Methods for solving quadratic equations. What is a puzzle toy

Appendix

The solution of any type of equations online to the site to consolidate the studied material by students and schoolchildren. Solving equations online. Equations online. There are algebraic, parametric, transcendental, functional, differential and other types of equations. Some classes of equations have analytical solutions, which are convenient in that they not only give the exact value of the root, but allow you to write the solution in the form of a formula that may include parameters. Analytic expressions allow not only to calculate the roots, but to analyze their existence and their number depending on the values ​​of the parameters, which is often even more important for practical application than specific root values. Solution of equations online. Equations online. The solution of the equation is the task of finding such values ​​of the arguments for which this equality is achieved. Additional conditions (integer, real, etc.) can be imposed on the possible values ​​of the arguments. Solution of equations online. Equations online. You can solve the equation online instantly and with high precision result. The arguments of the given functions (sometimes called "variables") in the case of an equation are called "unknowns". The values ​​of the unknowns for which this equality is achieved are called solutions or roots of the given equation. Roots are said to satisfy a given equation. Solving an equation online means finding the set of all its solutions (roots) or proving that there are no roots. Solution of equations online. Equations online. Equivalent or equivalent are called equations, the sets of roots of which coincide. Equivalent are also considered equations that do not have roots. The equivalence of equations has the property of symmetry: if one equation is equivalent to another, then the second equation is equivalent to the first. The equivalence of equations has the property of transitivity: if one equation is equivalent to another, and the second is equivalent to the third, then the first equation is equivalent to the third. The equivalence property of equations makes it possible to carry out transformations with them, on which the methods for solving them are based. Solution of equations online. Equations online. The site will allow you to solve the equation online. The equations for which analytical solutions are known include algebraic equations, not higher than the fourth degree: a linear equation, a quadratic equation, cubic equation and a fourth degree equation. Algebraic equations of higher degrees in general case do not have an analytical solution, although some of them can be reduced to the equations lower degrees. Equations that include transcendental functions are called transcendental. Among them, analytical solutions are known for some trigonometric equations, since the zeros trigonometric functions well known. In the general case, when an analytical solution cannot be found, numerical methods are used. Numerical Methods do not give an exact solution, but only allow narrowing the interval in which the root lies to a predetermined set value. Solving equations online.. Online equations.. Instead of an online equation, we will present how the same expression forms a linear dependence and not only along a straight tangent, but also at the very inflection point of the graph. This method is indispensable at all times in the study of the subject. It often happens that the solution of equations approaches the final value by means of infinite numbers and writing vectors. It is necessary to check the initial data and this is the essence of the task. Otherwise, the local condition is converted into a formula. Straight line inversion from given function, which the equation calculator will calculate without much delay in execution, the privilege of space will serve as a netting. It will be about student performance in a scientific environment. However, like all of the above, it will help us in the process of finding, and when you solve the equation completely, then save the resulting answer at the ends of the straight line segment. Lines in space intersect at a point, and this point is called intersected by lines. The interval on the line is marked as given earlier. The highest post on the study of mathematics will be published. Assigning an argument value from a parametrically defined surface and solving an equation online will be able to indicate the principles of a productive call to a function. The Möbius strip, or as it is called infinity, looks like a figure eight. This is a one-sided surface, not a two-sided one. According to the principle well-known to all, we will objectively accept linear equations for the basic designation as is and in the field of study. Only two values ​​of successively given arguments are able to reveal the direction of the vector. To assume that a different solution of the online equations is much more than just solving it means obtaining a full-fledged version of the invariant at the output. Without an integrated approach, it is difficult for students to learn this material. As before, for each special case, our convenient and smart online equation calculator will help everyone in a difficult moment, because you just need to specify the input parameters and the system will calculate the answer itself. Before we start entering data, we need an input tool, which can be done without much difficulty. The number of each response score will be a quadratic equation leading to our conclusions, but this is not so easy to do, because it is easy to prove the opposite. The theory, due to its features, is not supported practical knowledge. To see a fraction calculator at the stage of publishing an answer is not an easy task in mathematics, since the alternative of writing a number on a set increases the growth of the function. However, it would be incorrect not to say about the training of students, so we will express each as much as it is necessary to do. The previously found cubic equation will rightfully belong to the domain of definition, and contain the space numerical values, as well as symbolic variables. Having learned or memorized the theorem, our students will prove themselves only with better side and we will be happy for them. In contrast to the set of intersections of fields, our online equations are described by a plane of motion along the multiplication of two and three numerical combined lines. A set in mathematics is not uniquely defined. The best solution, according to the students, is the written expression completed to the end. As it was said scientific language, the abstraction of symbolic expressions is not included in the state of affairs, but the solution of equations gives an unambiguous result in all known cases. The duration of the teacher's session is based on the needs in this offer. The analysis showed the need for all computational techniques in many areas, and it is absolutely clear that the equation calculator is an indispensable tool in the gifted hands of a student. A loyal approach to the study of mathematics determines the importance of views of different directions. You want to designate one of the key theorems and solve the equation in such a way, depending on the answer of which there will be a further need for its application. Analytics in this area is gaining momentum. Let's start from the beginning and derive the formula. Having broken through the level of increase of the function, the tangent line at the inflection point will necessarily lead to the fact that solving the equation online will be one of the main aspects in constructing the same graph from the function argument. The amateur approach has the right to be applied if this condition does not contradict the conclusions of the students. It is the subtask that puts the analysis of mathematical conditions as linear equations in the existing domain of the object definition that is brought to the background. Offsetting in the direction of orthogonality cancels out the advantage of a lone absolute value. Modulo, solving equations online gives the same number of solutions, if you open the brackets first with a plus sign, and then with a minus sign. In this case, there are twice as many solutions, and the result will be more accurate. A stable and correct online equation calculator is a success in achieving the intended goal in the task set by the teacher. Required method it seems possible to choose due to the significant differences in the views of great scientists. The resulting quadratic equation describes the curve of the lines, the so-called parabola, and the sign will determine its convexity in square system coordinates. From the equation we obtain both the discriminant and the roots themselves according to the Vieta theorem. It is necessary to present the expression as a proper or improper fraction and use the fraction calculator at the first stage. Depending on this, a plan for our further calculations will be formed. Mathematics at theoretical approach useful at every stage. We will definitely present the result as a cubic equation, because we will hide its roots in this expression in order to simplify the task for a student at a university. Any methods are good if they are suitable for superficial analysis. Extra arithmetic operations will not lead to calculation errors. Determine the answer with a given accuracy. Using the solution of equations, let's face it - finding an independent variable from a given function is not so easy, especially during the study period parallel lines at infinity. In view of the exception, the need is very obvious. The polarity difference is unambiguous. From the experience of teaching in institutes, our teacher took main lesson, on which equations were studied online in the full mathematical sense. Here it was about higher efforts and special skills in the application of theory. In favor of our conclusions, one should not look through a prism. Until recently, it was believed that a closed set is growing rapidly over the area as it is, and the solution of equations simply needs to be investigated. At the first stage, we did not consider all possible options, but this approach is justified more than ever. Extra actions with brackets justify some advances along the ordinate and abscissa axes, which cannot be overlooked by the naked eye. There is an inflection point in the sense of a broad proportional increase of a function. Once again, we prove how necessary condition will be applied throughout the entire descending interval of one or another descending position of the vector. In a confined space, we will select a variable from the initial block of our script. The system built as a basis on three vectors is responsible for the absence of the main moment of force. However, the equation calculator deduced and helped in finding all the terms of the constructed equation, both above the surface and along parallel lines. Let's describe a circle around the starting point. Thus, we will begin to move up along the section lines, and the tangent will describe the circle along its entire length, as a result we will get a curve, which is called an involute. By the way, let's talk about this curve a little history. The fact is that historically in mathematics there was no concept of mathematics itself in the pure sense as it is today. Previously, all scientists were engaged in one common cause i.e. science. Later, several centuries later, when scientific world filled with a colossal amount of information, mankind still singled out many disciplines. They still