Quadratic equation - easy to solve! *Further in the text "KU". Friends, it would seem that in mathematics it can be easier than solving such an equation. But something told me that many people have problems with him. I decided to see how many impressions Yandex gives per request per month. Here's what happened, take a look:
What does it mean? This means that about 70,000 people a month are looking for this information, and this is summer, and what will happen during the school year - there will be twice as many requests. This is not surprising, because those guys and girls who have long graduated from school and are preparing for the exam are looking for this information, and schoolchildren are also trying to refresh their memory.
Despite the fact that there are a lot of sites that tell how to solve this equation, I decided to also contribute and publish the material. Firstly, I want visitors to come to my site on this request; secondly, in other articles, when the speech “KU” comes up, I will give a link to this article; thirdly, I will tell you a little more about his solution than is usually stated on other sites. Let's get started! The content of the article:
A quadratic equation is an equation of the form:
where coefficients a,band with arbitrary numbers, with a≠0.
In the school course, the material is given in the following form - the division of equations into three classes is conditionally done:
1. Have two roots.
2. * Have only one root.
3. Have no roots. It is worth noting here that they do not have real roots
How are roots calculated? Just!
We calculate the discriminant. Under this "terrible" word lies a very simple formula:
The root formulas are as follows:
*These formulas must be known by heart.
You can immediately write down and solve:
Example:
1. If D > 0, then the equation has two roots.
2. If D = 0, then the equation has one root.
3. If D< 0, то уравнение не имеет действительных корней.
Let's look at the equation:
On this occasion, when the discriminant is zero, the school course says that one root is obtained, here it is equal to nine. That's right, it is, but...
This representation is somewhat incorrect. In fact, there are two roots. Yes, yes, do not be surprised, it turns out two equal roots, and to be mathematically accurate, then two roots should be written in the answer:
x 1 = 3 x 2 = 3
But this is so - a small digression. At school, you can write down and say that there is only one root.
Now the following example:
As we know, the root of a negative number is not extracted, so there is no solution in this case.
That's the whole decision process.
Quadratic function.
Here is how the solution looks geometrically. This is extremely important to understand (in the future, in one of the articles, we will analyze in detail the solution of a quadratic inequality).
This is a function of the form:
where x and y are variables
a, b, c are given numbers, where a ≠ 0
The graph is a parabola:
That is, it turns out that by solving a quadratic equation with "y" equal to zero, we find the points of intersection of the parabola with the x-axis. There can be two of these points (the discriminant is positive), one (the discriminant is zero) or none (the discriminant is negative). More about the quadratic function You can view article by Inna Feldman.
Consider examples:
Example 1: Decide 2x 2 +8 x–192=0
a=2 b=8 c= -192
D = b 2 –4ac = 8 2 –4∙2∙(–192) = 64+1536 = 1600
Answer: x 1 = 8 x 2 = -12
* You could immediately divide the left and right sides of the equation by 2, that is, simplify it. The calculations will be easier.
Example 2: Decide x2–22 x+121 = 0
a=1 b=-22 c=121
D = b 2 –4ac =(–22) 2 –4∙1∙121 = 484–484 = 0
We got that x 1 \u003d 11 and x 2 \u003d 11
In the answer, it is permissible to write x = 11.
Answer: x = 11
Example 3: Decide x 2 –8x+72 = 0
a=1 b= -8 c=72
D = b 2 –4ac =(–8) 2 –4∙1∙72 = 64–288 = –224
The discriminant is negative, there is no solution in real numbers.
Answer: no solution
The discriminant is negative. There is a solution!
Here we will talk about solving the equation in the case when a negative discriminant is obtained. Do you know anything about complex numbers? I will not go into detail here about why and where they arose and what their specific role and necessity in mathematics is, this is a topic for a large separate article.
The concept of a complex number.
A bit of theory.
