Symmetric systems of alignment. §five. Homogeneous alignment and systems
1.
Rivnyannia are called symmetrical equals of the 3rd stage yakscho stink
ax 3 + bx 2 + bx + a = 0.
In order to successfully win such a kind of jealousy, it’s important to know that it’s worth victoriously attacking the simplest power of the rightful equals:
but) Have any kind of virtuous equivalence of the unpaired world, the root is the root, equivalence -1.
In fact, to group the left part of the addendum like this: a(x 3 + 1) + bx(x + 1) = 0, so that you can blame the big multiplier, so. (x + 1) (ax 2 + (b - a) x + a) \u003d 0, to that,
x + 1 = 0 or ah 2 + (b - a) x + a = 0
b) The root equal to zero has no root, which is equal to zero.
in) When the polynomial of the unpaired degree (x + 1) is split, it is better to use the rotary polynomial again and it is not necessary to introduce the induction.
Butt.
x 3 + 2x2 + 2x + 1 = 0.
Solution.
The output equalization of obov'yazkovo has root x = -1, so we divide x 3 + 2x 2 + 2x + 1 into (x + 1) according to Horner's scheme:
| . |
1 |
2 |
2 |
1 |
| -1 |
1 |
2 – 1 = 1 | 2 – 1 = 1 | 1 – 1 = 0 |
x 3 + 2x 2 + 2x + 1 = (x + 1)(x 2 + x + 1) = 0.
Square alignment x 2 + x + 1 = 0 has no root.
Suggestion: -1.
2.
Rivnyannia are called symmetrical equals of the 4th step yakscho stink
ax 4+bx3+cx2+bx+a=0.
Solution algorithm similar equals are:
but) Divide the insults in the part of the outward equivalence by x 2. Tsya diya is not created before the root is spent, even if x = 0 the solution of the given equalization is not є.
b) For additional grouping, bring equal to the point:
a(x 2 + 1/x 2) + b(x + 1/x) + c = 0.
in) Enter a new unknown: t = (x + 1/x).
We can see the transformation: t 2 \u003d x 2 +2 + 1 / x 2. How to express x 2 + 1/x 2, then t 2 - 2 = x 2 + 1/x 2.
G) Virishiti in the new otriman square alignment:
at 2 + bt + c - 2a = 0.
e) Zrobiti zvorotnu pіdstanovka.
butt.
6x 4 - 5x 3 - 38x 2 - 5x + 6 = 0.
Solution.
6x2 - 5x - 38 - 5/x + 6/x2 = 0.
6 (x 2 + 1 / x 2) - 5 (x + 1 / x) - 38 = 0.
Input t: substitution (x+1/x) = t. Replacement: (x 2 + 1 / x 2) \u003d t 2 - 2, maybe:
6t 2 - 5t - 50 = 0.
t = -5/2 or t = 10/3.
Let's turn to change. After the return of the replacement, we will untie two omissions of equality:
1) x + 1/x = -5/2;
x 2 + 5/2 x +1 = 0;
x = -2 chi x = -1/2.
2) x + 1/x = 10/3;
x 2 - 10/3 x + 1 = 0;
x = 3 chi x = 1/3.
Vidpovid: -2; -1/2; 1/3; 3.
Ways to untie the current types of equals of the higher steps
1. Rivnyannya, what to look at (x + a) n + (x + b) n = c, are changed by the substitution t = x + (a + b)/2. This method is called by the method of symmetry.
The butt of such an equalization can be equal to the form (x + a) 4 + (x + b) 4 \u003d c.
butt.
(x + 3) 4 + (x + 1) 4 = 272.
Solution.
Robimo substitution, more was said about the yaku:
t = x + (3 + 1) / 2 = x + 2
(t - 2 + 3) 4 + (t - 2 + 1) 4 = 272.
(t + 1) 4 + (t - 1) 4 = 272.
Having cleaned the arms for additional formulas, we take:
t 4 + 4t 3 + 6t 2 + 4t + 1 + t 4 - 4t 3 + 6t 2 - 4t + 1 = 272.
2t4 + 12t2 - 270 = 0.
t 4 + 6t 2 - 135 = 0.
t2=9 or t2=-15.
Another equal root does not give, and the z-axis of the first may t = ±3.
After the replacement, it is possible that x \u003d -5 or x \u003d 1.
Vidpovid: -5; one.
For the development of similar equals, they often appear to be effective and method of factorization of the left side of the equation
2. Equal to mind (x + a) (x + b) (x + c) (x + d) = A de a + d = c + b.
The method of rozvyazannya similar rivnyan polagat chastkovy rozkritti shackles, and then introduced a new zminnoy.
butt.
(x + 1) (x + 2) (x + 3) (x + 4) = 24.
Solution.
Counting: 1 + 4 = 2 + 3. Grouping the bows by pairs:
((x + 1) (x + 4)) ((x + 2) (x + 3)) = 24,
(X 2 + 5x + 4) (X 2 + 5x + 6) = 24.
Zrobivshi replacement x 2 + 5x + 4 \u003d t, maybe even
t(t + 2) = 24
t 2 + 2t - 24 = 0.
t = -6 chi t = 4.
After the victory of the victorious replacement, it is easy to know the root of the outward equivalence.
Vidpovid: -5; 0.
3. Equal to mind (x + a) (x + b) (x + c) (x + d) = Ax 2 de ad = cb.
The method of topping is polygay in the partial splitting of the arches, splitting both parts on x 2 and the topping of the square rows.
butt.
(x + 12) (x + 2) (x + 3) (x + 8) = 4x2.
Solution.
Multiplying in the left part of the first two and the remaining two arches, we subtract:
(x 2 + 14x + 24) (x 2 + 11x + 24) = 4x2. Divisible by x 2 ≠ 0.
(x + 14 + 24/x) (x + 11 + 24/x) = 4. Replacing (x + 24/x) = t comes to square alignment:
(t + 14) (t + 11) = 4;
t 2 + 25x + 150 = 0.
t = 10 chi t = 15.
Zrobivshi zavorotnu zaminu x + 24/x = 10 or x + 24/x = 15, obviously the root.
Vidpovid: (-15±129)/2; -4; -6.
4.
Razvyazati equalization (3x + 5) 4 + (x + 6) 3 = 4x2 + 1.
Solution.