remain unchanged. And yet, every year, scientists around the world try to prove that science is limitless, and you can't solve an equation unless you have knowledge of the natural sciences. It may not be possible to finally put an end to it. Thinking about it is as pointless as warming the air outside. Let's find the interval at which the argument, with its positive value, determines the modulus of the value in a sharply increasing direction. The reaction will help to find at least three solutions, but it will be necessary to check them. Let's start with the fact that we need to solve the equation online using the unique service of our website. Let's enter both parts of the given equation, press the "SOLVE" button and get the exact answer within just a few seconds. AT special occasions let's take a book on mathematics and double-check our answer, namely, let's look only at the answer and everything will become clear. The same project will fly out on an artificial redundant parallelepiped. There is a parallelogram with its own parallel sides, and it explains many principles and approaches to the study spatial relationship ascending process of accumulation of hollow space in natural formulas. Ambiguous linear equations show the dependence of the desired variable with our common this moment time by decision and it is necessary to somehow withdraw and bring improper fraction to a non-trivial case. We mark ten points on the straight line and draw a curve through each point in a given direction, and with a convexity upwards. Without much difficulty, our equation calculator will present an expression in such a form that its check for the validity of the rules will be obvious even at the beginning of the recording. The system of special representations of stability for mathematicians in the first place, unless otherwise provided by the formula. We will answer this with a detailed presentation of a report on the isomorphic state of a plastic system of bodies and the solution of equations online will describe the movement of each material point in this system. At the level of an in-depth study, it will be necessary to clarify in detail the question of inversions of at least the lower layer of space. In ascending order on the section of the discontinuity of the function, we will apply the general method of an excellent researcher, by the way, our fellow countryman, and we will tell below about the behavior of the plane. Due to the strong characteristics of the analytically given function, we only use the online equation calculator for its intended purpose within the derived limits of authority. Arguing further, we stop our review on the homogeneity of the equation itself, that is, its right side is equated to zero. Once again, we will verify the correctness of our decision in mathematics. To avoid getting trivial solution Let's make some adjustments to initial conditions on the problem of conditional stability of the system. Let's compose a quadratic equation, for which we write out two entries using the well-known formula and find negative roots. If one root exceeds the second and third roots by five units, then by making changes to the main argument, we thereby distort the initial conditions of the subproblem. At its core, something unusual in mathematics can always be described to the nearest hundredth of a positive number. The fraction calculator is several times superior to its counterparts on similar resources at the best moment of server load. On the surface of the velocity vector growing along the y-axis, we draw seven lines bent in opposite directions to each other. The commensurability of the assigned function argument leads the recovery balance counter. In mathematics, this phenomenon can be represented through a cubic equation with imaginary coefficients, as well as in a bipolar progress of decreasing lines. Critical points temperature difference in many of its meaning and progress describe the process of factoring a complex fractional function. If you are told to solve the equation, do not rush to do it this minute, definitely first evaluate the entire plan of action, and only then take the right approach. There will certainly be benefits. Ease in work is obvious, and in mathematics it is the same. Solve the equation online. All online equations are a certain kind an entry of numbers or parameters and a variable to be defined. Calculate this very variable, that is, find specific values ​​or intervals of a set of values ​​for which the identity will be satisfied. The initial and final conditions directly depend. AT common decision equations usually include some variables and constants, by setting which, we will get whole families of solutions for a given problem statement. In general, this justifies the efforts invested in the direction of increasing the functionality of a spatial cube with a side equal to 100 centimeters. You can apply a theorem or lemma at any stage of constructing an answer. The site gradually issues a calculator of equations, if necessary, at any interval of summation of products show smallest value. In half the cases, such a ball is hollow, not in more meets the requirements for setting an intermediate answer. At least on the y-axis in the direction of decreasing vector representation, this proportion will undoubtedly be more optimal than the previous expression. At the hour when linear functions will be a full point analysis, we will, in fact, bring together all of our complex numbers and bipolar plane spaces. By substituting a variable into the resulting expression, you will solve the equation in stages and give the most detailed answer with high accuracy. Once again, checking your actions in mathematics will be a good form on the part of a student. The proportion in the ratio of fractions fixed the integrity of the result in all important areas of activity of the zero vector. Triviality is confirmed at the end of the performed actions. With a simple task set, students cannot have difficulties if they solve the equation online in the shortest possible time periods, but do not forget about all kinds of rules. The set of subsets intersect in the area of ​​converging notation. AT different occasions the product is not erroneously factorized. You will be helped to solve the equation online in our first section on the basics of mathematical techniques for significant sections for students in universities and colleges. Answering examples will not make us wait for several days, since the process of the best interaction of vector analysis with sequential finding of solutions was patented at the beginning of the last century. It turns out that the efforts to connect with the surrounding team were not in vain, something else was obviously overdue in the first place. Several generations later, scientists all over the world led to believe that mathematics is the queen of sciences. Whether it is the left answer or the right answer, the exhaustive terms must still be written in three rows, since in our case we will talk definitely only about vector analysis matrix properties. Nonlinear and linear equations, along with biquadratic equations, have taken a special place in our book about the best methods for calculating the trajectory of motion in the space of all material points of a closed system. Help us to bring the idea to life linear analysis dot product three consecutive vectors. At the end of each setting, the task is made easier by introducing optimized numerical exceptions in the context of the numerical space overlays being performed. Another judgment will not oppose the found answer in an arbitrary form of a triangle in a circle. The angle between the two vectors contains the necessary percentage of margin and solving equations online often reveals a certain common root equations as opposed to initial conditions. The exception plays the role of a catalyst in the entire inevitable process of finding a positive solution in the field of function definition. If it is not said that you cannot use a computer, then the online equation calculator is just right for your difficult tasks. It is enough just to enter your conditional data in the correct format and our server will issue a full-fledged resulting response in the shortest possible time. Exponential function increases much faster than linear. This is evidenced by the Talmuds of clever library literature. Will perform the calculation in the general sense, as the given quadratic equation with three complex coefficients would do. The parabola in the upper part of the half-plane characterizes rectilinear parallel motion along the axes of the point. Here it is worth mentioning the potential difference in the working space of the body. In return for a suboptimal result, our fraction calculator rightfully occupies the first position in the mathematical rating of the review of functional programs on the back end. Ease of use this service appreciated by millions of Internet users. If you do not know how to use it, then we will be happy to help you. We also want to highlight and highlight the cubic equation from a number of primary schoolchildren's tasks, when you need to quickly find its roots and plot a function graph on a plane. higher degrees reproduction is one of the most difficult mathematical problems at the institute and a sufficient number of hours are allocated for its study. Like all linear equations, ours is no exception to many objective rules, take a look under different points vision, and it will be simple and sufficient to set the initial conditions. The interval of increase coincides with the interval of convexity of the function. Solution of equations online. The study of theory is based on equations online from numerous sections on the study core discipline. Due to this approach, uncertain tasks, it is very easy to present the solution of equations in a predetermined form and not only draw conclusions, but also predict the outcome of such a positive solution. learn subject area service will help us in the best traditions of mathematics, just as it is customary in the East. At the best moments of the time interval, similar tasks were multiplied by a common multiplier ten times. With an abundance of multiplications of multiple variables in the equation calculator, it started to multiply by quality, and not by quantitative variables, such values ​​as mass or body weight. To avoid imbalances material system, the derivation of the three-dimensional converter on the trivial convergence of nondegenerate mathematical matrices is quite obvious to us. Complete the task and solve the equation in given coordinates, since the output is unknown in advance, as well as all the variables included in the post-space time are unknown. For a short time, push the common factor out of the parentheses and divide by the largest common divisor both parts in advance. From under the resulting covered subset of numbers extract detailed way thirty-three points in a row in a short period. Insofar as in at its best it is possible for every student to solve the equation online, looking ahead, let's say one important, but key thing, without which we will not be easy to live in the future. In the last century, the great scientist noticed a number of regularities in the theory of mathematics. In practice, it turned out not quite the expected impression of the events. However, in principle, this very solution of equations online helps to improve the understanding and perception of a holistic approach to the study and practical consolidation of the past theoretical material at students. It is much easier to do this during your study time.