A complex number z is a number of the form
z = a + bi
where a and b are real numbers, i is the so-called imaginary unit.
a+bi is a SINGLE NUMBER, not an addition.
The imaginary unit is equal to the root of minus one:
Now consider the equation:
Get two conjugate roots.
Incomplete quadratic equation.
Consider special cases, this is when the coefficient "b" or "c" is equal to zero (or both are equal to zero). They are solved easily without any discriminants.
Case 1. Coefficient b = 0.
The equation takes the form:
Let's transform:
Example:
4x 2 -16 = 0 => 4x 2 =16 => x 2 = 4 => x 1 = 2 x 2 = -2
Case 2. Coefficient c = 0.
The equation takes the form:
Transform, factorize:
*The product is equal to zero when at least one of the factors is equal to zero.
Example:
9x 2 –45x = 0 => 9x (x–5) =0 => x = 0 or x–5 =0
x 1 = 0 x 2 = 5
Case 3. Coefficients b = 0 and c = 0.
Here it is clear that the solution to the equation will always be x = 0.
Useful properties and patterns of coefficients.
There are properties that allow solving equations with large coefficients.
Ax 2 + bx+ c=0 equality
a + b+ c = 0, That
— if for the coefficients of the equation Ax 2 + bx+ c=0 equality
a+ with =b, That
These properties help solve a certain kind of equation.
Example 1: 5001 x 2 –4995 x – 6=0
The sum of the coefficients is 5001+( – 4995)+(– 6) = 0, so
Example 2: 2501 x 2 +2507 x+6=0
Equality a+ with =b, Means
Regularities of coefficients.
1. If in the equation ax 2 + bx + c \u003d 0 the coefficient "b" is (a 2 +1), and the coefficient "c" is numerically equal to the coefficient "a", then its roots are
ax 2 + (a 2 +1) ∙ x + a \u003d 0 \u003d\u003e x 1 \u003d -a x 2 \u003d -1 / a.
Example. Consider the equation 6x 2 +37x+6 = 0.
x 1 \u003d -6 x 2 \u003d -1/6.
2. If in the equation ax 2 - bx + c \u003d 0, the coefficient "b" is (a 2 +1), and the coefficient "c" is numerically equal to the coefficient "a", then its roots are
ax 2 - (a 2 + 1) ∙ x + a \u003d 0 \u003d\u003e x 1 \u003d a x 2 \u003d 1 / a.
Example. Consider the equation 15x 2 –226x +15 = 0.
x 1 = 15 x 2 = 1/15.
3. If in the equation ax 2 + bx - c = 0 coefficient "b" equals (a 2 – 1), and the coefficient “c” numerically equal to the coefficient "a", then its roots are equal
ax 2 + (a 2 -1) ∙ x - a \u003d 0 \u003d\u003e x 1 \u003d - a x 2 \u003d 1 / a.
Example. Consider the equation 17x 2 + 288x - 17 = 0.
x 1 \u003d - 17 x 2 \u003d 1/17.
4. If in the equation ax 2 - bx - c \u003d 0, the coefficient "b" is equal to (a 2 - 1), and the coefficient c is numerically equal to the coefficient "a", then its roots are
ax 2 - (a 2 -1) ∙ x - a \u003d 0 \u003d\u003e x 1 \u003d a x 2 \u003d - 1 / a.
Example. Consider the equation 10x2 - 99x -10 = 0.
x 1 \u003d 10 x 2 \u003d - 1/10
Vieta's theorem.
Vieta's theorem is named after the famous French mathematician Francois Vieta. Using Vieta's theorem, one can express the sum and product of the roots of an arbitrary KU in terms of its coefficients.
45 = 1∙45 45 = 3∙15 45 = 5∙9.
In sum, the number 14 gives only 5 and 9. These are the roots. With a certain skill, using the presented theorem, you can solve many quadratic equations immediately orally.
