Methods for solving quadratic equations. Quadratic equations

The problem is well known from mathematics. The initial data here are the coefficients a, b, c. The solution in the general case are two roots x 1 and x 2, which are calculated by the formulas:

All values used in this program are of real type.

alg roots of a quadratic equation

thing a, b, c, x1, x2, d

early input a, b, c

x1:=(-b+Öd)/(2a)

x2:=(-b–Öd)/(2a)

output x1, x2

The weakness of such an algorithm is visible to the naked eye. He does not possess the most important property applied to qualitative algorithms: universality in relation to the initial data. Whatever the values of the initial data, the algorithm must lead to a certain result and reach the end. The result may be a numerical answer, but it may also be a message that with such data the problem has no solution. Stops in the middle of the algorithm due to the impossibility of performing some operation are not allowed. The same property in the literature on programming is called the effectiveness of the algorithm (in any case, some result must be obtained).

In order to build a universal algorithm, it is first necessary to carefully analyze the mathematical content of the problem.

The solution of the equation depends on the values of the coefficients a, b, c. Here is an analysis of this problem (we restrict ourselves only to finding real roots):

if a=0, b=0, c=0, then any x is a solution to the equation;

if a=0, b=0, c¹0, then the equation has no solutions;

if a=0, b¹0, then this linear equation, which has one solution: x=–c/b;

if a¹0 and d=b 2 -4ac³0, then the equation has two real roots (the formulas are given above);

if a¹0 and d<0, то уравнение не имеет вещественных корней.

Block diagram of the algorithm:

The same algorithm in algorithmic language:

alg roots of a quadratic equation

thing a, b, c, d, x1, x2

early input a, b, c

if a=0

then if b=0

then if c=0

then output "any x is a solution"

otherwise output "no solutions"

otherwise x:= -c/b

otherwise d:=b2–4ac

if and d<0

then output "no real roots"

otherwise e x1:=(-b+Öd)/(2a); x2:=(-b–Öd)/(2a)

output “x1=”,x1, “x2=”,x2

This algorithm reuses branch structure command. The general view of the branch command in flowcharts and in the algorithmic language is as follows:

First, the “condition” is checked (the relation, the logical expression is calculated). If the condition is true, then "series 1" is executed - the sequence of commands indicated by the arrow with the inscription "yes" (positive branch). Otherwise, "series 2" (negative branch) is executed. In EL, the condition is written after the service word "if", the positive branch - after the word "then", the negative branch - after the word "otherwise". The letters "kv" indicate the end of the branch.

If the branches of one branch contain other branches, then such an algorithm has the structure nested branches. It is this structure that the algorithm "roots of a quadratic equation" has. In it, for brevity, instead of the words "yes" and "no", respectively, "+" and "-" are used.

Consider the following problem: given a positive integer n. It is required to calculate n! (n-factorial). Recall the definition of factorial.

Below is a block diagram of the algorithm. It uses three integer type variables: n is an argument; i is an intermediate variable; F is the result. A trace table was built to check the correctness of the algorithm. In such a table, for specific values of the initial data, the changes in the variables included in the algorithm are traced by steps. This table is compiled for the case n=3.

The trace proves the correctness of the algorithm. Now let's write this algorithm in algorithmic language.

alg Factorial

whole n, i, F

early input n

F:=1; i:=1

till i£n, repeat

nc F:=F´i

This algorithm has a cyclic structure. The algorithm uses the "loop-while" or "loop with precondition" structural command. The general view of the “loop-bye” command in flowcharts and in EL is as follows:

The execution of a series of commands (loop body) is repeated while the loop condition is true. When the condition becomes false, the loop terminates. The service words "nts" and "kts" denote the beginning of the cycle and the end of the cycle, respectively.

A loop with a precondition is the main, but not the only form of organization of cyclic algorithms. Another option is loop with postcondition. Let's return to the algorithm for solving a quadratic equation. It can be approached from this position: if a=0, then this is no longer a quadratic equation and it can be ignored. In this case, we will assume that the user made a mistake when entering data and should be prompted to repeat the entry. In other words, the algorithm will provide for the control of the reliability of the initial data, providing the user with the opportunity to correct the error. The presence of such control is another sign of good program quality.