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The human intellect needs constant training no less than the body needs physical activity. The best way to develop, expand the abilities of this quality of the psyche - to solve crossword puzzles and solve puzzles, the most famous of which, of course, is the Rubik's Cube. However, not everyone manages to collect it. Knowledge of the schemes and formulas for solving the assembly of this intricate toy will help to cope with this task.

What is a puzzle toy

Mechanical cube made of plastic, the outer faces of which consist of small cubes. The size of the toy is determined by the number of small elements:

• 2 x 2;
• 3 x 3 (the original version of the Rubik's Cube was exactly 3 x 3);
• 4 x 4;
• 5 x 5;
• 6 x 6;
• 7 x 7;
• 8 x 8;
• 9 x 9;
• 10 x 10;
• 11 x 11;
• 13 x 13;
• 17 x 17.

Any of the small cubes can rotate in three directions along the axes, represented as protrusions of a fragment of one of the three cylinders of the large cube. So the design has the ability to rotate freely, but at the same time, small parts do not fall out, but hold on to each other.

Each side of the toy includes 9 elements, painted in one of six colors, opposite each other in pairs. The classic combination of shades is:

• red opposite orange;
• white opposite yellow;
• blue opposite green.

However, modern versions may be colored in other combinations.

Today you can find Rubik's Cubes different color and forms

It is interesting. The Rubik's Cube even exists in a version for the blind. There, instead of color squares, there is a relief surface.

The goal of assembling the puzzle is to arrange the small squares so that they form the face of a large cube of the same color.

History of appearance

The idea of ​​​​creation belongs to the Hungarian architect Erne Rubik, who, in fact, did not create a toy, but a visual aid for his students. In such an interesting way, the resourceful teacher planned to explain the theory of mathematical groups (algebraic structures). It happened in 1974, and a year later the invention was patented as a puzzle toy - future architects (and not only them) got so attached to the intricate and bright manual.

The release of the first series of the puzzle was timed to coincide with the new year 1978, but the toy entered the world thanks to the entrepreneurs Tibor Lakzi and Tom Kremer.

It is interesting. Since the appearance of the Rubik's Cube ("magic cube", "magic cube"), about 350 million copies have been sold worldwide, which puts the puzzle in first place in popularity among toys. Not to mention dozens computer games based on this assembly principle.

The Rubik's Cube is an iconic toy for many generations

In the 80s, the inhabitants of the USSR met the Rubik's Cube, and in 1982, the first world championship in assembling a puzzle for speed, speedcubing, was organized in Hungary. Then best result was 22.95 seconds (for comparison: in 2017 a new world record was set: 4.69 seconds).