Vieta's theorem, moreover. convenient because after solving the quadratic equation in the usual way (through the discriminant), the resulting roots can be checked. I recommend doing this all the time.
TRANSFER METHOD
With this method, the coefficient "a" is multiplied by the free term, as if "transferred" to it, which is why it is called 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.
If A± b+c≠ 0, then the transfer technique is used, for example:
2X 2 – 11x+ 5 = 0 (1) => X 2 – 11x+ 10 = 0 (2)
According to the Vieta theorem in equation (2), it is easy to determine that x 1 \u003d 10 x 2 \u003d 1
The resulting roots of the equation must be divided by 2 (since the two were “thrown” from x 2), we get
x 1 \u003d 5 x 2 \u003d 0.5.
What is the rationale? See what's happening.
The discriminants of equations (1) and (2) are:
If you look at the roots of the equations, then only different denominators are obtained, and the result depends precisely on the coefficient at x 2:
The second (modified) roots are 2 times larger.
Therefore, we divide the result by 2.
*If we roll three of a kind, then we divide the result by 3, and so on.
Answer: x 1 = 5 x 2 = 0.5
sq. ur-ie and the exam.
I will say briefly about its importance - YOU SHOULD BE ABLE TO DECIDE quickly and without thinking, you need to know the formulas of the roots and the discriminant by heart. A lot of the tasks that are part of the USE tasks come down to solving a quadratic equation (including geometric ones).
What is worth noting!
1. The form of the equation can be "implicit". For example, the following entry is possible:
15+ 9x 2 - 45x = 0 or 15x+42+9x 2 - 45x=0 or 15 -5x+10x 2 = 0.
You need to bring it to a standard form (so as not to get confused when solving).
2. Remember that x is an unknown value and it can be denoted by any other letter - t, q, p, h and others.
With this math program you can solve quadratic equation.
The program not only gives the answer to the problem, but also displays the solution process in two ways:
- using the discriminant
- using the Vieta theorem (if possible).
Moreover, the answer is displayed exact, not approximate.
For example, for the equation \(81x^2-16x-1=0\), the answer is displayed in this form:
This program can be useful for high school students in preparation for tests and exams, when testing knowledge before the Unified State Examination, for parents to control the solution of many problems in mathematics and algebra. Or maybe it's too expensive for you to hire a tutor or buy new textbooks? Or do you just want to get your math or algebra homework done as quickly as possible? In this case, you can also use our programs with a detailed solution.
In this way, you can conduct your own training and/or the training of your younger brothers or sisters, while the level of education in the field of tasks to be solved is increased.
If you are not familiar with the rules for entering a square polynomial, we recommend that you familiarize yourself with them.
Rules for entering a square polynomial
Any Latin letter can act as a variable.
For example: \(x, y, z, a, b, c, o, p, q \) etc.
Numbers can be entered as integers or fractions.
Moreover, fractional numbers can be entered not only in the form of a decimal, but also in the form of an ordinary fraction.
Rules for entering decimal fractions.
In decimal fractions, the fractional part from the integer can be separated by either a dot or a comma.
For example, you can enter decimals like this: 2.5x - 3.5x^2
Rules for entering ordinary fractions.
Only a whole number can act as the numerator, denominator and integer part of a fraction.
The denominator cannot be negative.
When entering a numerical fraction, the numerator is separated from the denominator by a division sign: /
The integer part is separated from the fraction by an ampersand: &
Input: 3&1/3 - 5&6/5z +1/7z^2
Result: \(3\frac(1)(3) - 5\frac(6)(5) z + \frac(1)(7)z^2 \)
When entering an expression you can use brackets. In this case, when solving a quadratic equation, the introduced expression is first simplified.
For example: 1/2(y-1)(y+1)-(5y-10&1/2)
Decide
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A bit of theory.
Quadratic equation and its roots. Incomplete quadratic equations
Each of the equations
\(-x^2+6x+1,4=0, \quad 8x^2-7x=0, \quad x^2-\frac(4)(9)=0 \)
has the form
\(ax^2+bx+c=0, \)
where x is a variable, a, b and c are numbers.