In general, the structural command "loop with postcondition" or "loop-before" is represented as follows:

This is where the loop termination condition is used. When it becomes true, the loop terminates.

Let us compose an algorithm for solving the following problem: given two natural numbers M and N. It is required to calculate their greatest common divisor - gcd(M,N).

This problem is solved using a method known as Euclid's algorithm. His idea is based on the property that if M>N, then gcd(M

1) if the numbers are equal, then take their total value as an answer; otherwise, continue the execution of the algorithm;

2) determine the larger of the numbers;

3) replace the larger number with the difference between the larger and smaller values;

4) return to the implementation of paragraph 1.

The block diagram and algorithm in AL will be as follows:

The algorithm has a loop structure with nested branching. Do your own tracing of this algorithm for the case M=18, N=12. The result is gcd=6, which is obviously true.

Bibliographic description: Gasanov A. R., Kuramshin A. A., Elkov A. A., Shilnenkov N. V., Ulanov D. D., Shmeleva O. V. Solutions quadratic equations// Young scientist. - 2016. - No. 6.1. - S. 17-20..04.2019).

Our project is dedicated to the ways of solving quadratic equations. The purpose of the project: to learn how to solve quadratic equations in ways that are not included in the school curriculum. Task: find all possible ways to solve quadratic equations and learn how to use them yourself and introduce classmates to these methods.

What are "quadratic equations"?

Quadratic equation- equation of the form ax2 + bx + c = 0, where a, b, c- some numbers ( a ≠ 0), x- unknown.

The numbers a, b, c are called the coefficients of the quadratic equation.

a is called the first coefficient;
b is called the second coefficient;
c - free member.

And who was the first to "invent" quadratic equations?

Some algebraic techniques for solving linear and quadratic equations were known as early as 4000 years ago in Ancient Babylon. The found ancient Babylonian clay tablets, dated somewhere between 1800 and 1600 BC, are the earliest evidence of the study of quadratic equations. The same tablets contain methods for solving certain types of quadratic equations.

The need to solve equations not only of the first, but also of the second degree in ancient times was caused by the need to solve problems related to finding the areas of land and earthworks of a military nature, as well as the development of astronomy and mathematics itself.

The rule for solving these equations, stated in the Babylonian texts, coincides essentially with the modern one, but it is not known how the Babylonians came to this rule. Almost all the cuneiform texts found so far give only problems with solutions stated in the form of recipes, with no indication of how they were found. Despite the high level of development of algebra in Babylon, the cuneiform texts lack the concept of a negative number and general methods for solving quadratic equations.

Babylonian mathematicians from about the 4th century B.C. used the square complement method to solve equations with positive roots. Around 300 B.C. Euclid came up with a more general geometric solution method. The first mathematician who found solutions to an equation with negative roots in the form of an algebraic formula was an Indian scientist. Brahmagupta(India, 7th century AD).

Brahmagupta outlined a general rule for solving quadratic equations reduced to a single canonical form:

ax2 + bx = c, a>0

In this equation, the coefficients can be negative. Brahmagupta's rule essentially coincides with ours.

In India, public competitions in solving difficult problems were common. In one of the old Indian books, the following is said about such competitions: “As the sun outshines the stars with its brilliance, so a learned person will outshine the glory in public meetings, proposing and solving algebraic problems.” Tasks were often dressed in poetic form.

In an algebraic treatise Al-Khwarizmi a classification of linear and quadratic equations is given. The author lists 6 types of equations, expressing them as follows:

1) “Squares are equal to roots”, i.e. ax2 = bx.

2) “Squares are equal to number”, i.e. ax2 = c.

3) "The roots are equal to the number", i.e. ax2 = c.

4) “Squares and numbers are equal to roots”, i.e. ax2 + c = bx.

5) “Squares and roots are equal to number”, i.e. ax2 + bx = c.

6) “Roots and numbers are equal to squares”, i.e. bx + c == ax2.

For Al-Khwarizmi, who avoided the use of negative numbers, the terms of each of these equations are addends, not subtractions. In this case, equations that do not have positive solutions are obviously not taken into account. The author outlines the methods for solving these equations, using the methods of al-jabr and al-muqabala. His decision, of course, does not completely coincide with ours. Not to mention the fact that it is purely rhetorical, it should be noted, for example, that when solving an incomplete quadratic equation of the first type, Al-Khwarizmi, like all mathematicians before the 17th century, does not take into account the zero solution, probably because in specific practical tasks, it does not matter. When solving complete quadratic equations, Al-Khwarizmi sets out the rules for solving them using particular numerical examples, and then their geometric proofs.