It is interesting. Fans of assembling a multi-colored puzzle are so attached to the toy that they find it not enough for them to assemble for speed alone. Therefore, in last years there were championships for solving puzzles with closed eyes, one hand, legs.

What are the formulas for the Rubik's Cube

Collecting a magic cube means arranging all the little details so that you get a whole face of the same color, you need to use God's algorithm. This term refers to a set of minimum actions that will solve a puzzle that has finite number moves and combinations.

It is interesting. In addition to the Rubik's Cube, God's algorithm is applied to puzzles such as Meffert's pyramid, Taken, Tower of Hanoi, etc.

Since the Rubik's Magic Cube was created as mathematical aid, then its assembly is decomposed according to the formulas.

The assembly of the Rubik's cube is based on the use of special formulas

Important definitions

In order to learn how to understand the schemes for solving the puzzle, you need to get acquainted with the names of its parts.

1. An angle is a combination of three colors. The 3 x 3 cube will have 3, the 4 x 4 version will have 4, and so on. The toy has 12 corners.
2. An edge denotes two colors. There are 8 of them in a cube.
3. The center contains one color. There are 6 in total.
4. Facets, as already mentioned, are simultaneously rotating elements of the puzzle. They are also called "layers" or "slices".

Values ​​in formulas

It should be noted that the assembly formulas are written in Latin - these are the schemes that are widely presented in various manuals for working with the puzzle. But there are also Russified versions. The list below shows both options.

1. The front face (front or facade) is the front face, which is in color to us [Ф] (or F - front).
2. The back face is the face that is centered away from us [З] (or B - back).
3. Right Edge - the edge that is on the right [P] (or R - right).
4. Left Edge - the edge that is on the left [L] (or L - left).
5. Bottom Face - the face that is below [H] (or D - down).
6. Upper Face - the face that is at the top [B] (or U - up).

Photo gallery: parts of the Rubik's cube and their definitions

To clarify the notation in the formulas, we use the Russian version - it will be clearer for beginners, but for those who want to switch to professional level speedcubing without international notation on English language not enough.

It is interesting. International system designation adopted by the World Cube Association (WCA).

1. The central cubes are indicated in the formulas of one lower case- f, t, p, l, v, n.
2. Corner - in three letters according to the name of the faces, for example, fpv, flni, etc.
3. Capital letters Ф, Т, П, Л, В, Н denote elementary operations of rotation of the corresponding face (layer, slice) of the cube by 90° clockwise.
4. Designations Ф, Т, П, Л, В, Н" correspond to the rotation of faces by 90° counterclockwise.
5. The designations Ф 2 , П 2 , etc., indicate a double rotation of the corresponding face (Ф 2 = FF).
6. The letter C denotes the rotation of the middle layer. The subscript shows which side of the face to look at to make that turn. For example, C P - from the side of the right side, C N - from the bottom side, C "L" - from the left side, counterclockwise, etc. It is clear that C N \u003d C "B, C P \u003d C" L and etc.
7. The letter O is the rotation (revolution) of the entire cube around its axis. О Ф - from the side of the front face clockwise, etc.

Recording the process (F "P") N 2 (PF) means: rotate the front face counterclockwise by 90 °, the same - the right side, rotate the bottom face twice (that is, by 180 °), rotate the right side by 90 ° along clockwise, rotate the front face 90° clockwise.

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http://dedfoma.ru/kubikrubika/kak-sobrat-kubik-rubika-3x3x3.htm

It is important for beginners to learn to understand the formulas

As a rule, instructions for building a puzzle in classic colors recommend holding the puzzle with the yellow center up. This advice is especially important for beginners.

It is interesting. There are websites that visualize formulas. Moreover, the speed of the assembly process can be set independently. For example, alg.cubing.net

How to solve a Rubik's puzzle

There are two types of schemas:

• for newbies;
• for professionals.

Their difference is in the complexity of the formulas, as well as the assembly speed. For beginners, of course, instructions appropriate to their level of knowledge of the puzzle will be more useful. But even they, after training, after a while will be able to fold the toy in 2-3 minutes.

How to build a standard 3 x 3 cube

Let's start by building a classic 3 x 3 Rubik's Cube using a 7-step pattern.

The classic version of the puzzle is the Rubik's Cube 3 x 3

It is interesting. The reverse process used to solve certain irregularly placed cubes is the reverse sequence of the action described by the formula. That is, the formula must be read from right to left, and the layers must be rotated counterclockwise if direct movement was indicated, and vice versa: direct if the opposite is described.

Assembly instructions

1. We start by assembling the cross of the upper face. We lower the required cube down by turning the corresponding side face (P, T, L) and bring it to the front face with the operation N, N "or H 2. We finish the stage of the removal by mirroring (reverse) the same side face, restoring the original position of the affected edge cube of the upper layer. After that, we perform operation a) or b) of the first stage. In case a) the cube came to the front face so that the color of its front face matches the color of the facade. In case b) the cube must not only be moved up, but also unfolded so that it is correctly oriented, standing in its place.

   

   We collect the cross of the upper line

2. The required corner cube is found (having the colors of the faces F, V, L) and, using the same technique that is described for the first stage, it is displayed in the left corner of the selected front face (or yellow). There can be three cases of orientation of this cube. We compare our case with the picture and apply one of the operations of the second stage a, beat c. The dots on the diagram mark the place where the desired cube should be placed. We look for the remaining three corner cubes on the cube and repeat the described technique to move them to their places on the top face. Result: the top layer is picked up. The first two stages cause almost no difficulty for anyone: it is quite easy to follow your actions, since all attention is paid to one layer, and what is done in the remaining two is not at all important.

   

   Choosing the top layer

3. Our goal: to find the desired cube and first bring it down to the front face. If it is at the bottom - by simply turning the bottom face until it matches the color of the facade, and if it is in the middle layer, then you must first lower it down using any of the operations a) or b), and then match it in color with the color of the facade face and perform the operation of the third stage a) or b). Result: two layers collected. The formulas given here are mirror formulas in the full sense of the word. You can clearly see this if you put a mirror to the right or left of the cube (with an edge towards you) and do any of the formulas in the mirror: we will see the second formula. That is, operations with the front, bottom, top (not involved here), and back (also not involved) faces change sign to the opposite: it was clockwise, it became counterclockwise, and vice versa. And the left side changes from the right one, and, accordingly, changes the direction of rotation to the opposite.