In the first equation a = -1, b = 6 and c = 1.4, in the second a = 8, b = -7 and c = 0, in the third a = 1, b = 0 and c = 4/9. Such equations are called quadratic equations.
Definition.
quadratic equation an equation of the form ax 2 +bx+c=0 is called, where x is a variable, a, b and c are some numbers, and \(a \neq 0 \).
The numbers a, b and c are the coefficients of the quadratic equation. The number a is called the first coefficient, the number b is the second coefficient and the number c is the intercept.
In each of the equations of the form ax 2 +bx+c=0, where \(a \neq 0 \), the largest power of the variable x is a square. Hence the name: quadratic equation.
Note that a quadratic equation is also called an equation of the second degree, since its left side is a polynomial of the second degree.
A quadratic equation in which the coefficient at x 2 is 1 is called reduced quadratic equation. For example, the given quadratic equations are the equations
\(x^2-11x+30=0, \quad x^2-6x=0, \quad x^2-8=0 \)
If in the quadratic equation ax 2 +bx+c=0 at least one of the coefficients b or c is equal to zero, then such an equation is called incomplete quadratic equation. So, the equations -2x 2 +7=0, 3x 2 -10x=0, -4x 2 =0 are incomplete quadratic equations. In the first of them b=0, in the second c=0, in the third b=0 and c=0.
Incomplete quadratic equations are of three types:
1) ax 2 +c=0, where \(c \neq 0 \);
2) ax 2 +bx=0, where \(b \neq 0 \);
3) ax2=0.
Consider the solution of equations of each of these types.
To solve an incomplete quadratic equation of the form ax 2 +c=0 for \(c \neq 0 \), its free term is transferred to the right side and both parts of the equation are divided by a:
\(x^2 = -\frac(c)(a) \Rightarrow x_(1,2) = \pm \sqrt( -\frac(c)(a)) \)
Since \(c \neq 0 \), then \(-\frac(c)(a) \neq 0 \)
If \(-\frac(c)(a)>0 \), then the equation has two roots.
If \(-\frac(c)(a) To solve an incomplete quadratic equation of the form ax 2 +bx=0 for \(b \neq 0 \) factorize its left side and obtain the equation
\(x(ax+b)=0 \Rightarrow \left\( \begin(array)(l) x=0 \\ ax+b=0 \end(array) \right. \Rightarrow \left\( \begin (array)(l) x=0 \\ x=-\frac(b)(a) \end(array) \right. \)
Hence, an incomplete quadratic equation of the form ax 2 +bx=0 for \(b \neq 0 \) always has two roots.
An incomplete quadratic equation of the form ax 2 \u003d 0 is equivalent to the equation x 2 \u003d 0 and therefore has a single root 0.
The formula for the roots of a quadratic equation
Let us now consider how quadratic equations are solved in which both coefficients of the unknowns and the free term are nonzero.
We solve the quadratic equation in general form and as a result we obtain the formula of the roots. Then this formula can be applied to solve any quadratic equation.
Solve the quadratic equation ax 2 +bx+c=0
Dividing both its parts by a, we obtain the equivalent reduced quadratic equation
\(x^2+\frac(b)(a)x +\frac(c)(a)=0 \)
We transform this equation by highlighting the square of the binomial:
\(x^2+2x \cdot \frac(b)(2a)+\left(\frac(b)(2a)\right)^2- \left(\frac(b)(2a)\right)^ 2 + \frac(c)(a) = 0 \Rightarrow \)
The root expression is called discriminant of a quadratic equation ax 2 +bx+c=0 (“discriminant” in Latin - distinguisher). It is denoted by the letter D, i.e.