Forms for solving quadratic equations on the model of Al-Khwarizmi in Europe were first described in the "Book of the Abacus", written in 1202. Italian mathematician Leonard Fibonacci. The author independently developed some new algebraic examples of problem solving and was the first in Europe to approach the introduction of negative numbers.

This book contributed to the spread of algebraic knowledge not only in Italy, but also in Germany, France and other European countries. Many tasks from this book were transferred to almost all European textbooks of the 14th-17th centuries. The general rule for solving quadratic equations reduced to a single canonical form x2 + bx = c with all possible combinations of signs and coefficients b, c, was formulated in Europe in 1544. M. Stiefel.

Vieta has a general derivation of the formula for solving a quadratic equation, but Vieta recognized only positive roots. Italian mathematicians Tartaglia, Cardano, Bombelli among the first in the 16th century. take into account, in addition to positive, and negative roots. Only in the XVII century. thanks to the work Girard, Descartes, Newton and other scientists, the way of solving quadratic equations takes on a modern form.

Consider several ways to solve quadratic equations.

Standard ways to solve quadratic equations from the school curriculum:

Factorization of the left side of the equation.
Full square selection method.
Solution of quadratic equations by formula.
Graphical solution of a quadratic equation.
Solution of equations using Vieta's theorem.

Let us dwell in more detail on the solution of reduced and non-reduced quadratic equations using the Vieta theorem.

Recall that to solve the given quadratic equations, it is enough to find two numbers such that the product of which is equal to the free term, and the sum is equal to the second coefficient with the opposite sign.

Example.x 2 -5x+6=0

You need to find numbers whose product is 6 and the sum is 5. These numbers will be 3 and 2.

Answer: x 1 =2, x 2 =3.

But you can use this method for equations with the first coefficient not equal to one.

Example.3x 2 +2x-5=0

We take the first coefficient and multiply it by the free term: x 2 +2x-15=0

The roots of this equation will be numbers whose product is - 15, and the sum is - 2. These numbers are 5 and 3. To find the roots of the original equation, we divide the obtained roots by the first coefficient.

Answer: x 1 =-5/3, x 2 =1

6. Solution of equations by the method of "transfer".

Consider the quadratic equation ax 2 + bx + c = 0, where a≠0.

Multiplying both its parts by a, we get the equation a 2 x 2 + abx + ac = 0.

Let ax = y, whence x = y/a; then we arrive at the equation y 2 + by + ac = 0, which is equivalent to the given one. We find its roots at 1 and at 2 using the Vieta theorem.

Finally we get x 1 = y 1 /a and x 2 = y 2 /a.

With this method, the coefficient a is multiplied by the free term, as if "transferred" to it, therefore it is called the "transfer" method. This method is used when it is easy to find the roots of an equation using Vieta's theorem and, most importantly, when the discriminant is an exact square.

Example.2x 2 - 11x + 15 = 0.

Let's "transfer" the coefficient 2 to the free term and making the replacement we get the equation y 2 - 11y + 30 = 0.

According to Vieta's inverse theorem

y 1 = 5, x 1 = 5/2, x 1 = 2.5; y 2 = 6, x 2 = 6/2, x 2 = 3.

Answer: x 1 =2.5; X 2 = 3.

7. Properties of the coefficients of a quadratic equation.

Let the quadratic equation ax 2 + bx + c \u003d 0, a ≠ 0 be given.

1. If a + b + c \u003d 0 (i.e., the sum of the coefficients of the equation is zero), then x 1 \u003d 1.

2. If a - b + c \u003d 0, or b \u003d a + c, then x 1 \u003d - 1.

Example.345x 2 - 137x - 208 = 0.

Since a + b + c \u003d 0 (345 - 137 - 208 \u003d 0), then x 1 \u003d 1, x 2 \u003d -208/345.

Answer: x 1 =1; X 2 = -208/345 .