   

   We find the desired cube and bring it down to the front face

4. The goal is achieved by operations that move the side cubes of one face, without ultimately violating the order in the collected layers. One of the processes that allows you to pick up all the side faces is shown in the figure. It also shows what happens in this case with other face cubes. By repeating the process, choosing a different front face, you can put all four cubes in place. Result: the rib pieces are in place, but two of them, or even all four, may be incorrectly oriented. Important: before proceeding with this formula, we look at which cubes are already in place - they may be incorrectly oriented. If there is none or one, then we try to rotate the upper face so that the two that are on two adjacent side faces (fv + pv, pv + tv, tv + lv, lv + fv) fall into place, after that we orient the cube like this , as shown in the figure, and execute the formula given at this stage. If it is not possible to combine the details belonging to adjacent faces by turning the top face, then we execute the formula for any position of the cubes of the top face once and try again by turning the top face to put 2 details located on two adjacent side faces in their places.

   

   It is important to check the orientation of the cubes at this stage

5. We take into account that the unfolded cube should be on the right side, in the figure it is marked with arrows (cube pv). Figures a, b, and c show possible cases of location of incorrectly oriented cubes (marked with dots). Using the formula in case a), we perform an intermediate rotation B "to bring the second cube to the right side, and the final rotation B, which will return the upper face to its original position, in case b) an intermediate rotation B 2 and the final one also B 2, and in case c) intermediate rotation B must be performed three times, after turning each cube and also completed with rotation B. Many are confused by the fact that after the first part of the process (PS N) 4, the desired cube unfolds as it should, but the order in the collected layers is violated. confuses and makes some people throw an almost completed cube halfway through. Having completed an intermediate turn, ignoring the “breakage” of the lower layers, we perform operations (PS N) 4 with the second cube (the second part of the process), and everything falls into place. Result: assembled cross.

   

   The result of this stage will be an assembled cross

6. We put the corners of the last face into place using an easy-to-remember 8-way process - forward, rearranging the three corner pieces in a clockwise direction, and reverse, rearranging the three dice in a counterclockwise direction. After the fifth stage, as a rule, at least one cube will sit in its place, even if it is incorrectly oriented. (If after the fifth stage none of the corner cubes has sat down in its place, then we apply any of the two processes for any three cubes, after that exactly one cube will be in its place.). Result: all the corner cubes are in place, but two of them (maybe four) may not be oriented correctly.

   

   Corner cubes sit in their places

7. We repeatedly repeat the sequence of turns PF "P" F. Rotate the cube so that the cube we want to rotate is on the right upper corner facade. An 8-way process (2 x 4 turns) will rotate it 1/3 turn clockwise. If at the same time the cube has not yet oriented, repeat the 8-move again (in the formula this is reflected by the index “N”). We do not pay attention to the fact that the lower layers will become a mess. The figure shows four cases of incorrectly oriented cubes (they are marked with dots). In case a) an intermediate turn B and a final B" are required, in case b) - an intermediate and final turn B 2, in case c) - turn B is performed after each cube is rotated to the correct orientation, and the final B 2, in case d) - intermediate turn B is also performed after turning each cube to the correct orientation, and the final turn in this case will also be turn B. Result: the last face is assembled.

   

   Possible errors are shown with dots

Formulas for correcting the placement of cubes can be shown like this.

Formulas for Correcting Misaligned Cubes in the Last Step

The essence of Jessica Friedrich's method

There are several ways to assemble the puzzle, but one of the most memorable is the one developed by Jessica Friedrich, a professor at the University of Binghamton, New York, who develops techniques for hiding data in digital images. While still a teenager, Jessica became so fascinated with the cube that in 1982 she became the world champion in speed cubing and subsequently did not leave her hobby, developing formulas for quickly assembling the "magic cube". One of the most popular options for folding a cube is called CFOP - after the first letters of the four assembly steps.

Instruction:

1. We collect the cross on the upper face, which is made up of cubes on the edges of the lower face. This stage is called Cross - cross.
2. We collect the lower and middle layers, that is, the face on which the cross is located, and the intermediate layer, consisting of four side parts. The name of this step is F2L (First two layers) - the first two layers.
3. We collect the remaining face, not paying attention to the fact that not all the details are in place. The stage is called OLL (Orient the last layer), which translates as “orientation of the last layer”.
4. The last level - PLL (Permute the last layer) - consists in correct placement top layer cubes.

Friedrich Method Video Instructions

The speedcubers liked the method proposed by Jessica Friedrich so much that the most advanced amateurs develop their own methods to speed up the assembly of each of the stages proposed by the author.

Video: accelerating the assembly of the cross

Video: collecting the first two layers

Video: working with the last layer

Video: last build level by Friedrich

2 x 2

The 2 x 2 Rubik's Cube or mini Rubik's Cube is also stacked in layers, starting from the bottom level.

The mini-dice is a lighter version of the classic puzzle

Easy Assembly Instructions for Beginners

1. We assemble the bottom layer so that the colors of the last four cubes match, and the remaining two colors are the same as the colors of the neighboring parts.
2. Let's start organizing the top layer. Please note that on this stage the goal is not to match the colors, but to put the cubes in their places. We start by determining the color of the top. Everything is simple here: it will be the color that did not appear in the bottom layer. Rotate any of the top cubes so that it gets to the position where the three colors of the element intersect. Having fixed the corner, we arrange the elements of the remaining ones. We use two formulas for this: one for changing diagonal cubes, the other for neighboring ones.
3. We complete the top layer. We carry out all operations in pairs: we rotate one corner, and then the other, but in the opposite direction (for example, the first one is clockwise, the second is counterclockwise). You can work with three angles at once, but in this case there will be only one combination: either clockwise or counterclockwise. Between rotations of the corners, we rotate the upper face so that the corner being worked out is in the upper right corner. If we work with three corners, then we put the correctly oriented one at the back left.

Formulas for rotating angles:

• (VFPV P"V"F")² (5);
• V²F V²F "V"F V"F"(6);
• FVF² LFL² VLV² (7).

To rotate three corners at once:

• (FVPV "P" F "V")² (8);
• FV F "V FV² F" V² (9);
• V²L"V"L²F"L"F²V"F" (10).

Photo Gallery: Building a 2 x 2 Cube

Video: Friedrich method for a 2 x 2 cube

Collecting the most difficult versions of the cube

These include toys with a number of parts from 4 x 4 and up to 17 x 17.

Models of a cube for many elements usually have rounded corners for ease of manipulation with a toy

It is interesting. AT this moment A 19 x 19 version is being developed.

At the same time, it should be remembered that they were created on the basis of a 3 x 3 cube, therefore the assembly is built in two directions.

1. We assemble the center so that the elements of the 3 x 3 cube remain.
2. We work according to the schemes for assembly original version toys (most often cubers use the Jessica Friedrich method).

4 x 4

This version is called "Rubik's Revenge".

Instruction:

The assembly of models 5 x 5, 6 x 6 and 7 x 7 is similar to the previous one, only we take the center as the basis large quantity cubes.

Video: Rubik's Cube 5 x 5

Working on solving the 6 x 6 puzzle

This cube is rather inconvenient to work with: a large number of small details required special attention. Therefore, we will divide the video instructions into four parts: for each assembly step.

Video: how to solve the center in a 6 x 6 cube, part 1

Video: pairing edge elements in a 6 x 6 cube, part 2

Video: pairing four elements of the 6 x 6 puzzle, part 3

Video: final assembly of the Rubik's Cube 6 x 6, part 4

Video: putting together a 7 x 7 puzzle

How to solve the pyramid puzzle

This puzzle is mistakenly considered a variation of the Rubik's Cube. But in fact, Meffert's toy, which is also called the "Japanese tetrahedron" or "Moldavian pyramid", appeared several years earlier visual aid architect teacher.

Meffert's pyramid is mistakenly called a Rubik's puzzle.

To work with this puzzle, it is important to know its structure, because the mechanism of work plays a key role in the assembly. The Japanese tetrahedron consists of:

• four axis elements;
• six costal;
• four corners.

Each part of the axis has small triangles facing three adjacent faces. That is, each element can be rotated without the threat of falling out of the structure.

It is interesting. There are 75,582,720 options for the arrangement of the elements of the pyramid. Unlike the Rubik's Cube, it's not that much. The classic version of the puzzle has 43,252,003,489,856,000 options configurations.

Instruction and diagram

Video: a simple technique for assembling a pyramid completely

Method for children

Using formulas and applying ways to speed up assembly for children just starting to get acquainted with the puzzle will be too difficult task. Therefore, the task of adults is to simplify the explanation as much as possible.

The Rubik's Cube is not only an opportunity to entertain a child with useful and an interesting activity but also a way to develop patience, perseverance

It is interesting. It is better to start teaching children with the 3 x 3 model.

Instructions (cube 3 x 3):

1. We decide on the color of the upper face and take the toy so that the central cube of the desired color is at the top.
2. We collect the upper cross, but at the same time the second color of the middle layer was the same as the color of the side faces.
3. Set the corners of the top face. Let's move on to the second layer.
4. We collect the last layer, but we start by restoring the sequence of the first ones. Then we set the corners so that they coincide with the central details of the faces.
5. We check the location of the middle parts of the last face, changing their location if necessary.

Solving the Rubik's cube in any of its variations is a great exercise for the mind, a way to relieve stress and distract yourself. Even a child can learn how to solve a puzzle using an age-friendly explanation. Gradually, you can master more intricate assembly methods, improve your own time indicators, and then it’s not far from speedcubing competitions. The main thing is perseverance and patience.

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Goals:

1. To systematize and generalize knowledge and skills on the topic: Solutions of equations of the third and fourth degree.
2. To deepen knowledge by completing a series of tasks, some of which are not familiar either in their type or in the method of solving.
3. Formation of interest in mathematics through the study of new heads of mathematics, education of graphic culture through the construction of graphs of equations.

Lesson type: combined.

Equipment: graph projector.

Visibility: table "Vieta's theorem".

During the classes

1. Mental account

a) What is the remainder of the division of the polynomial p n (x) \u003d a n x n + a n-1 x n-1 + ... + a 1 x 1 + a 0 by the binomial x-a?

b) How many roots can a cubic equation have?

c) With what help do we solve the equation of the third and fourth degree?

d) If b is an even number in the quadratic equation, then what is D and x 1; x 2

2. Independent work(in groups)

Make an equation if the roots are known (answers to tasks are coded) Use the "Vieta Theorem"

1 group

Roots: x 1 = 1; x 2 \u003d -2; x 3 \u003d -3; x 4 = 6

Write an equation:

B=1 -2-3+6=2; b=-2

c=-2-3+6+6-12-18=-23; c= -23

d=6-12+36-18=12; d=-12

e=1(-2)(-3)6=36

x 4 -2 x 3 - 23 x 2 - 12 x + 36 = 0(this equation is then solved by group 2 on the board)

Decision . We are looking for integer roots among the divisors of the number 36.

p = ±1; ±2; ±3; ±4; ±6…

p 4 (1)=1-2-23-12+36=0 The number 1 satisfies the equation, therefore =1 is the root of the equation. Horner's scheme

p 3 (x) = x 3 -x 2 -24x -36

p 3 (-2) \u003d -8 -4 +48 -36 \u003d 0, x 2 \u003d -2

p 2 (x) \u003d x 2 -3x -18 \u003d 0

x 3 \u003d -3, x 4 \u003d 6

Answer: 1; -2; -3; 6 the sum of the roots 2 (P)

2 group

Roots: x 1 \u003d -1; x 2 = x 3 =2; x 4 \u003d 5

Write an equation:

B=-1+2+2+5-8; b=-8

c=2(-1)+4+10-2-5+10=15; c=15

D=-4-10+20-10=-4; d=4

e=2(-1)2*5=-20;e=-20

8 + 15 + 4x-20 \u003d 0 (group 3 solves this equation on the board)

p = ±1; ±2; ±4; ±5; ±10; ±20.

p 4 (1)=1-8+15+4-20=-8

p 4 (-1)=1+8+15-4-20=0

p 3 (x) \u003d x 3 -9x 2 + 24x -20

p 3 (2) \u003d 8 -36 + 48 -20 \u003d 0

p 2 (x) \u003d x 2 -7x + 10 \u003d 0 x 1 \u003d 2; x 2 \u003d 5

Answer: -1;2;2;5 sum of roots 8(P)