\(D = b^2-4ac\)
Now, using the notation of the discriminant, we rewrite the formula for the roots of the quadratic equation:
\(x_(1,2) = \frac( -b \pm \sqrt(D) )(2a) \), where \(D= b^2-4ac \)
It's obvious that:
1) If D>0, then the quadratic equation has two roots.
2) If D=0, then the quadratic equation has one root \(x=-\frac(b)(2a)\).
3) If D Thus, depending on the value of the discriminant, the quadratic equation can have two roots (for D > 0), one root (for D = 0) or no roots (for D When solving a quadratic equation using this formula, it is advisable to do the following way:
1) calculate the discriminant and compare it with zero;
2) if the discriminant is positive or equal to zero, then use the root formula, if the discriminant is negative, then write down that there are no roots.
Vieta's theorem
The given quadratic equation ax 2 -7x+10=0 has roots 2 and 5. The sum of the roots is 7, and the product is 10. We see that the sum of the roots is equal to the second coefficient, taken with the opposite sign, and the product of the roots is equal to the free term. Any reduced quadratic equation that has roots has this property.
The sum of the roots of the given quadratic equation is equal to the second coefficient, taken with the opposite sign, and the product of the roots is equal to the free term.
Those. Vieta's theorem states that the roots x 1 and x 2 of the reduced quadratic equation x 2 +px+q=0 have the property:
\(\left\( \begin(array)(l) x_1+x_2=-p \\ x_1 \cdot x_2=q \end(array) \right. \)
Type equation
Expression D= b 2
- 4ac called discriminant quadratic equation. IfD = 0, then the equation has one real root; if D> 0, then the equation has two real roots.
In case when D = 0
, it is sometimes said that a quadratic equation has two identical roots.
Using the notation D= b 2
- 4ac, formula (2) can be rewritten as
If b= 2k, then formula (2) takes the form:
Where k= b / 2
.
The last formula is especially convenient when b / 2
is an integer, i.e. coefficient b- even number.
Example 1: solve the equation 2
x 2
-
5 x +
2
=
0
. Here a=2, b=-5, c=2. We have D= b 2
-
4ac =
(-5) 2-
4*2*2
=
9
. Because D >
0
, then the equation has two roots. Let's find them by the formula (2)
So x 1
=(5 + 3) / 4 = 2,x 2
=(5 - 3) / 4 = 1 / 2
,
that is x 1
=
2
And x 2
=
1
/
2
are the roots of the given equation.
Example 2: solve the equation 2
x 2
- 3 x + 5 = 0
. Here a=2, b=-3, c=5. Finding the discriminant D= b 2
-
4ac =
(-3) 2- 4*2*5 = -31
. Because D 0
, then the equation has no real roots.
Incomplete quadratic equations.
If in a quadratic equation ax 2
+bx+c =0
second factor b or free member c equals zero, then the quadratic equation is called incomplete. Incomplete equations are distinguished because to find their roots, you can not use the formula for the roots of a quadratic equation - it is easier to solve the equation by factoring its left side into factors.
Example 1: solve the equation 2
x 2
- 5 x = 0
.
We have x(2 x - 5) = 0
. So either x = 0
, or 2
x - 5 = 0
, that is x =
2.5
. So the equation has two roots: 0
And 2.5
Example 2: solve the equation 3
x 2
- 27 = 0
.
We have 3
x 2
= 27
. Therefore, the roots of this equation are 3
And -3
.
Vieta's theorem. If the given quadratic equation x 2 +px+ q =0 has real roots, then their sum is equal to - p, and the product is q, that is
x 1 + x 2 \u003d -p,
x 1 x 2 = q
(the sum of the roots of the given quadratic equation is equal to the second coefficient, taken with the opposite sign, and the product of the roots is equal to the free term).
