2.

Before you figure it out how to solve quadratic inequality, let's consider what inequality is called square.

Remember!

The inequality is called square, if the highest (greatest) power of the unknown "x" is equal to two.

Let's practice determining the type of inequality using examples.

How to solve a quadratic inequality

In previous lessons, we discussed how to solve linear inequalities. But unlike linear inequalities, square inequalities are solved in a completely different way.

Important!

It is impossible to solve a quadratic inequality in the same way as a linear one!

To solve a quadratic inequality, a special method is used, which is called interval method.

What is the interval method

interval method called a special way of solving quadratic inequalities. Below we will explain how to use this method and why it is so named.

Remember!

To solve a quadratic inequality using the interval method, you need:

We understand that the rules described above are difficult to perceive only in theory, so we will immediately consider an example of solving a quadratic inequality using the algorithm above.

It is required to solve a quadratic inequality.

Now, as said in , draw "arches" over the intervals between the marked points.

Let's put signs inside the intervals. From right to left, alternating, starting with "+", we note the signs.

We just have to execute , that is, select the desired intervals and write them down in response. Let's return to our inequality.

Since in our inequality x 2 + x − 12 ", so we need negative intervals. Let's shade all negative areas on a numerical axis and we will write out them in the answer.

Only one interval turned out to be negative, which is between the numbers " −3" and "4", so we write it in response as a double inequality
"-3".

Let's write down the answer of the quadratic inequality.

Answer: -3

By the way, it is precisely because we consider the intervals between numbers when solving a quadratic inequality that the method of intervals got its name.

After receiving the answer, it makes sense to check it to make sure the solution is correct.

Let's choose any number that is in the shaded area of ​​the received answer " −3" and substitute it instead of "x" in the original inequality. If we get the correct inequality, then we have found the answer to the quadratic inequality is correct.

Take, for example, the number "0" from the interval. Substitute it into the original inequality "x 2 + x − 12".

X 2 + x − 12
0 2 + 0 − 12 −12 (correct)

We got the correct inequality when substituting a number from the solution area, which means that the answer was found correctly.

Brief notation of the solution by the method of intervals

Abbreviated record of the solution of the quadratic inequality " x 2 + x − 12 ” method of intervals will look like this:

X 2 + x − 12
x2 + x − 12 = 0

x 1 = 1+ 7 2

x 2 = 1 − 7 2

x 1 = 8 2

x 2 = x 1 = 1+ 1 4 x 2 = 1 − 1 4 x 1 = 2 4 x 2 = 0 4 x 1 = 1 2 x2 = 0 Answer: x ≤ 0 ; x ≥ 1 2 Consider an example where there is a negative coefficient in front of "x 2" in a square inequality. In this section, we have collected information about quadratic inequalities and the main approaches to their solution. We will consolidate the material with an analysis of examples. What is a quadratic inequality Let's see how to distinguish between different types of inequalities by the type of record and select square ones among them. Definition 1 Square inequality is an inequality that looks like a x 2 + b x + c< 0 , where a , b and c are some numbers, and a not equal to zero. x is a variable, and in place of the sign < can be any other inequality sign. The second name of quadratic equations is the name of "inequality of the second degree". The existence of the second name can be explained as follows. On the left side of the inequality is a polynomial of the second degree - a square trinomial. The application of the term "quadratic inequalities" to quadratic inequalities is incorrect, since quadratic functions are given by equations of the form y = a x 2 + b x + c. Here is an example of a quadratic inequality: Example 1 Let's take 5 x 2 − 3 x + 1 > 0. In this case a = 5 , b = − 3 and c = 1. Or this inequality: Example 2 − 2 , 2 z 2 − 0 , 5 z − 11 ≤ 0, where a = − 2 , 2 , b = − 0 , 5 and c = − 11. Let's show some examples of quadratic inequalities: Example 3 Particular attention should be paid to the fact that the coefficient x2 considered to be zero. This is explained by the fact that otherwise we get a linear inequality of the form b x + c > 0, since the quadratic variable, when multiplied by zero, will itself become equal to zero. At the same time, the coefficients b and c can be equal to zero both together and separately. Example 4 An example of such an inequality x 2 − 5 ≥ 0. Ways to solve quadratic inequalities There are three main methods: Definition 2 graphic; interval method; through the selection of the square of the binomial on the left side. Graphic method The method involves the construction and analysis of a graph of a quadratic function y = a x 2 + b x + c for square inequalities a x 2 + b x + c< 0 (≤ , >, ≥) . The solution of a quadratic inequality is the intervals or intervals on which the specified function takes positive and negative values. Spacing Method You can solve a quadratic inequality with one variable using the interval method. The method is applicable to solve any kind of inequalities, not just square ones. The essence of the method is to determine the signs of the intervals into which the coordinate axis is divided by the zeros of the trinomial a x 2 + b x + c if available. For inequality a x 2 + b x + c< 0 the solutions are intervals with a minus sign, for the inequality a x 2 + b x + c > 0, intervals with a plus sign. If we are dealing with non-strict inequalities, then the solution becomes an interval that includes points that correspond to the zeros of the trinomial. Selection of the square of the binomial The principle of selecting the square of the binomial on the left side of the quadratic inequality is to perform equivalent transformations that allow us to go to the solution of an equivalent inequality of the form (x − p) 2< q (≤ , >, ≥) , where p and q- some numbers. It is possible to come to quadratic inequalities with the help of equivalent transformations from inequalities of other types. This can be done in different ways. For example, by rearranging terms in a given inequality or transferring terms from one part to another. Let's take an example. Consider an equivalent transformation of the inequality 5 ≤ 2 x − 3 x2. If we transfer all the terms from the right side to the left side, then we get a quadratic inequality of the form 3 x 2 − 2 x + 5 ≤ 0. Example 5 It is necessary to find a set of solutions to the inequality 3 (x − 1) (x + 1)< (x − 2) 2 + x 2 + 5 . Decision To solve the problem, we use the formulas of abbreviated multiplication. To do this, we collect all the terms on the left side of the inequality, open the brackets and give similar terms: 3 (x − 1) (x + 1) − (x − 2) 2 − x 2 − 5< 0 , 3 · (x 2 − 1) − (x 2 − 4 · x + 4) − x 2 − 5 < 0 , 3 · x 2 − 3 − x 2 + 4 · x − 4 − x 2 − 5 < 0 , x 2 + 4 · x − 12 < 0 . We have obtained an equivalent quadratic inequality, which can be solved graphically by determining the discriminant and intersection points. D ’ = 2 2 − 1 (− 12) = 16 , x 1 = − 6 , x 2 = 2 Having built a graph, we can see that the set of solutions is the interval (− 6 , 2) . Answer: (− 6 , 2) . Irrational and logarithmic inequalities are an example of inequalities that often reduce to squares. So, for example, the inequality 2 x 2 + 5< x 2 + 6 · x + 14 is equivalent to the quadratic inequality x 2 − 6 x − 9< 0 , and the logarithmic inequality log 3 (x 2 + x + 7) ≥ 2 to the inequality x 2 + x − 2 ≥ 0. If you notice a mistake in the text, please highlight it and press Ctrl+Enter In this lesson, we will continue to consider rational inequalities and their systems, namely: a system of linear and quadratic inequalities. Let us first recall what a system of two linear inequalities with one variable is. Next, we consider a system of quadratic inequalities and a method for solving them using the example of specific problems. Let's take a closer look at the so-called roof method. We will analyze typical solutions of systems and at the end of the lesson we will consider the solution of a system with linear and quadratic inequalities. 2. Electronic educational and methodological complex for preparing grades 10-11 for entrance exams in computer science, mathematics, Russian language (). 3. Education Center "Technology of Education" (). 4. College.ru section on mathematics (). 1. Mordkovich A.G. et al. Algebra Grade 9: Taskbook for students of educational institutions / A. G. Mordkovich, T. N. Mishustina et al. - 4th ed. - M .: Mnemosyne, 2002.-143 p.: ill. No. 58 (a, c); 62; 63. Definition of quadratic inequality Remark 1 Square inequality is called because. variable is squared. Also called quadratic inequalities inequalities of the second degree. Example 1 Example. $7x^2-18x+3 0$, $11z^2+8 \le 0$ are quadratic inequalities. As can be seen from the example, not all elements of the inequality of the form $ax^2+bx+c > 0$ are present. For example, in the inequality $\frac(5)(11) y^2+\sqrt(11) y>0$ there is no free term (term $c$), but in the inequality $11z^2+8 \le 0$ there is no term with coefficient $b$. Such inequalities are also square inequalities, but they are also called incomplete quadratic inequalities. It only means that the coefficients $b$ or $c$ are equal to zero. Methods for solving quadratic inequalities When solving quadratic inequalities, the following basic methods are used: graphic; interval method; selection of the square of the binomial. Graphical way Remark 2 A graphical way to solve square inequalities $ax^2+bx+c > 0$ (or with $ sign These intervals are solution of the quadratic inequality. Spacing Method Remark 3 The interval method for solving square inequalities of the form $ax^2+bx+c > 0$ (the inequality sign can also be $ Solutions of the quadratic inequality with sign $""$ - positive intervals, with signs $"≤"$ and $"≥"$ - negative and positive intervals (respectively), including points that correspond to zeros of the trinomial. Selection of the square of the binomial The method for solving a quadratic inequality by selecting the square of a binomial is to pass to an equivalent inequality of the form $(x-n)^2 > m$ (or with the sign $ Inequalities that reduce to square Remark 4 Often, when solving inequalities, they need to be reduced to quadratic inequalities of the form $ax^2+bx+c > 0$ (the inequality sign can also be $ inequalities that reduce to square ones. Remark 5 The simplest way to reduce inequalities to square ones can be to rearrange the terms in the original inequality or transfer them, for example, from the right side to the left. For example, when transferring all the terms of the inequality $7x > 6-3x^2$ from the right side to the left side, a quadratic inequality of the form $3x^2+7x-6 > 0$ is obtained. If we rearrange the terms on the left side of the inequality $1.5y-2+5.3x^2 \ge 0$ in descending order of the degree of the variable $y$, then this will lead to an equivalent quadratic inequality of the form $5.3x^2+1.5y-2 \ge $0. When solving rational inequalities, one often uses their reduction to quadratic inequalities. In this case, it is necessary to transfer all the terms to the left side and convert the resulting expression to the form of a square trinomial. Example 2 Example. Square the inequality $7 \cdot (x+0,5) \cdot x > (3+4x)^2-10x^2+10$. Decision. We transfer all the terms to the left side of the inequality: $7 \cdot (x+0.5) \cdot x-(3+4x)^2+10x^2-10 > 0$. Using the abbreviated multiplication formulas and expanding the brackets, we simplify the expression on the left side of the inequality: $7x^2+3.5x-9-24x-16x^2+10x^2-10 > 0$; $x^2-21.5x-19 > 0$. Answer: $x^2-21.5x-19 > 0$. RELATED ARTICLES Feedback site `s map About the site