It is very important to classify and choose the method of rozvyazannya. Let's make a handful of this, vikoristuuchi the difference of squares and the difference of cubes:
((3x + 5) 2 - 4x 2) + ((x + 6) 3 - 1) \u003d 0. Then, after the fault of the global multiplier, we will come to a simple equal:
(X + 5) (X 2 + 18X + 48) = 0.
Vidpovid: -5; -9±33.
Manager.
Fold the polynomial of the third degree, which may have one root, equal 4, may be a multiplicity of 2 and root, equal -2.
Solution.
f(x)/((x - 4) 2 (x + 2)) = q(x) or f(x) = (x - 4) 2 (x + 2)q(x).
Multiplying the first two arches and adding similar additions, we take away: f (x) \u003d (x 3 - 6x 2 + 32) q (x).
x 3 - 6x 2 + 32 - rich term of the third stage, also, q (x) - deike number s R(Tobto diysne). Let q(x) be one, then f(x) = x 3 - 6x 2 + 32.
Suggestion: f(x) = x 3 - 6x 2 + 32.
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Looking at the supplementary literature about the perfection of equal systems, I studied a new type of systems - symmetrical. I put myself for the meta:
Uzagalniti naukovі v_domosti on the topic "Systems of Rivnyan".
Rozіbratisya and learn how to virіshuvati way of zaprovadzhennya new zminnyh;
3) Look at the main theories related to symmetrical systems of equalities
4) Learn to rozv'yazuvati symmetrical systems equal.
The history of the development of the river systems.
It has been a long time since the exclusion of non-domicants from the linear rivnas has been stagnant. At 17-18 Art. in. Fermat, Newton, Leibniz, Euler, Bezout, Lagrange took turns switching off.
In the current record, the system of two linear lines from two is invisible: a1x + b1y \u003d c1, a2x + b2x \u003d c2 x \u003d c1b1 - c2b; y \u003d a1c2 - a2c1 The solutions of the system are expressed by formulas.
a1b2 – a2b1 a1b2 – a2b1
Zavdyaki method of coordinates, created in the 17th century. Farm and Descartes, it became possible to visualize the system of equalities graphically.
In ancient Babylonian texts, written in 3-2 thousand BC. i.e., to take revenge for a few days, as if they are violating for the help of the folding systems of the equal, in the yak it is to introduce the equal of another level.
Stock #1:
The area of two of its squares I clave: 25. the side of the other square is the same as the side of the first one 5. The other system is equal to the different record may look like: x2 + y2 = 25, y = x = 5
Diophantus, who did not designate for the wealthy of the unknown, reported chimalo zusil for the choice of the unknown in such a rank, to lead the decisions of the system to the completion of one equal.
Stock #2:
"Know two natural numbers, knowing that their sum is 20, and the sum of their squares is 208.
The supervisors also checked the folding of the equal system, x + y = 20, and ale virishuvav x2 + y2 = 208
Diophantus, choosing like an unknown half of the roaring numbers, tobto.
(x - y) = z, + (x + y) = 10
2z2 + 200 \u003d 208 z \u003d + 2z \u003d -2- does not satisfy the mind tasks, so z \u003d 2x \u003d 12, and y \u003d 8
Understanding the system of equalization of algebra.
It is necessary to know a lot of unknown quantities, knowing that others, adopted from an additional value (functions of unknown ones) are equal to one and the same as given values. Let's look at the simplest example.
Rectangular plot of land with an area of 2400 m2 is fenced with a 200m parking lot. know the length and width of the plot. In fact, the “algebraic model” of this task is a system of two equals and one unevenness.
Possibility of obmezhennya-irrevnosti is necessary for the mother to take care of. If you fail, the task of folding the systems is equal. But all the same, it’s a smut - virishity yourself is equal. About the method, how to get stuck, I'll tell you.
Let's get back to you.
The system of equals is called a set of equals (more than one) equals, with a figured bow.
The figurative arch means that all the alignment of the system is due to coincide with one hour, and it shows that it is necessary to know such a pair of numbers (x; y), so that I transform the leather alignment into the correct alignment.
The solutions of the system name such a pair of numbers х і y, yakі when substantiating the system in qi, they turn the skin її rivnyan y to the correct numerical equivalence.
Break the system of equals - tse means, know all and її rozv'yazannya chi to install, what її ї nemє.
Substitution method.
The substitution method is based on the fact that in one of the equals one change is expressed through the other. Otrimane viraz to put on another equal, as if after that one turns to equal with one snake, and then we vilify yoga. The meaning of the value of the change, which is to appear, is presented in whether it is equal to the outer system and to know the other change.
Algorithm
1. Virazity through x s one alignment of the system.
2. Submit the elimination of viraz zamіst in another equal system.
3. Virishiti otrimane equal shdo x.
4. Submit on the basis of the skin from the knowledge on the third croc of the roots equal to x in viraz y through x, deletions on the first croc.
5) Write down the value of the pairs (x; y).
Butt No. 1 y \u003d x - 1,
Let's imagine before another equalization y \u003d x - 1, take 5x + 2 (x - 1) \u003d 16, stars x \u003d 2. Imagine subtracting viraz y before equaling: y \u003d 2 - 1 \u003d 1.
Suggestion: (2; 1).
Stock #2:
8y - x \u003d 4, 1) 2 (8y - 4) - 21y \u003d 2
2x - 21y \u003d 2 16y - 8 - 21y \u003d 2
5y \u003d 10 x \u003d 8y - 4, y \u003d -2
2x - 21y = 2
2) x \u003d 8 * (-2) - 4 x \u003d 8y - 4, x \u003d -20
2 (8y - 4) - 21y \u003d 2 x \u003d 8y - 4, y \u003d -2 x \u003d -20, y \u003d -2
Response: (-20; -2).
Stock #3: x2 + y +8 = xy, 1) x2 + 2x + 8 = x * 2x y - 2x = 0 x2 + 2x + 8 = 2x2
x2 + 2x + 8 = 0 x2 + y + 8 = xy, x2 - 2x - 8 = 0 (-2) y = 2x y1 = -4 y2 = 2 * 4 x1 = -2 y2 = 8 x2 = 4 y = 2x x1 = -2, x2 = 4 y1 = -4, y2 = 8
Old (-2; -4); (4; 8) - solutions to the problem of the system.
Adding method.