Kopyevskaya rural secondary school
10 Ways to Solve Quadratic Equations
Head: Patrikeeva Galina Anatolyevna,
mathematic teacher
s.Kopyevo, 2007
1. History of the development of quadratic equations
1.1 Quadratic equations in ancient Babylon
1.2 How Diophantus compiled and solved quadratic equations
1.3 Quadratic equations in India
1.4 Quadratic equations in al-Khwarizmi
1.5 Quadratic equations in Europe XIII - XVII centuries
1.6 About Vieta's theorem
2. Methods for solving quadratic equations
Conclusion
Literature
1. History of the development of quadratic equations
1.1 Quadratic equations in ancient Babylon
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. Quadratic equations were able to solve about 2000 BC. e. Babylonians.
Applying modern algebraic notation, we can say that in their cuneiform texts there are, in addition to incomplete ones, such, for example, complete quadratic equations:
X 2 + X = ¾; X 2 - X = 14,5
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.
1.2 How Diophantus compiled and solved quadratic equations.
Diophantus' Arithmetic does not contain a systematic exposition of algebra, but it contains a systematic series of problems, accompanied by explanations and solved by drawing up equations of various degrees.
When compiling equations, Diophantus skillfully chooses unknowns to simplify the solution.
Here, for example, is one of his tasks.
Task 11."Find two numbers knowing that their sum is 20 and their product is 96"
Diophantus argues as follows: it follows from the condition of the problem that the desired numbers are not equal, since if they were equal, then their product would not be 96, but 100. Thus, one of them will be more than half of their sum, i.e. . 10+x, the other is smaller, i.e. 10's. The difference between them 2x .
Hence the equation:
(10 + x)(10 - x) = 96
100 - x 2 = 96
x 2 - 4 = 0 (1)
From here x = 2. One of the desired numbers is 12 , other 8 . Solution x = -2 for Diophantus does not exist, since Greek mathematics knew only positive numbers.
If we solve this problem by choosing one of the desired numbers as the unknown, then we will come to the solution of the equation
y(20 - y) = 96,
y 2 - 20y + 96 = 0. (2)
It is clear that Diophantus simplifies the solution by choosing the half-difference of the desired numbers as the unknown; he manages to reduce the problem to solving an incomplete quadratic equation (1).
1.3 Quadratic equations in India
Problems for quadratic equations are already found in the astronomical tract "Aryabhattam", compiled in 499 by the Indian mathematician and astronomer Aryabhatta. Another Indian scientist, Brahmagupta (7th century), outlined the general rule for solving quadratic equations reduced to a single canonical form:
ah 2+ b x = c, a > 0. (1)
In equation (1), the coefficients, except for A, can also be negative. Brahmagupta's rule essentially coincides with ours.
In ancient 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 of another in public meetings, proposing and solving algebraic problems.” Tasks were often dressed in poetic form.
Here is one of the problems of the famous Indian mathematician of the XII century. Bhaskara.
Task 13.
“A frisky flock of monkeys And twelve in vines ...
Having eaten power, had fun. They began to jump, hanging ...
Part eight of them in a square How many monkeys were there,
Having fun in the meadow. You tell me, in this flock?
Bhaskara's solution indicates that he knew about the two-valuedness of the roots of quadratic equations (Fig. 3).
The equation corresponding to problem 13 is:
( x /8) 2 + 12 = x
Bhaskara writes under the guise of:
x 2 - 64x = -768
and, to complete the left side of this equation to a square, he adds to both sides 32 2 , getting then:
x 2 - 64x + 32 2 = -768 + 1024,
(x - 32) 2 = 256,
x - 32 = ± 16,
x 1 = 16, x 2 = 48.
1.4 Quadratic equations in al-Khorezmi
Al-Khorezmi's algebraic treatise gives a classification of linear and quadratic equations. The author lists 6 types of equations, expressing them as follows:
1) "Squares are equal to roots", i.e. ax 2 + c = b X.
2) "Squares are equal to number", i.e. ax 2 = s.
3) "The roots are equal to the number", i.e. ah = s.