3 group

Roots: x 1 \u003d -1; x 2 =1; x 3 \u003d -2; x 4 \u003d 3

Write an equation:

B=-1+1-2+3=1;b=-1

s=-1+2-3-2+3-6=-7; s=-7

D=2+6-3-6=-1; d=1

e=-1*1*(-2)*3=6

x 4 - x 3- 7x 2 + x + 6 = 0(this equation is solved later on the board by group 4)

Decision. We are looking for integer roots among the divisors of the number 6.

p = ±1; ±2; ±3; ±6

p 4 (1)=1-1-7+1+6=0

p 3 (x) = x 3 - 7x -6

p 3 (-1) \u003d -1 + 7-6 \u003d 0

p 2 (x) = x 2 -x -6=0; x 1 \u003d -2; x 2 \u003d 3

Answer: -1; 1; -2; 3 The sum of the roots 1 (O)

4 group

Roots: x 1 = -2; x 2 \u003d -2; x 3 \u003d -3; x 4 = -3

Write an equation:

B=-2-2-3+3=-4; b=4

c=4+6-6+6-6-9=-5; c=-5

D=-12+12+18+18=36; d=-36

e=-2*(-2)*(-3)*3=-36; e=-36

x 4+4x 3 - 5x 2 - 36x -36 = 0(this equation is then solved by group 5 on the board)

Decision. We are looking for integer roots among the divisors of the number -36

p = ±1; ±2; ±3…

p(1)= 1 + 4-5-36-36 = -72

p 4 (-2) \u003d 16 -32 -20 + 72 -36 \u003d 0

p 3 (x) \u003d x 3 + 2x 2 -9x-18 \u003d 0

p 3 (-2) \u003d -8 + 8 + 18-18 \u003d 0

p 2 (x) = x 2 -9 = 0; x=±3

Answer: -2; -2; -3; 3 Sum of roots-4 (F)

5 group

Roots: x 1 \u003d -1; x 2 \u003d -2; x 3 \u003d -3; x 4 = -4

Write an equation

x 4+ 10x 3 + 35x 2 + 50x + 24 = 0(this equation is then solved by the 6th group on the board)

Decision . We are looking for integer roots among the divisors of the number 24.

p = ±1; ±2; ±3

p 4 (-1) = 1 -10 + 35 -50 + 24 = 0

p 3 (x) \u003d x- 3 + 9x 2 + 26x + 24 \u003d 0

p 3 (-2) \u003d -8 + 36-52 + 24 \u003d O

p 2 (x) \u003d x 2 + 7x + 12 \u003d 0

Answer: -1; -2; -3; -4 sum-10 (I)

6 group

Roots: x 1 = 1; x 2 = 1; x 3 \u003d -3; x 4 = 8

Write an equation

B=1+1-3+8=7;b=-7

c=1 -3+8-3+8-24= -13

D=-3-24+8-24=-43; d=43

x 4 - 7x 3- 13x 2 + 43x - 24 = 0 (this equation is then solved by 1 group on the board)

Decision . We are looking for integer roots among the divisors of the number -24.

p 4 (1)=1-7-13+43-24=0

p 3 (1)=1-6-19+24=0

p 2 (x) \u003d x 2 -5x - 24 \u003d 0

x 3 \u003d -3, x 4 \u003d 8

Answer: 1; 1; -3; 8 sum 7 (L)

3. Solution of equations with a parameter

1. Solve the equation x 3 + 3x 2 + mx - 15 = 0; if one of the roots is (-1)

Answer in ascending order

R=P 3 (-1)=-1+3-m-15=0

x 3 + 3x 2 -13x - 15 = 0; -1+3+13-15=0

By condition x 1 = - 1; D=1+15=16

P 2 (x) \u003d x 2 + 2x-15 \u003d 0

x 2 \u003d -1-4 \u003d -5;

x 3 \u003d -1 + 4 \u003d 3;

Answer: - 1; -5; 3

In ascending order: -5;-1;3. (b n s)

2. Find all the roots of the polynomial x 3 - 3x 2 + ax - 2a + 6, if the remainders of its division into binomials x-1 and x + 2 are equal.

Solution: R \u003d R 3 (1) \u003d R 3 (-2)

P 3 (1) \u003d 1-3 + a- 2a + 6 \u003d 4-a

P 3 (-2) \u003d -8-12-2a-2a + 6 \u003d -14-4a

x 3 -3x 2 -6x + 12 + 6 \u003d x 3 -3x 2 -6x + 18

x 2 (x-3)-6(x-3) = 0

(x-3)(x 2 -6) = 0

3) a \u003d 0, x 2 -0 * x 2 +0 \u003d 0; x 2 =0; x 4 \u003d 0

a=0; x=0; x=1

a>0; x=1; x=a ± √a

2. Write an equation

1 group. Roots: -4; -2; one; 7;

2 group. Roots: -3; -2; one; 2;

3 group. Roots: -1; 2; 6; ten;

4 group. Roots: -3; 2; 2; 5;

5 group. Roots: -5; -2; 2; 4;

6 group. Roots: -8; -2; 6; 7.

Quadratic equations.

Quadratic equation- algebraic equation general view

where x is a free variable,

a, b, c, - coefficients, and

Expression called a square trinomial.

Solutions quadratic equations.

1. METHOD : Factorization of the left side of the equation.

Let's solve the equation x 2 + 10x - 24 = 0. Let's factorize the left side:

x 2 + 10x - 24 \u003d x 2 + 12x - 2x - 24 \u003d x (x + 12) - 2 (x + 12) \u003d (x + 12) (x - 2).

Therefore, the equation can be rewritten as:

(x + 12)(x - 2) = 0

Since the product is zero, then at least one of its factors is zero. Therefore, the left side of the equation vanishes at x = 2, as well as at x = - 12. This means that the number 2 and - 12 are the roots of the equation x 2 + 10x - 24 = 0.

2. METHOD : Full square selection method.

Let's solve the equation x 2 + 6x - 7 = 0. Highlight on the left side full square.

To do this, we write the expression x 2 + 6x in following form:

x 2 + 6x = x 2 + 2 x 3.

In the resulting expression, the first term is the square of the number x, and the second is double product x by 3. Therefore, to get a full square, you need to add 3 2, since

x 2+ 2 x 3 + 3 2 \u003d (x + 3) 2.