The method of folding is based on the fact that as the system is given, it is folded from equals, if, with the addition of a new one, it is equal to one change, then violating the purpose of the equal, we take away the significance of one of the changes. The meaning of another change is known, as well as ways of substitution.
Algorithm for the decoupling of systems by the method of adding.
1. Check the modules of coefficients for one of the unknown.
2. Putting together or looking at otrimani equal, to know one is not at home.
3. Presenting the known values of one of the equals of the external system, it is unknown to know the other.
Example number 1. Untie the equalization system by adding: x + y \u003d 20, x - y \u003d 10
Vіdnіmemo from the first equal friend, otrimaemo
Virazimo from another virazu x = 20 - y
Imagine y = 5 in cei viraz: x \u003d 20 - 5 x \u003d 15.
Vidpovid: (15; 5).
Stock #2:
It is imagined that in the eyes of retailers the proponation system is equal, it is taken
7y = 21, stars y = 3
Assume that the value of the expression from another system equals x = subtract x = 4.
Suggestion: (4; 3).
Stock #3:
2x + 11y = 15,
10x - 11y = 9
Squeeze the data equal, maybe:
2x + 10x = 15 + 9
12x = 24x = 2
10 * 2 - 11y \u003d 9, stars y \u003d 1.
Solutions for the system є pair: (2; 1).
Graphical way of rozv'yazannya systems rivnyannya.
Algorithm
1. Encourage the schedule of the skin care system.
2. Finding the coordinates of the crossing points of the inciting lines.
vipadok mutual distribution straight on the flat.
1. As if they are straight ahead, so that they make one point, then the system of equals can be one solution.
2. As straight lines are parallel, so that do not mix dots, then the system of equalities cannot be broken.
3. As straight as possible, to make an impersonal point, then the system equals an impersonal decision.
Stock #1:
Expand graphically the system of equals x - y \u003d -1,
Virazimo from the first and the other equals y: y \u003d 1 + x, y \u003d 4 - 2x
Let's take a look at the schedules of the dermal system:
1) y \u003d 1 + x - graph of the function є straight line x 0 1 (1; 2) y 1 2
2) y \u003d 4 - 2x - graph of the function є straight line x 0 1 y 4 2
Suggestion: (1; 2).
Butt number 2: y x + 2y \u003d 6,
4y \u003d 8 - 2x x y \u003d, y \u003d y \u003d - graph of the function є straight x 0 2 y 3 2 y \u003d - graph of the function є straight x 0 2 y 2 1
Suggestion: there is no solution.
Butt number 3: y x - 2y \u003d 2,
3x - 6y \u003d 6 x - 2y \u003d 2, x - 2y \u003d 2 x y \u003d - Graph of the function є straight x 0 2 y -1 0
Vidpovid: the system is impersonal solution.
The method of requesting new changes.
The method of introducing new changes is based on the fact that a new change is introduced only in one line or two new changes for both lines, another line or a change in the number of new changes, in case of which you need to leave more simple system rіvnyan, z yakikh know shukane solution.
Stock #1:
x + y = 5
Significantly = z, then =.
The first time I look at the future z + =, it is even stronger 6z - 13 + 6 = 0. The difference is equal, what happened, maybe z = ; z=. Todi = or = , then first the equalization split into two equal, then, perhaps, two systems:
x + y = 5 x + y = 5
Versions of these systems are the solutions of these systems.
The decisions of the first system are a pair: (2; 3), and the other is a bet (3; 2).
Also, by the solutions of the system + = , x + y = 5
Є bet (2; 3); (3; 2)
Stock #2:
Come on = X, a = Y.
X \u003d, 5 * - 2Y \u003d 1
5X - 2Y \u003d 1 2.5 (8 - 3Y) - 2Y \u003d 1
20 - 7.5 - 2U \u003d 1
X \u003d, -9.5Y \u003d -19
5 * - 2Y = 1 Y = 2
Let's do a reverse change.
2 x = 1, y = 0.5
Vidpovid: (1; 0.5).
Symmetric systems of alignment.
The system s n is invariably called symmetrical, because it does not change at the hour of permutation of the unknown.
The system of two equals from two is symmetrical, x and y are not equal to the substitution u = x + y, v = xy. Respectfully, what virazi, what zustrіchayutsya in symmetrical systems virazhayutsya through u and v. We will introduce a number of such applications, which will become of great interest for the development of rich symmetrical systems: x2 + y2 \u003d (x + y) 2 - 2xy \u003d u2 - 2v, x3 + y3 \u003d (x + y) (x2 - xy + y2) = u (u2 - 2v - v) = u3 - 3uv, x4 + y4 = (x2 + y2)2 - 2x2y2 = (u2 - 2v)2 - 2v2 = u4 - 4u2v + 2v2, x2 + xy + y2 = u2 - 2v + v = u2 - v i etc.
The system of triox equals is symmetrical, but invariably x y z is violated by the substitution x + y + z = u, xy + yz + xz = w. If u, v, w are found, the cubic equalization t2 - ut2 + vt - w = 0 is added, the root of this t1, t2, t3 in different permutations is the solutions of the external system. The most common virazi in such systems are virazhayutsya through u, v, w offensive rank: x2 + y2 + z2 \u003d u2 - 2v x3 + y3 + z3 \u003d u3 - 3uv + 3w
Stock #1: x2 + xy + y2 = 13, x + y = 4
Say x + y = u, xy = v.
u2 – v = 13, u = 4
16 - v = 13, u = 4 v = 3, u = 4
Let's do a reverse change.
Suggestion: (1; 3); (3; 1).
Stock #2: x3 + y3 = 28, x + y = 4
Say x + y = u, xy = v.
u3 - 3uv = 28, u = 4
64 - 12 v = 28, u = 4
12v = -36 u = 4 v = 3, u = 4
Let's do a reverse change.
x + y = 4, xy = 3 x = 4 - y xy = 3 x = 4 - y,
(4 – y) y = 3 x = 4 – y, y1 = 3; y2 = 1 x1 = 1, x2 = 3, y1 = 3, y2 = 1
Suggestion: (1; 3); (3; 1).