4) "Squares and numbers are equal to roots", i.e. ax 2 + c = b X.
5) "Squares and roots are equal to the number", i.e. ah 2+ bx = s.
6) "Roots and numbers are equal to squares", i.e. bx + c \u003d ax 2.
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 decisions, of course, do 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-Khorezmi, like all mathematicians before the 17th century, does not take into account the zero solution, probably because it does not matter in specific practical problems. When solving complete quadratic equations, al-Khorezmi sets out the rules for solving, and then geometric proofs, using particular numerical examples.
Task 14.“The square and the number 21 are equal to 10 roots. Find the root" (assuming the root of the equation x 2 + 21 = 10x).
The author's solution goes something like this: divide the number of roots in half, you get 5, multiply 5 by itself, subtract 21 from the product, 4 remains. Take the root of 4, you get 2. Subtract 2 from 5, you get 3, this will be the desired root. Or add 2 to 5, which will give 7, this is also a root.
Treatise al-Khorezmi is the first book that has come down to us, in which the classification of quadratic equations is systematically stated and formulas for their solution are given.
1.5 Quadratic equations in Europe XIII - XVII centuries
Formulas for solving quadratic equations on the model of al - Khorezmi in Europe were first set forth in the "Book of the Abacus", written in 1202 by the Italian mathematician Leonardo Fibonacci. This voluminous work, which reflects the influence of mathematics, both the countries of Islam and Ancient Greece, is distinguished by both completeness and clarity of presentation. The author independently developed some new algebraic examples of problem solving and was the first in Europe to approach the introduction of negative numbers. His book contributed to the spread of algebraic knowledge not only in Italy, but also in Germany, France and other European countries. Many problems from the Book of the Abacus passed into almost all European textbooks of the 16th-17th centuries. and partly XVIII.
The general rule for solving quadratic equations reduced to a single canonical form:
x 2+ bx = with,
for all possible combinations of signs of the coefficients b , With was formulated in Europe only in 1544 by M. Stiefel.
Vieta has a general derivation of the formula for solving a quadratic equation, but Vieta recognized only positive roots. The Italian mathematicians Tartaglia, Cardano, Bombelli were 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 of Girard, Descartes, Newton and other scientists, the way to solve quadratic equations takes on a modern look.
1.6 About Vieta's theorem
The theorem expressing the relationship between the coefficients of a quadratic equation and its roots, bearing the name of Vieta, was formulated by him for the first time in 1591 as follows: “If B + D multiplied by A - A 2 , equals BD, That A equals IN and equal D ».
To understand Vieta, one must remember that A, like any vowel, meant for him the unknown (our X), the vowels IN, D- coefficients for the unknown. In the language of modern algebra, Vieta's formulation above means: if
(a + b )x - x 2 = ab ,
x 2 - (a + b )x + a b = 0,
x 1 = a, x 2 = b .
Expressing the relationship between the roots and coefficients of equations by general formulas written using symbols, Viet established uniformity in the methods of solving equations. However, the symbolism of Vieta is still far from its modern form. He did not recognize negative numbers, and therefore, when solving equations, he considered only cases where all roots are positive.
2. Methods for solving quadratic equations
Quadratic equations are the foundation on which the majestic edifice of algebra rests. Quadratic equations are widely used in solving trigonometric, exponential, logarithmic, irrational and transcendental equations and inequalities. We all know how to solve quadratic equations from school (grade 8) until graduation.
Having received a general idea of \u200b\u200bequalities, and having become acquainted with one of their types - numerical equalities, you can start talking about another form of equality that is very important from a practical point of view - about equations. In this article, we will analyze what is the equation, and what is called the root of the equation. Here we give the corresponding definitions, and also give various examples of equations and their roots.
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What is an equation?
Purposeful familiarity with equations usually begins in math classes in grade 2. At this time the following equation definition:
Definition.
The equation is an equality containing an unknown number 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 standpoint of the form of the notation. In other words, equality is an equation when it obeys the specified notation 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 like x=8 , y=3 , etc. Equations that contain signs of arithmetic operations 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 acquaintance with - equations with brackets begin to appear, for example, 2 (x−1)=18 and x+3 (x+2 (x−2))=3 . An unknown letter can appear multiple times in an equation, for example, x+3+3 x−2−x=9 , and 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 integers, rational, real numbers, new mathematical objects are studied: degrees, roots, logarithms, etc., while more and more new types of equations appear that contain these things. Examples can be found in the article. main types of equations studied at school.
In grade 7, 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). In this case, the word “variable” is introduced into the definition of the equation, and it becomes like this:
Definition.
Equation name an equality containing a variable whose value is to be found.
For example, the equation x+3=6 x+7 is an equation with variable x , and 3 z−1+z=0 is an equation with variable z .
In algebra lessons in the same 7th grade, there is a meeting with equations containing in their record not one, but two different unknown variables. They are called equations with two variables. In the future, the presence of three or more variables in the equation record is allowed.
Definition.
Equations with one, two, three, etc. variables- these are equations containing one, two, three, ... unknown variables in their record, 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 the equation?
The definition of the equation's root is directly related to the definition of the equation. We will carry out some reasoning that will help us understand what the root of the equation is.
Suppose we have an equation with one letter (variable). If instead of the letter included in the record of this equation, a certain number is substituted, then the equation will turn into a numerical equality. Moreover, the resulting equality can be both true and false. For example, if instead of the letter a in the equation a+1=5 we substitute the number 2 , then we get an incorrect numerical equality 2+1=5 . If we substitute the number 4 instead of a in this equation, then we get the correct equality 4+1=5.
In practice, in the vast majority of cases, of interest are such values of the variable, the substitution of which into the equation gives the correct equality, these values are called the roots or solutions of this equation.
Definition.
Root of the equation- this is the value of the letter (variable), when substituting which the equation turns into the correct numerical equality.
Note that the root of an equation with 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, we return to the equation written above a+1=5 . According to the voiced definition of the root of the 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 with roots and equations without roots. For example, the equation x+1=5 has a root 4, and the equation 0 x=5 has no roots, since no matter what number we substitute into this equation instead of the variable x, we will get the wrong equality 0=5.
As for the number of roots of an equation, there are both equations that have some finite number of roots (one, two, three, etc.) and equations that have infinitely many 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 of the roots of the equation. If the equation has no roots, then usually they write “the equation has no roots” or use the sign of the empty set ∅. If the equation has roots, then they are written separated by commas, or written as set elements 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 possible to write the roots of the equation in the form of simple equalities. For example, if the letter x enters the equation, 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 numbers the roots of the equation. An infinite set of roots of an equation is usually written in the form, also, if possible, the notation of sets of natural numbers N, integers Z, real numbers R is used. For example, if the root of the equation with the variable x is any integer, then write , and if the roots of the equation with the 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 “equation root” is not used, in these cases they say “solution of the equation”. What is called the solution of equations with several variables? Let us give an appropriate definition.
Definition.
Solving an equation with two, three, etc. variables call a pair, three, etc. values of the variables, which turns this equation into a true numerical equality.
We will show explanatory examples. Consider an equation with two variables x+y=7 . We substitute the number 1 instead of x, and the number 2 instead of y, while 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 come to 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 multiple variables, like equations with one variable, may have no roots, may have a finite number of roots, or may have infinitely many roots.
Pairs, triples, fours, etc. variable values 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 beginning of analysis is given to finding the roots of equations with one variable. We will analyze the rules of this process in great detail in the article. solution of equations.
Bibliography.
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- Algebra: textbook for 7 cells. general education institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; ed. S. A. Telyakovsky. - 17th ed. - M. : Education, 2008. - 240 p. : ill. - ISBN 978-5-09-019315-3.
- 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.