Example.132x 2 + 247x + 115 = 0

Because a-b + c \u003d 0 (132 - 247 + 115 \u003d 0), then x 1 \u003d - 1, x 2 \u003d - 115/132

Answer: x 1 = - 1; X 2 =- 115/132

There are other properties of the coefficients of a quadratic equation. but their usage is more complicated.

8. Solving quadratic equations using a nomogram.

Fig 1. Nomogram

This is an old and currently forgotten method for solving quadratic equations, placed on p. 83 of the collection: Bradis V.M. Four-digit mathematical tables. - M., Education, 1990.

Table XXII. Nomogram for Equation Solving z2 + pz + q = 0. This nomogram allows, without solving the quadratic equation, to determine the roots of the equation by its coefficients.

The curvilinear scale of the nomogram is built according to the formulas (Fig. 1):

Assuming OS = p, ED = q, OE = a(all in cm), from Fig. 1 similarity of triangles SAN And CDF we get the proportion

whence, after substitutions and simplifications, the equation follows z 2 + pz + q = 0, and the letter z means the label of any point on the curved scale.

Rice. 2 Solving a quadratic equation using a nomogram

Examples.

1) For the equation z 2 - 9z + 8 = 0 the nomogram gives the roots z 1 = 8.0 and z 2 = 1.0

Answer: 8.0; 1.0.

2) Solve the equation using the nomogram

2z 2 - 9z + 2 = 0.

Divide the coefficients of this equation by 2, we get the equation z 2 - 4.5z + 1 = 0.

The nomogram gives the roots z 1 = 4 and z 2 = 0.5.

Answer: 4; 0.5.

9. Geometric method for solving quadratic equations.

Example.X 2 + 10x = 39.

In the original, this problem is formulated as follows: "The square and ten roots are equal to 39."

Consider a square with side x, rectangles are built on its sides so that the other side of each of them is 2.5, therefore, the area of \u200b\u200beach is 2.5x. The resulting figure is then supplemented to a new square ABCD, completing four equal squares in the corners, the side of each of them is 2.5, and the area is 6.25

Rice. 3 Graphical way to solve the equation x 2 + 10x = 39

The area S of square ABCD can be represented as the sum of the areas: the original square x 2, four rectangles (4∙2.5x = 10x) and four attached squares (6.25∙4 = 25), i.e. S \u003d x 2 + 10x \u003d 25. Replacing x 2 + 10x with the number 39, we get that S \u003d 39 + 25 \u003d 64, which implies that the side of the square ABCD, i.e. segment AB \u003d 8. For the desired side x of the original square, we get

10. Solution of equations using Bezout's theorem.

Bezout's theorem. The remainder after dividing the polynomial P(x) by the binomial x - α is equal to P(α) (that is, the value of P(x) at x = α).

If the number α is the root of the polynomial P(x), then this polynomial is divisible by x -α without remainder.

Example.x²-4x+3=0

Р(x)= x²-4x+3, α: ±1,±3, α=1, 1-4+3=0. Divide P(x) by (x-1): (x²-4x+3)/(x-1)=x-3

x²-4x+3=(x-1)(x-3), (x-1)(x-3)=0

x-1=0; x=1, or x-3=0, x=3; Answer: x1 =2, x2 =3.

Output: The ability to quickly and rationally solve quadratic equations is simply necessary for solving more complex equations, for example, fractional rational equations, equations of higher powers, biquadratic equations, and in high school trigonometric, exponential and logarithmic equations. Having studied all the methods found for solving quadratic equations, we can advise classmates, in addition to standard methods, to solve by the transfer method (6) and solve equations by the property of coefficients (7), since they are more accessible for understanding.

Literature:

Bradis V.M. Four-digit mathematical tables. - M., Education, 1990.
Algebra grade 8: textbook for grade 8. general education institutions Makarychev Yu. N., Mindyuk N. G., Neshkov K. I., Suvorova S. B. ed. S. A. Telyakovsky 15th ed., revised. - M.: Enlightenment, 2015
https://en.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%B4%D1%80%D0%B0%D1%82%D0%BD%D0%BE%D0 %B5_%D1%83%D1%80%D0%B0%D0%B2%D0%BD%D0%B5%D0%BD%D0%B8%D0%B5
Glazer G.I. History of mathematics at school. A guide for teachers. / Ed. V.N. Younger. - M.: Enlightenment, 1964.