We now transform the left side of the equation

x 2 + 6x - 7 = 0,

adding to it and subtracting 3 2 . We have:

x 2 + 6x - 7 = x 2+ 2 x 3 + 3 2 - 3 2 - 7 = (x + 3) 2 - 9 - 7 = (x + 3) 2 - 16.

Thus, this equation can be written as follows:

(x + 3) 2 - 16 = 0, (x + 3) 2 = 16.

Hence, x + 3 - 4 = 0, x 1 = 1, or x + 3 = -4, x 2 = -7.

3. METHOD :Solution of quadratic equations by formula.

Multiply both sides of the equation

ax 2 + bx + c \u003d 0, a ≠ 0

on 4a and successively we have:

4a 2 x 2 + 4abx + 4ac = 0,

((2ax) 2 + 2ax b + b 2) - b 2 + 4ac \u003d 0,

(2ax + b) 2 = b 2 - 4ac,

2ax + b \u003d ± √ b 2 - 4ac,

2ax \u003d - b ± √ b 2 - 4ac,

Examples.

a) Let's solve the equation: 4x2 + 7x + 3 = 0.

a = 4, b = 7, c = 3, D = b 2 - 4ac = 7 2 - 4 4 3 = 49 - 48 = 1,

D > 0 two different roots;

Thus, in the case of a positive discriminant, i.e. at

b 2 - 4ac >0, the equation ax 2 + bx + c = 0 has two different root.

b) Let's solve the equation: 4x 2 - 4x + 1 = 0,

a \u003d 4, b \u003d - 4, c \u003d 1, D \u003d b 2 - 4ac \u003d (-4) 2 - 4 4 1= 16 - 16 \u003d 0,

D=0 one root;

So, if the discriminant is zero, i.e. b 2 - 4ac = 0, then the equation

ax 2 + bx + c = 0 has a single root

in) Let's solve the equation: 2x 2 + 3x + 4 = 0,

a = 2, b = 3, c = 4, D = b 2 - 4ac = 3 2 - 4 2 4 = 9 - 32 = - 13, D< 0.

This equation has no roots.

So, if the discriminant is negative, i.e. b2-4ac< 0 , the equation

ax 2 + bx + c = 0 has no roots.

Formula (1) of the roots of the quadratic equation ax 2 + bx + c = 0 allows you to find the roots any quadratic equation (if any), including reduced and incomplete. Formula (1) is expressed verbally as follows: the roots of a quadratic equation are equal to a fraction whose numerator is equal to the second coefficient, taken from opposite sign, plus minus the square root of the square of this coefficient without quadruple the product of the first coefficient and the free term, and the denominator is twice the first coefficient.

4. METHOD: Solution of equations using Vieta's theorem.

As is known, the given quadratic equation has the form

x 2 + px + c = 0.(1)

Its roots satisfy the Vieta theorem, which, when a =1 has the form

x 1 x 2 = q,

x 1 + x 2 = - p

From this we can draw the following conclusions (the signs of the roots can be predicted from the coefficients p and q).

a) If the summary term q of the reduced equation (1) is positive ( q > 0), then the equation has two roots of the same sign and this is the envy of the second coefficient p. If a R< 0 , then both roots are negative if R< 0 , then both roots are positive.

For example,

x 2 - 3x + 2 = 0; x 1 = 2 and x 2 \u003d 1, as q = 2 > 0 and p=-3< 0;

x2 + 8x + 7 = 0; x 1 = - 7 and x 2 \u003d - 1, as q = 7 > 0 and p=8 > 0.

b) If a free member q of the reduced equation (1) is negative ( q< 0 ), then the equation has two roots of different sign, and the larger root in absolute value will be positive if p< 0 , or negative if p > 0 .

For example,

x 2 + 4x - 5 = 0; x 1 = - 5 and x 2 \u003d 1, as q= - 5< 0 and p = 4 > 0;

x 2 - 8x - 9 \u003d 0; x 1 = 9 and x 2 \u003d - 1, as q = - 9< 0 and p=-8< 0.

Examples.

1) Solve the equation 345x 2 - 137x - 208 = 0.

Decision. As a + b + c \u003d 0 (345 - 137 - 208 \u003d 0), then

x 1 = 1, x 2 = c / a = -208/345.

Answer: 1; -208/345.

2) Solve the equation 132x 2 - 247x + 115 = 0.

Decision. As a + b + c \u003d 0 (132 - 247 + 115 \u003d 0), then

x 1 \u003d 1, x 2 \u003d c / a \u003d 115/132.

Answer: 1; 115/132.

B. If the second coefficient b = 2k is an even number, then the formula of the roots

Example.

Let's solve the equation 3x2 - 14x + 16 = 0.

Decision. We have: a = 3, b = - 14, c = 16, k = - 7;

D \u003d k 2 - ac \u003d (- 7) 2 - 3 16 \u003d 49 - 48 \u003d 1, D\u003e 0, two different roots;

Answer: 2; 8/3

AT. Reduced Equation

x 2 + px + q \u003d 0

coincides with the general equation, in which a = 1, b = p and c = q. Therefore, for the reduced quadratic equation, the formula for the roots

Takes the form:

Formula (3) is especially convenient to use when R- even number.

Example. Let's solve the equation x 2 - 14x - 15 = 0.

Decision. We have: x 1.2 \u003d 7 ±

Answer: x 1 = 15; x 2 \u003d -1.

5. METHOD: Solving equations graphically.

Example. Solve the equation x2 - 2x - 3 = 0.

Let's plot the function y \u003d x2 - 2x - 3

1) We have: a = 1, b = -2, x0 = 1, y0 = f(1)= 12 - 2 - 3= -4. This means that the point (1; -4) is the vertex of the parabola, and the straight line x \u003d 1 is the axis of the parabola.

2) Take two points on the x-axis that are symmetrical about the axis of the parabola, for example, the points x \u003d -1 and x \u003d 3.

We have f(-1) = f(3) = 0. Let's construct points (-1; 0) and (3; 0) on the coordinate plane.

3) Through the points (-1; 0), (1; -4), (3; 0) we draw a parabola (Fig. 68).

The roots of the equation x2 - 2x - 3 = 0 are the abscissas of the points of intersection of the parabola with the x-axis; so the roots of the equation are: x1 = - 1, x2 - 3.