Stock #3: x + y + xy = 7, x2 + y2 + xy = 13
Let x=y=u, xy=v.
u + v = 7, u2 - v = 13 u2 - v = 13 u2 - 7 + u = 13 u2 + u = 20 v = 7 - u, u (u + 1) = 20 u2 - v = 13 u = 4 v = 7 - u, u = 4 v = 3, u = 4
Let's do a reverse change.
x + y = 4, xy = 3 x = 4 - y xy = 3 x = 4 - y,
(4 – y) y = 3 x = 4 – y, y1 = 3; y2 = 1 x1 = 1, x2 = 3, y1 = 3, y2 = 1
Suggestion: (1; 3); (3; 1).
Stock #4: x + y = 5, x3 + y3 = 65
Say x + y = u, xy = v.
u = 5, u3 - 3uv = 65 u3 - 3uv = 65 125 - 15v = 65
15v = -60 u = 5, v = 4 v = 4
Let's do a reverse change.
x + y = 5, xy = 4 x = 5 - y, xy = 4 x = 5 - y, y (5 - y) = 4 x = 5 - y y1 = 1, y2 = 4 x1 = 4, x2 = 1, y1 = 1, y2 = 4
Suggestion: (4; 1); (fourteen).
Stock #5: x2 + xy + y2 = 49, x + y + xy = 23
I’ll replace the unknown, the system will wake up looking u2 + v = 49, u + v = 23
Having added qi equal, we take u2 + u - 72 = 0 from the roots u1 = 8, u2 = -9. Vidpovidno v1 = 15, v2 = 32. The virishity of the systems is no longer sufficient x + y = 8, x + y = -9, xy = 15 xy = 32
System x + y \u003d 8 maє solution x1 \u003d 3, y1 \u003d 5; x2 = 5, y2 = 3.
System x + y \u003d -9, there is no real solution.
Suggestion: (3; 5), (5; 3).
Example number 6. Unleash the rivnyan system.
2x2 - 3xy + 2y2 = 16, x + xy + y + 3 = 0
Vykoristovuyuchi main symmetrical rich segments u = y + x і v = xy, we will take the equalization system
2u2 - 7v = 16, u + v = -3
Substituting another equalization of the viraz system v = -3 - u in the first equal, we take the advance equal 2u2 + 7u + 5 = 0, the roots of which є u1 = -1 and u2 = -2.5; appropriate value v1 = -2 і v2 = -0.5 go out v = -3 – u.
Now there is no more virishity to come the aggregation of systems x + y = -1, і x + y = -2.5, xy = -2 xy = -0.5
The solution to the combination of systems, and therefore the totality of systems (due to their equivalence), are: (1; -2), (-2; 1), (;).
Stock #7:
3x2y - 2xy + 3x2 = 78,
2x - 3xy + 2y + 8 = 0
For the help of the main symmetrical rich sections, the system can be written in the offensive
3uv - 2v = 78,
Inverting from another equalization u = і substituting yogo in the first equal, we take 9v2 - 28v - 156 = 0. The root of the equal equal v1 = 6 and v2 = - allow you to know the basic values u1 = 5, u2 = - due to u =.
Virishimo now comes the aggregation of systems x + y = 5, i x + y = -, xy = 6 xy = -.
x \u003d 5 - y, i y \u003d -x -, xy \u003d 6 xy \u003d -.
x \u003d 5 - y, i y \u003d -x -, y (5 - y) \u003d 6 x (-x -) \u003d -.
x = 5 - y, i y = -x - , y1 = 3, y2 = 2 x1 = , x2 = - x1 = 2, x2 = 3, i x1 = , x2 = - y1 = 3, y2 = 2 y1 = -, y2 =
Suggestion: (2; 3), (3; 2), (; -), (-;).
Visnovok.
At the process of writing the article, I got to know different types systems of equal algebra. Uzagalnila scientific vіdomosti on the topic "Systems of Rivnyan".
Rose, she learned how to virishuvate the way of arranging new chicks;
Looked at the main theories related to symmetrical systems of equalities
Learned how to develop symmetrical systems of alignment.
Entry
Symmetry ... є ієyu іdeєyu, for the help of such a person, for a long time, he tried to touch and create order, beauty and perfection.
Understanding the symmetry to go through the crisis in the entire history of mankind. Vono zustrіchaetsya vzhe dzherel dzherel human knowledge. Vyniklo out of the stars of a living organism, the people themselves, and was victorious by sculptors from the 5th century to the stars. e.
The word "symmetry" is Greek. Vono means “proportionality”, “proportionality”, however, the roztashuvannya of parts. Yogo widely vikoristovuyut usі without blame directly modern science.
A lot of great people thought about this law. For example, L.N. Tolstoy, saying: “Standing in front of a black doshka and little at a new creed of different positions, I raptly spoke with a thought: why did the symmetry understand the eye? What is symmetry? Tse vrodzhene pochutya. What is it based on?
True, symmetry is a reception for the eye. Who adores the symmetry of the creations of nature: leaves, flowers, birds, creatures; but by the creations of people: life, technology, we will let us know that we are away from children, Tim, how beautiful that harmony is.
Symmetry (Inn.-Greek συμμετρία - “proportionality”), for a wide sense - immutability in case of any transformations. So, for example, the spherical symmetry of the body means that the appearance of the body does not change, as if wrapping it around space good kuti(reserving one point of the house). Bilateral symmetry means that the right and left sides should look the same on the same plane.
With the symmetry of mi zustrіchaєmosya skrіz - in nature, technіtsi, mystekstvі, science. Significantly, for example, the symmetry, the dominating blizzard and the maple leaf, the symmetry of the car and the aircraft, the symmetry of the rhythmic verse and the musical phrase, the symmetry of ornaments and borders, the symmetry of the atomic structure of molecules and crystals. The understanding of symmetry is to pass through all the rich history of human creativity. Vono zustrіchaєtsya vzhe dzherel dzherel human knowledge; yogo widely vikoristovuyut usі without blame directly modern science. The principles of symmetry play an important role in physics and mathematics, chemistry and biology, technology and architecture, painting and sculpture, poetry and music. Law of nature, that they guard the inexhaustible picture of phenomena in their own way, in their own hands, adhere to the principles of symmetry.
Qile:
Look at the type of symmetry;
Analyze how and de victorious symmetry;
Take a look, how symmetry wins in school course algebra
symmetry.