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Quadratic equations cycle of algebra lessons in the 8th grade according to the textbook by A.G. Mordkovich

Teacher MBOU Grushevskaya secondary school Kireeva T.A.

slide 3

Objectives: to introduce the concepts of a quadratic equation, the root of a quadratic equation; show solutions of quadratic equations; to form the ability to solve quadratic equations; show a way to solve complete quadratic equations using the formula of the roots of a quadratic equation.

slide 4

slide 5

A bit of history Quadratic equations in Ancient Babylon. The need to solve equations not only of the first, but also of the second degree, even in antiquity was caused by the need to solve problems related to finding the areas of land and earthworks of a military nature, as well as the development of astronomy and mathematics itself. The Babylonians knew how to solve quadratic equations about 2000 years before our faith. Applying modern algebraic notation, one can say that in their cuneiform texts there are, in addition to incomplete ones, such, for example, complete quadratic equations.

slide 6

The rule for solving these equations, set forth in the Babylonian texts, coincides with the modern one, but it is not known how the Babylonians came to this rule. Almost all the cuneiform texts found so far give only problems with solutions set out in the form of recipes, with no indication of how they were found. Despite the high level of development of algebra in Babylonia, the concept of a negative number and general methods for solving quadratic equations are absent in cuneiform texts.

Slide 7

Definition 1. A quadratic equation is an equation of the form where the coefficients a, b, c are any real numbers, and the Polynomial is called a square trinomial. a is the first or highest coefficient c is the second coefficient c is a free term

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Definition 2. A quadratic equation is called reduced if its leading coefficient is equal to 1; a quadratic equation is called unreduced if the leading coefficient is different from 1. Example. 2 - 5 + 3 = 0 - unreduced quadratic equation - reduced quadratic equation

Slide 9

Definition 3. A complete quadratic equation is a quadratic equation in which all three terms are present. a + in + c \u003d 0 An incomplete quadratic equation is an equation in which not all three terms are present; is an equation for which at least one of the coefficients in, with zero.

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Methods for solving incomplete quadratic equations.

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Solve tasks No. 24.16 (a, b) Solve the equation: or Answer. or Answer.

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Definition 4 The root of a quadratic equation is any value of the variable x at which the square trinomial vanishes; such a value of the variable x is also called the root of a square trinomial. Solving a quadratic equation means finding all its roots or establishing that there are no roots.

slide 13

The discriminant of a quadratic equation D 0 D=0 The equation has no roots The equation has two roots The equation has one root Formulas for the roots of a quadratic equation

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D>0 the quadratic equation has two roots, which are found by the formulas Example. Solve the equation Solution. a \u003d 3, b \u003d 8, c \u003d -11, Answer: 1; -3

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Algorithm for solving a quadratic equation 1. Calculate the discriminant D using the formula D = 2. If D 0, then the quadratic equation has two roots.

Quadratic equations are studied in grade 8, so there is nothing complicated here. The ability to solve them is essential.

A quadratic equation is an equation of the form ax 2 + bx + c = 0, where the coefficients a , b and c are arbitrary numbers, and a ≠ 0.

Before studying specific solution methods, we note that all quadratic equations can be divided into three classes:

Have no roots;
They have exactly one root;
They have two different roots.

This is an important difference between quadratic and linear equations, where the root always exists and is unique. How to determine how many roots an equation has? There is a wonderful thing for this - discriminant.

Discriminant

Let the quadratic equation ax 2 + bx + c = 0 be given. Then the discriminant is simply the number D = b 2 − 4ac .

This formula must be known by heart. Where it comes from is not important now. Another thing is important: by the sign of the discriminant, you can determine how many roots a quadratic equation has. Namely:

If D< 0, корней нет;
If D = 0, there is exactly one root;
If D > 0, there will be two roots.

Please note: the discriminant indicates the number of roots, and not at all their signs, as for some reason many people think. Take a look at the examples and you will understand everything yourself:

A task. How many roots do quadratic equations have:

x 2 - 8x + 12 = 0;

5x2 + 3x + 7 = 0;

x 2 − 6x + 9 = 0.

We write the coefficients for the first equation and find the discriminant:
a = 1, b = −8, c = 12;
D = (−8) 2 − 4 1 12 = 64 − 48 = 16

So, the discriminant is positive, so the equation has two different roots. We analyze the second equation in the same way:
a = 5; b = 3; c = 7;
D \u003d 3 2 - 4 5 7 \u003d 9 - 140 \u003d -131.