The word "symmetry" can be ambiguous. In one sense, symmetrical means even more proportional, balanced; Symmetry shows that way of using rich elements, with the help of which they unite in a whole. Another meaningful word is jealous. Even Aristotle, speaking about symmetry, is such a camp, which is characterized by extreme extremes. Why is it that Aristotle, perhaps, is closest to discovering one of the fundamental laws of Nature - laws about її duality.
The following aspects are visible, without any symmetry it is impossible:
1) object - nose symmetry; in the role of symmetrical objects, speeches, processes, geometric positions, mathematical speech, living organisms can be spoken.
2) signs of deeds - magnitude, power, visibility, processes, manifestations - objects, which, when the symmetry is transformed, become immutable; They are called invariant chi invariants.
3) change (of the object), as if the object is overwritten by the same self for invariant signs; such changes are called transformations of symmetry;
4) the power of the object is transformed by seeing the signs on itself after the changes.
In this rank, the symmetry shows the savings for some change, or the savings for no change. Symmetry conveys the immutability of the object itself, and if there are any other powers, they are completely transformed, vikoanimous over the object. The immutability of other objects can be guarded by changing before various operations - before turning, transferring, mutually replacing parts, refinishing, etc. You can see the link with zim different types symmetry.
Asymmetry
Asymmetry - lack of symmetry or broken symmetry.
In architecture, symmetry and asymmetry are two protracted methods and regular organization of spacious forms. Asymmetric compositions in the process of development of architecture have become like an incorporation of foldable life processes and the minds of an indispensable middle.
Disymetry
Destroyed, often tormented by the symmetry of my name dissymmetry
.
Disymmetry is a phenomenon, wider in living nature. Vaughn and the people. The person is dissymmetrical, unimportant to those whose contours of the body can create a plane of symmetry. Dissymmetry is indicated on
to the shortest one of the hands, to the asymmetrical position of the heart and richness of other organs, to the other organs.
The disymmetries of the human body are similar to the heights of the exact symmetry in architecture. Sound the stench of a practical need, tim, that different functions are not included in the inter-zhorst laws of symmetry. Sometimes such inspiration gives the basis for a sharp emotional effect.
^ Types of symmetry that are used in mathematics in the natural sciences:
Bilateral symmetry- symmetry of the mirror-image, when such an object has one plane of symmetry, while two halves of it are mirror-symmetrical. In creatures, bilateral symmetry is manifested in the similarity or may be the sameness of the left and right halves of the body. When tsomu zavzhdi іsnuyut vipadkovі vіdhilennya od simetrії (napriklad, vіdmіnnostі in papіlyarnih lіnіyah, rozgaluzhennі sudin. Іsnuyut nevelikі Often, ale zakonomіrnі vіdmіnnostі in zovnіshnіy budovі that bіlsh suttєvі vіdmіnnostі mіzh his right that half lіvoyu tіla in roztashuvannі internal organs. For example, the hearts of the savants sound asymmetrically placed, with the shifted to the left.
In creatures of the appearance of bilateral symmetry in evolution, it is tied to the substrate (along the bottom of the water), the connection between which is the spinal cord, and also the right side of the left half of the body. In the middle of the creatures, bilateral symmetry is more pronounced in actively limp forms, lower in sessile ones. In roslin, the bilateral symmetry may not sound the whole organism, but only parts of it - leafing or leafing. Bilaterally symmetrical flowers of botany are called zygomorphic.
^ Symmetry of the nth order- symmetricality of turns around 360 ° / n along any axis. Described by a Zn edge.
Axial symmetry(Radial symmetry, change of symmetry) - a form of symmetry, with a body (or a figure) escaping itself with an object wrapping around a single point or a straight line. Often this point runs around the center of symmetry of the object, so that point
there is an infinite number of axes of bilateral symmetry. Radial symmetry can be made by such geometric objects, like a colo, a sump, a cylinder or a cone. Described by the SO(2) group.
^ Spherical symmetry- symmetry to wrap around a trivi-worldly expanse on a pretty kut. Described by the SO(3) group. Local spherical symmetry to the space of the middle is also called isotropy.
^ Obertal symmetry- a term that means the symmetry of the object to all or any of the powerful wrappings of the m-peaceful Euclidean space.
^ Symmetry in creatures and people.
Symmetry is an important sign of life, which reflects the peculiarities of life, the way of life and the behavior of the creature. The symmetry of the form is needed for ribs, for weaving; birds fly. Otzhe, symmetry in nature is not without reason: outwardly corisna, but, otherwise, apparently, dotsilna. Biology has a center of symmetry: flowers, jellyfish, sea stars are thin. e. The manifestation of symmetry forms is already evident in the simplest - single-cliten (infusoria, amoeba). The human body is inspired by the principle of bilateral symmetry. Brain subdivided into two halves. In the modern appearance to the global symmetry of the human body, the skin pivkulya may be more accurate mirror images of the other. The control of the main body movements of the human body and її sensory functions is equally divided between the two brain vesicles. Lіva pіvkulya kontrolє to the right side of the brain, and to the right - to the left side. Conducted investigations showed that the person is symmetrical and more accommodating. So doslidniki affirm that a person with ideal proportions is a sign that the body of the yogi Volodar is well prepared to fight infections. Zvichayna cold, asthma and flu with a high level of immunity come before people whose left side is exactly similar to the right side. And in clothes, a person, as a rule, tries to improve the symmetry of the image: the right sleeve is in line with the left, the right leg is left. The cuddles on the kurtz and on the shirt sit evenly in the middle, and if they step into it, then on the symmetrical stand. At one time, people are encouraged to reinforce, to strengthen the difference between living and ruling. In the middle ages, people at their time wore pantaloons with trousers of different colors (for example, one red, and another black or white). ale
such fashion is forever unpredictable. Less tact, modest breaths in symmetry are left for long hours.
Symmetry in the art
Symmetry in the arts vzagali and in the image-creating zokrema takes its cob from the real activity, like a symmetrically powerful forms.
The symmetric organization of the composition is characterized by the vrіvnovazhenіst її parts for masses, for tone, color and shape. At times, one part of the mayzhe is mirror-like to the other. In symmetrical compositions, the center is most clearly pronounced. As a rule, the veins zbіgaєtsya from the geometric center of the picture area. Like a point immediately shifted to the center, one of the parts is more fascinated by masses, otherwise the image will be diagonal, all the same proving the dynamism of the composition and destroying the ideal equivalence with a singing world.
The rule of symmetry was also used by sculptors Ancient Greece. The butt can be the composition of the front pediment of the temple of Zeus and Olympia. In її the basis was laid for the struggle of the Lapiths (Greeks) with the centaurs in the presence of the god Apollo. Move step by step from the edges to the center. It is within reach of the boundary virility in the images of two young men, as if they swung at the centaurs. The growing ruh is shaved on the steps to the figure of Apollo, which is calmly and majestically standing at the center of the pediment.
Statement about the use of the works of famous artists of the 5th century BC. e. can be folded according to the ancient vase painting and Pompeian frescoes, inspired by the works of the Greek masters of the era of the classics.
The symmetrical compositions were afraid that the Greek maestros IV-III could reach the stars. e. You can judge from copies of frescoes. In the Pompeian frescoes, the head figures perebuvayut at the center of the pyramidal composition, which looks like symmetry.
Before the rules of symmetry, artists often went into the hour of depicting the tracts of rich people gatherings, parades, meetings at the great halls.
Great respect for the rule of symmetry was attached by the artists of the early Renaissance, which was celebrated by monumental painting (for example, frescoes by Giotto). During the period of the High Renaissance, the Italian composition reached maturity. For example, in the painting “Holy Hannah with Mary and the Unloved Christ” by Leonardo da Vinci, he composes three posts on the burn-out trikutnik. At the lower right fold of the veins, I give a figurine of a lamb, which the little Christ holds. Everything is arranged in such a way that the tricoutnik is less likely to be guessed under the volumetric-spacious group of figures.
A symmetrical composition can also be called "The Last Supper" by Leonardo da Vinci. This fresco shows a dramatic moment, if
Christ having recited his teachings: "One of you will heal me." The psychological reaction of the apostles to the words of the prophets evoke the characters from the compositional center, in whom they try to place Christ. Emotional solidity in the center of the pre-center composition is illustrated by the fact that the artist, having shown the refectory in perspective, with the point of convergence of parallel lines at the middle of the window, the head of Christ is clearly depicted on the aphids. In this rank, the look of the peeping glance is miraculously straight to the central figure of the picture.
Among the works that demonstrate the possibility of symmetry, one can also call Raphael's “Bailing Mary”, de nayshli the most pronounced expression of the composition, characteristic of the Age of Renaissance.
The painting by V. M. Vasnetsov "The Bogatyrs" was also inspired on the basis of the rule of symmetry. The center of the composition is the figure of Illy Muromets. Livoruch and right-handed, like a mirror image, placed Alosha Popovich and Dobrinya Mykitovich. The figures are ruffled in the air of the picture area, so it’s easy to sit on horseback. The symmetrical composition of Budova conveys a breathtaking calmness. The left and right of the figures behind the masses are not the same, according to the author's idea. Ale, insulting the stench, it’s less hard to match with the figure of Muromets and swear to put on a new equal composition.
The stamina of the composition reminds the starer of the impermeability of the rich, the defenders of the Russian land. Moreover, at the “Bogatiry” the camp of strained calm was transferred on the border of the transition to the day. And tse means that symmetry carries in itself the germ of a dynamic movement in the hour of that space.
Symmetry in algebra.
The simplest symmetric lines are similar to the roots of the square equalization, which are the same as the Viet theorem. Tse allow you to beat them at the ceremony of deyah zavdans, which lie up to square equals. Let's take a look at the low applications.
Example 1:
Square alignment 
maє the root is that. Not viruchuyuchi tsgogo rivnyannya, virazimo i sumi,. Viraz symmetrical shodo i. We can prove їх through + і , and then we will prove Vietta's theorem.
Rational equivalence and nervousness
I. Rational equivalence.
Linear alignment.
Systems of linear lines.
Rotary alignment.
Vieta's formula for the rich segments of the higher steps.
Systems equal to another level.
The method of introduction of new unknowns in the case of violations of rіvnyan and systems of rіvnyan.
Uniform rivnyannia.
Razvyazannya symmetrical systems rivnyan.
Equalization of that system is equalized by parameters.
Graphical method of rozvyazannya systems of non-linear alignments.
Rivnyannya to replace the sign of the module.
Basic methods of developing rational equalities
II. Rational inconsistencies.
The dominance of equally strong irritability.
Algebraic irregularities.
Interval method.
Fractional-rational unevenness.
Ner_vnostі, scho avenge nevіdome under the sign of the absolute value.
Irregularities in parameters.
Systems of rational inequalities.
Graphic rozv'yazannya nerіvіvnosti.
III. Perevirochny test.
Rational equivalence
mind function
P(x) \u003d a 0 x n + a 1 x n - 1 + a 2 x n - 2 + ... + a n - 1 x + a n,
de n - natural, a 0, a 1, ..., a n - deakі dіysnі numbers is called a whole rational function.
Equal to the form P(x) = 0, where P(x) is the number of a rational function, is called a whole rational equal.
Equal to mind
P 1 (x) / Q 1 (x) + P 2 (x) / Q 2 (x) + ... + P m (x) / Q m (x) = 0,
de P 1 (x), P 2 (x), ..., P m (x), Q 1 (x), Q 2 (x), ..., Q m (x) - the number of rational functions, is called rational equal.
The decision of the rational parity P(x) / Q(x) = 0, where P(x) and Q(x) are multiterms (Q(x) 0), is reduced to the perfection of the parity P(x) = 0 and reversing that what root pleases the mind Q (x) 0.
Linear alignment.
Equal to the mind ax+b=0 de а і b - deyakі fastіynі, called linear rivnyann.
Yakshcho a0, linear equal may have a single root: x \u003d -b / a.
How a = 0; b0, linear solution cannot be solved.
How a = 0; b=0, then, having rewritten the visual alignment as ax = -b, it is easy to figure out if x is the solution of the linear alignment.
Alignment of straight lines can be seen: y \u003d ax + b.
If a straight line passes through a point with coordinates X 0 and Y 0, then the coordinates satisfy the alignment of the straight line, then Y 0 = aX 0 + b.
Butt 1.1. Rozvyazati rivnyannia
2x - 3 + 4 (x - 1) = 5.
Solution. Sequentially opening the arches, we introduce such terms and know x: 2x - 3 + 4x - 4 \u003d 5, 2x + 4x \u003d 5 + 4 + 3,
butt 1.2. Rozvyazati rivnyannia
2x - 3 + 2 (x - 1) = 4 (x - 1) - 7.
Solution. 2x + 2x - 4x = 3 +2 - 4 - 7, 0x = - 6.
Suggestion: .
Butt 1.3. Rozvyazati rivnyannya.
2x + 3 - 6 (x - 1) = 4 (x - 1) + 5.
Solution. 2x - 6x + 3 + 6 = 4 - 4x + 5,
- 4x + 9 = 9 - 4x,
4x + 4x = 9 - 9,
Vidpovid: Be it a number.
Systems of linear lines.
Equal to mind
a 1 x 1 + a 2 x 2 + … + a n x n = b,
de a 1, b 1, ..., a n, b - deacers are constant, are called linear equals with n nevidomim x 1, x 2, ..., x n.
The equalization system is called linear, because all equals that enter the system are linear. As a system of s n nevіdomih, then there are three possible ways:
the system has no solution;
the system can be exactly one solution;
the system may be an impersonal solution.
Example 2.4. rozvyazat system rivnyan
Solution. You can change the system of linear alignments by the method of substantiation, which means that whether the system is equal to one another, through another unknown, and then put the meaning of the unknown into another alignment.
From the first equalization, it is shown: x = (8 - 3y) / 2.
X \u003d (8 - 3y) / 2, 3 (8 - 3y) / 2 + 2y \u003d 7. For the other equal, we can take y \u003d 2. For the improvement of the first equal, x \u003d 1. Indication: (1; 2). 2.5. Unleash the rivnyan system
Solution. There is no solution for the system, but the two equal parts of the system are impossible to be satisfied at the same time (from the first equal x + y = 3, and from the second equal x + y = 3.5).
Suggestion: There is no solution.
Example 2.6. rozvyazat system rivnyan
Solution. The system can be an impersonal decision, but other shards are equal to go out of the first way of multiplication by 2 (that is, in fact, only one equal to two nevidomimi).
Suggestion: Without a doubt, a rich decision.
Example 2.7. rozvyazat system rivnyan
x + y - z = 2,
2x - y + 4z = 1,
Solution. When modifying systems of linear lines, it is easy to coristuate using the Gaus method, which is similar to the converted system to a tricot look.
Multiplying the first equalization of the system by – 2 і adding subtracting the result from other equals, we subtract – 3y + 6z = – 3. The equalization can be rewritten at the sight of y – 2z = 1. Adding the first equalization to the third, we subtract 7y = 7, otherwise y = one.
In this rank, the system of the nabul of the tricot look
x + y - z = 2,
Substituting y = 1 for the other equal, we know z = 0. Substituting y = 1 and z = 0 for the previous equal, we know x = 1. Suggestion: (1; 1; 0). Example 2.8. for any values of the parameter a, the system equals
2x + ay = a + 2,
(a + 1) x + 2ay = 2a + 4
can you make a big decision? Solution. From the first level, it can be seen x:
x = - (a / 2) y + a / 2 +1.
Submitting this virase to a friend is equal, otrimuemo
(a + 1) (- (a / 2) y + a / 2 + 1) + 2ay = 2a + 4.
(a + 1)(a + 2 – ay) + 4ay = 4a + 8,
4ay – a(a + 1)y = 4(a + 2) – (a + 1)(a + 2),
ya(4 – a – 1) = (a + 2)(4 – a – 1),
ya(3 – a) = (a + 2) (3 – a).
Analyzing the rest of the equation, it is significant that for a = 3 it may look like 0y = 0, then. one is satisfied with whatever the value of y is. Suggestion: 3.
Kvadratnі rіvnyannya that rіvnyannya, scho zvoditsya before them.
Equal to the form ax 2 + bx + c = 0, de a, b and c - decimal numbers (a0);
x - change, called square equals.
The formula for splitting a square alignment.
On the back of the hand, we divide the offending parts of the equal ax 2 + bx + c = 0 into a - the second root does not change. For the sake of seeing, what happened
x 2 + (b/a) x + (c/a) = 0
visible at the left part of the povny square
x 2 + (b / a) + (c / a) = (x 2 + 2(b / 2a)x + (b / 2a) 2) - (b / 2a) 2 + (c / a) =
= (x + (b / 2a)) 2 - (b 2) / (4a 2) + (c / a) = (x + (b / 2a)) 2 - ((b 2 - 4ac) / (4a 2 )).
Viraz (b 2 - 4ac) through D is significant for style.
There are three possibilities:
if the number D is positive (D > 0), then which way can be overcome from D square root and write D at a glance D = (D) 2 . Todi
D / (4a 2) = (D) 2 / (2a) 2 = (D / 2a) 2
x 2 + (b / a) x + (c / a) = (x + (b / 2a)) 2 - (D / 2a) 2 .
Following the formula for the difference of squares, the following numbers are displayed:
x 2 + (b / a) x + (c / a) = (x + (b / 2a) – (D / 2a))(x + (b / 2a) + (D / 2a)) =
= (x - ((-b + D) / 2a)) (x - ((- b - D) / 2a)).
Theorem: How to win the sameness
ax 2 + bx + c \u003d a (x - x 1) (x - x 2),
then square alignment ax 2 + bx + c \u003d 0 for X 1 X 2 there are two roots X 1 and X 2, and for X 1 \u003d X 2 - only one root X 1.
By virtue of the theorem of the above, the sameness follows that the equality
x 2 + (b/a) x + (c/a) = 0,
and by the same token i equal ax 2 + bx + c = 0, there are two roots:
X 1 \u003d (-b + D) / 2a; X 2 \u003d (-b - D) / 2a.
Also x 2 + (b / a)x + (c / a) \u003d (x - x1) (x - x2).
Sound the root and write it down with one formula:
de b 2 - 4ac = D.
if the number D is equal to zero (D = 0), then the identity
x 2 + (b / a) x + (c / a) = (x + (b / 2a)) 2 - (D / (4a 2))
looks at x 2 + (b / a) x + (c / a) = (x + (b / 2a)) 2 .
Zvіdsi viplivaє scho D \u003d 0 equal ax 2 + bx + c \u003d 0 maє root of the multiplicity 2: X 1 \u003d - b / 2a
3) How is the number D є negative (D< 0), то – D >0, and to that
x 2 + (b / a) x + (c / a) = (x + (b / 2a)) 2 - (D / (4a 2))
It is the sum of two dodankiv, one of which is non-negative, otherwise positive. Such a sum cannot be equal to zero, that is equal
x 2 + (b/a) x + (c/a) = 0
I don't have real roots. Can't match ax2+bx+c=0.
In this way, to complete the square alignment, calculate the discriminant
D \u003d b 2 - 4ac.
Even though D = 0, the square may be the same solution:
If D > 0, then there are two square roots:
X 1 \u003d (-b + D) / (2a); X 2 \u003d (-b - D) / (2a).
Yakscho D< 0, то квадратное уравнение не имеет корней.
If one of the coefficients b or c equals zero, then square equalization can be corrected without counting the discriminant:
b = 0; c 0; c/a<0; X1,2 = (-c / a)
b 0; c = 0; X1 = 0, X2 = -b/a.
The root of the square equalization of the embodied form ax 2 + bx + c \u003d 0 is behind the formula
Square alignment, for which the coefficient at x 2 is equal to 1, is called hover. Sound the squared alignment to mean like this:
x 2 + px + q = 0.
Vietta's theorem.
We have created identity
x 2 + (b / a) x + (c / a) = (x - x1) (x - x2),
de X 1 and X 2 are the roots of the square alignment ax 2 + bx + c = 0. We cut the arches at the right part of the identity.
x 2 + (b / a) x + (c / a) = x 2 - x 1 x - x 2 x + x 1 x 2 = x 2 - (x 1 + x 2) x + x 1 x 2.
Sounds like X 1 + X 2 \u003d - b / a and X 1 X 2 \u003d c / a. We brought the offensive theorem, first introduced by the French mathematician F. Viet (1540 - 1603):
Theorem 1 (Vieta). The sum of the square root equals the coefficient at X, taken with the opposite sign and divided by the coefficient at X 2; dobutok the root of the same equal to the free member, divided by the coefficient at X 2.
Theorem 2 (reverse). Yakshcho vykonuyutsya jealousy
X 1 + X 2 \u003d - b / a і X 1 X 2 \u003d c / a,
the numbers X 1 and X 2 are the roots of the square alignment ax 2 + bx + c = 0.
Respect. Formulas X 1 + X 2 \u003d - b / a that X 1 X 2 \u003d c / a are filled with variances and y in the fall, if the equality ax 2 + bx + c \u003d 0 can have one root X 1 of multiplicity 2, so put in the significant formulas X 2 = X1. Therefore, it is accepted that when D = 0, equal ax 2 + bx + c = 0 can have two falling roots one from one.
In case of virishenni zavdan, tied with the theorem of Vієta, to vindicate spіvvіdnoshennia
(1 / X 1) + (1 / X 2) = (X 1 + X 2) / X 1 X 2;
X 1 2 + X 2 2 = (X 1 + X 2) 2 - 2 X 1 X 2;
X 1 / X 2 + X 2 / X 1 = (X 1 2 + X 2 2) / X 1 X 2 = ((X 1 + X 2) 2 - 2X 1 X 2) / X 1 X 2;
X 1 3 + X 2 3 = (X 1 + X 2) (X 1 2 - X 1 X 2 + X 2 2) =
= (X 1 + X 2) ((X 1 + X 2) 2 - 3X 1 X 2).
Example 3.9. Untie equalization 2x2 + 5x - 1 = 0.
Solution. D \u003d 25 - 42 (-1) \u003d 33> 0;
X 1 \u003d (-5 + 33) / 4; X 2 \u003d (-5-33) / 4.
Vidpovid: X 1 \u003d (-5 + 33) / 4; X 2 \u003d (-5-33) / 4.
Example 3.10. Razvyazati rivnyanna x 3 – 5x2 + 6x = 0
Solution. Let's split the left part of the equation into multipliers x(x 2 - 5x + 6) = 0,
stars x \u003d 0 chi x 2 - 5x + 6 \u003d 0.
Virishyuyuchi square equal, otrimuemo X 1 \u003d 2, X 2 \u003d 3.
Response: 0; 2; 3.
Example 3.11.
x 3 - 3x + 2 = 0. Solution. Let's rewrite equal, writing -3x = - x - 2x, x 3 - x - 2x + 2 = 0, and now group x (x2 - 1) - 2 (x - 1) = 0, (x - 1) (x( x + 1) - 2) = 0, x - 1 = 0, x 1 = 1, x 2 + x - 2 = 0, x 2 = - 2, x 3 = 1. Suggestion: x 1 = x 3 = 1 , x 2 = - 2. Butt 3.12. Rozvyazati rivnyanya7
(x - 1) (x - 3) (x - 4)
(2x - 7) (x + 2) (x - 6) We know the range of admissible values x:X + 2 0; x – 6 0; 2x – 7 0 or x – 2; x 6; x 3.5. Aligned to the form (7x - 14) (x 2 - 7x + 12) \u003d (14 - 4x) (x 2 - 4x - 12), arched arches. 7x 3 - 49x 2 + 84x - 14x 2 + 98x – 168 + 4x 3 – 16x 2 – 48x – 14x 2 + 56x + 168 = 0.11x 3 – 93x 2 + 190x = 0.x(11x 2 – 93x + 190) = 0.x 1 93x + 190 = 0.93(8649 - 8360) 93 17 x 2.3 = = ,
Tobto. x 1 = 5; x2 = 38/11.
Known values satisfy the ODZ.
Response: x1=0; x 2 \u003d 5; x 3 = 38/11.
Example 3.13. Razvyazati rivnyanna x 6 - 5x 3 + 4 = 0
Solution. Significantly y = x 3
y 2 - 5y + 4 \u003d 0, having broken it down, it is possible Y 1 \u003d 1; Y2 = 4.
In this rank, vihіdne equal equivalent to marriage
equal: x 3 \u003d 1 or x 3 \u003d 4, then X 1 \u003d 1 or X 2 \u003d 3 4
Suggestion: 1; 3 4.
Example 3.14. Razvyazati equalization (x 3 - 27) / (x - 3) = 27
Solution. Let's spread the number for multipliers (according to the formula for the difference of cubes):
DopovіdScientific scientist: Kulabukhov Sergiy Yuriyovich, Candidate of Physical and Mathematical Sciences, teacher supplementary education MOU DOD DTDiM, m. Rostov-on-Don.