The discriminant is negative, there are no roots. The last equation remains:
a = 1; b = -6; c = 9;
D = (−6) 2 − 4 1 9 = 36 − 36 = 0.

The discriminant is equal to zero - the root will be one.

Note that coefficients have been written out for each equation. Yes, it's long, yes, it's tedious - but you won't mix up the odds and don't make stupid mistakes. Choose for yourself: speed or quality.

By the way, if you “fill your hand”, after a while you will no longer need to write out all the coefficients. You will perform such operations in your head. Most people start doing this somewhere after 50-70 solved equations - in general, not so much.

The roots of a quadratic equation

Now let's move on to the solution. If the discriminant D > 0, the roots can be found using the formulas:

The basic formula for the roots of a quadratic equation

When D = 0, you can use any of these formulas - you get the same number, which will be the answer. Finally, if D< 0, корней нет — ничего считать не надо.

x 2 - 2x - 3 = 0;

15 - 2x - x2 = 0;

x2 + 12x + 36 = 0.

First equation:
x 2 - 2x - 3 = 0 ⇒ a = 1; b = −2; c = -3;
D = (−2) 2 − 4 1 (−3) = 16.

D > 0 ⇒ the equation has two roots. Let's find them:

Second equation:
15 − 2x − x 2 = 0 ⇒ a = −1; b = −2; c = 15;
D = (−2) 2 − 4 (−1) 15 = 64.

D > 0 ⇒ the equation again has two roots. Let's find them

\[\begin(align) & ((x)_(1))=\frac(2+\sqrt(64))(2\cdot \left(-1 \right))=-5; \\ & ((x)_(2))=\frac(2-\sqrt(64))(2\cdot \left(-1 \right))=3. \\ \end(align)\]

Finally, the third equation:
x 2 + 12x + 36 = 0 ⇒ a = 1; b = 12; c = 36;
D = 12 2 − 4 1 36 = 0.

D = 0 ⇒ the equation has one root. Any formula can be used. For example, the first one:

As you can see from the examples, everything is very simple. If you know the formulas and be able to count, there will be no problems. Most often, errors occur when negative coefficients are substituted into the formula. Here, again, the technique described above will help: look at the formula literally, paint each step - and get rid of mistakes very soon.

Incomplete quadratic equations

It happens that the quadratic equation is somewhat different from what is given in the definition. For example:

x2 + 9x = 0;
x2 − 16 = 0.

It is easy to see that one of the terms is missing in these equations. Such quadratic equations are even easier to solve than standard ones: they do not even need to calculate the discriminant. So let's introduce a new concept:

The equation ax 2 + bx + c = 0 is called an incomplete quadratic equation if b = 0 or c = 0, i.e. the coefficient of the variable x or the free element is equal to zero.

Of course, a very difficult case is possible when both of these coefficients are equal to zero: b \u003d c \u003d 0. In this case, the equation takes the form ax 2 \u003d 0. Obviously, such an equation has a single root: x \u003d 0.

Let's consider other cases. Let b \u003d 0, then we get an incomplete quadratic equation of the form ax 2 + c \u003d 0. Let's slightly transform it:

Because arithmetic Square root exists only from a non-negative number, the last equality makes sense only for (−c /a ) ≥ 0. Conclusion:

If an incomplete quadratic equation of the form ax 2 + c = 0 satisfies the inequality (−c / a ) ≥ 0, there will be two roots. The formula is given above;
If (−c / a )< 0, корней нет.

As you can see, the discriminant was not required - there are no complex calculations at all in incomplete quadratic equations. In fact, it is not even necessary to remember the inequality (−c / a ) ≥ 0. It is enough to express the value of x 2 and see what is on the other side of the equal sign. If there is a positive number, there will be two roots. If negative, there will be no roots at all.

Now let's deal with equations of the form ax 2 + bx = 0, in which the free element is equal to zero. Everything is simple here: there will always be two roots. It is enough to factorize the polynomial:

Taking the common factor out of the bracket

The product is equal to zero when at least one of the factors is equal to zero. This is where the roots come from. In conclusion, we will analyze several of these equations: