![](data:image/svg+xml;utf8,<svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" viewBox="0 0 72 71" width="72"></svg>)
Who can help me write an equation with quadratic coefficient of 1 and sum of two real roots of 3?
x²-3x+2=0
Write a linear equation with one variable, if the coefficient of X is 1 / 2 and the solution of the equation is 3
X = 3, then 1 / 2 * x = 3 / 2
So it's (1 / 2) x-3 / 2 = 0
Please write a quadratic equation of one variable with quadratic coefficient 1 and real root 2______ The answer is not unique
Let two of the equations be 1 and 2, so the equation is (x-1) (X-2) = 0, that is, x2-3x + 2 = 0
If 1 + I is a root of the real coefficient equation x ^ 2 + box + C = 0, then the other root of the equation is
The process of solving the problem. It's BX
1-i
If a real coefficient polynomial has a complex root, then the conjugate complex of the complex is also the root of the polynomial
Answer these four questions, use arithmetic, disable equation
1. The road repair team plans to build a 91 kilometer long highway, in which the kilometers that have been repaired are 1 / 6 of those that have not been repaired. How many kilometers are there?
2. There are two baskets of fruits. Basket a is 30kg more than basket B. basket B sells 18kg, and the remaining kilogram is only 40% of basket A. how many kilogram of fruits are there in basket B?
3. Yongxin Chemical plant has a total of 960 employees. The chemical plant is divided into three workshops: A, B and C. The ratio of employees is 4:5:3. How many employees are there in each workshop?
4. The southern transportation team has to transport 1024 tons of goods, 25% of the total on the first day and 3% of the total on the second day. How many tons are left?
1. The team plans to build a 91 kilometer long highway, in which the kilometers that have been repaired are 1 / 6 of the kilometers that have not been repaired, and how many kilometers are there? If the kilometers that have not been repaired are regarded as unit 1, the roads that have been repaired are 1 / 6, the total roads are 1 + 1 / 6 = 7 / 6, and the corresponding actual kilometers are 91 kilometers
In the travel problem (3), the midpoint meets somewhere,
2. At the same time, the passenger car and the freight car leave each other from a and B. the passenger car travels 54 kilometers per hour and the freight car 48 kilometers per hour. After the two cars meet, they continue to move forward at the same speed. The passenger car returns immediately after arriving at B, and the freight car returns immediately after arriving at a. the two cars meet again at 108 kilometers away from the midpoint. Question: how many kilometers are there between a and B?
Distance between a and B:
108*2/(54-48)*(54+48)/3
=108*1/3*102/3
=1224km
The two cars run back and meet each other, that is, they run a whole course separately, and then they run a whole course together. They run a total of three whole courses. When they meet 108 meters away from the midpoint, the truck is slow, and it is no doubt that they have not reached the midpoint. Naturally, the bus has passed the midpoint, which is 108 * 2000 meters more than the truck
Practice of solving binary linear equations
1) Concept: an equation with two unknowns and the degree of the term containing the unknowns is 1 is called bivariate linear equation. Can you distinguish these equations? 5x + 3Y = 75 (bivariate linear equation); 3x + 1 = 8x (univariate linear equation); + y = 2 (univariate linear equation); 2XY = 9 (bivariate linear equation) (1) the algebraic expressions on both sides of the equal sign are integers; (2) in the equation, "Yuan" refers to the unknowns, and "binary" refers to two unknowns in the equation; (3) the degree of the term of the unknowns is 1, which actually means that the degree of the highest term in the equation is 1, which can be compared with the degree of the polynomial, (2) the value of a group of unknowns whose solution makes both sides of the equation equal is called a solution of the equation. The understanding of the solution of the equation should pay attention to the following points: (1) generally, there are countless solutions of a equation, And each solution refers to a pair of numerical values, rather than the value of a single unknown number; ② a solution of binary linear equation refers to the value of a pair of unknowns that make the left and right sides of the equation equal; conversely, if a group of numerical values can make the left and right sides of binary linear equation equal, then this group of numerical values is the solution of the equation; ③ when solving the solution of binary linear equation, The usual way is to use an unknown to express another unknown, and then give a value of this unknown to get another unknown correspondingly. In this way, we can get a solution of the quadratic equation of two variables. Can you try to solve the equation 3x-y = 6?
2. Binary linear equations
(1) bivariate linear equations: a group of equations composed of two bivariate linear equations is called bivariate linear equations. (2) the common solution of two equations in bivariate linear equations of bivariate linear equations is called the solution of bivariate linear equations, The same letter must represent the same number, otherwise two equations can not be combined. ② how to test whether a group of numerical values is the solution of a system of linear equations with two variables, the common methods are as follows: substituting the group of numerical values into each equation in the system of equations, only when the group of numerical values satisfy all the equations, can the group of numerical values be said to be the solution of the system of equations, If the set of values does not satisfy any of the equations, then it is not the solution of the equations
3. Substitution elimination method
(1) concept: to express an unknown number of an equation in a system of equations with an algebraic formula containing another unknown number, substitute it into another equation, eliminate an unknown number, obtain a linear equation of one variable, and finally obtain the solution of the system of equations. This method of solving the system of equations is called substitution elimination method, which is called substitution method for short. (2) the steps of substitution method for solving the system of linear equations of two variables Firstly, a binary linear equation with simple coefficients is selected to be transformed, and another unknown is represented by an algebraic formula containing an unknown; secondly, the transformed equation is substituted into another equation, and an unknown is eliminated to obtain a one variable linear equation (when substituting, it should be noted that it can not be substituted into the original equation, but can only be substituted into another equation without deformation, (3) solve the equation of one variable and find out the value of the unknown number; (4) substitute the value of the unknown number into the transformed equation in (1) and find out the value of the other unknown number; (5) use "{" to set up the values of the two unknowns, It is the solution of the system of equations; 6. Finally, check whether the result is correct (put it into the original system of equations to check whether the equation satisfies the left = right)
4. Addition, subtraction and elimination
(1) concept: when the coefficients of an unknown number of two equations in the equation are equal or opposite to each other, the two sides of the two equations are added or subtracted to eliminate the unknown number, so that the binary linear equation can be transformed into a unary linear equation, and finally the solution of the equation system can be obtained. This method of solving the equation system is called addition subtraction elimination method, It is called addition and subtraction method for short. (2) the steps of addition and subtraction method for solving binary linear equations are as follows: (1) using the basic properties of the equation, the coefficient of an unknown number in the original equation system is transformed into the form of equal or opposite number; (2) using the basic properties of the equation, the two transformed equations are added or subtracted to eliminate an unknown number, Get a one variable linear equation (be sure to multiply both sides of the equation by the same number, never multiply only one side, then use subtraction if the coefficients of the unknowns are equal, and use addition if the coefficients of the unknowns are opposite to each other); ③ solve the one variable linear equation and find out the value of the unknowns; ④ substitute the value of the unknowns into any one of the original equations, Find out the value of another unknown number; ⑤ use "{" to set up the value of two unknowns, that is the solution of the equation system; ⑥ finally check whether the result is correct (substitute into the original equation system to test whether the equation meets the left = right)
Key points and difficulties
This section focuses on the concept of binary linear equations and how to use substitution method and addition and subtraction method to solve binary linear equations
Typical examples
Example 1. Which of the following equations is a bivariate linear equation? (1) 8x-y = y; (2) xy = 3; (3) 2x-y = 9; (4) 8x-3 = 2. Analysis: the judgment of this problem is based on the definition of bivariate linear equation. Because the degree of the term XY with unknowns in equation (2) is 2, not 1, the degree of the term XY with unknowns in equation (2) is 2, Therefore, xy = 3 is not a bivariate linear equation; 2x-y = 9 is a bivariate linear equation; and because equation (4) is not an integral, so = 2 is not a bivariate linear equation. Equation 8x-y = y, 2x-y = 9 is a bivariate linear equation; xy = 3, = 2 is not a bivariate linear equation, Then judge according to the definition. Example 2. It is known that - 1 is the solution of the system of equations, and the value of M + n is obtained. Analysis: because it is the solution of the system of equations, it satisfies equation ① and equation ② at the same time. If it is substituted into equation ① and equation ② respectively, the values of M and N can be obtained from ③ and ④. Because it is the solution of the system of equations, the two equations substituted into the original system of equations still hold, The solution is m + n = - 1 + 0 = - 1 For example 3, write out all positive integer solutions of the quadratic equation 4x + y = 20. Analysis: in order to solve the problem conveniently, first transform the original equation into y = 20-4x, because the solution required in the problem is limited to "positive integer solution", so the values of X and y must be positive integers, Because X and y are positive integers, X can only take positive integers less than 5. When x = 1, y = 16; when x = 2, y = 12; when x = 3, y = 8; when x = 4, y = 4. That is to say, all positive integer solutions of 4X + y = 20 are:, The criterion of "correct" is that the values of two unknowns must be positive integers, which is suitable for this equation. Example 4. Given 5 x + Y-3 + (x-2y) = 0, find the values of X and Y. analysis: according to the meaning of absolute value and square, 5 x + Y-3 ≥ 0, (x-2y) ≥ 0, From the known condition 5 x + Y-3 + (x-2y) = 0, the values of X and y can be obtained. From the meaning of the problem, the solution can be obtained. Comment: non negative values add up to zero, and only they are zero at the same time. Example 5. Solving equations by substitution method: Analysis: select one of the equations and transform it into the form of y = ax + B or x = ay + B, The coefficients of X and Y in equation (1) are relatively small. Considering that x = 3-y and y =, it is obvious that in the following calculation, x = 3-y can be substituted into equation (2). From (1), x = 3-y (3) can be substituted into (2), 8 (3-y) + 3Y + 1 = 0 can be obtained, y = 125 can be substituted into (3), The result is: x = - 47, so the solution of this system of equations is evaluation: when using substitution method to solve the system of equations, (1) choose the deformed equation as simple as possible, and the algebraic expression should be as simple as possible. (2) predict and estimate the following calculation, so as to choose a better method, The coefficients of X are 4 and 6, the coefficients of Y are 3 and - 4, and their least common multiple are 12, which can be changed into 12 or - 12. It is quite difficult to choose whether to eliminate X or y
RELATED INFORMATIONS
- 1. The real coefficient equation f (x) = x ^ 2 + ax + 2B = 0 has one root in (0,1) and the other root in (1,2) The range of (b-2) / (A-1)
- 2. If 2-I is a root of the real coefficient equation x & # 178; + ax + B = 0 about X, then the sum of the modules of the two equations is?
- 3. The chemical reaction equation of Si and NaOH is Si and O2... Cl2 and NaBr
- 4. Is the basic reaction type of SiO2 + 2C = Si + 2CO reduction reaction or displacement reaction
- 5. In the reaction of SiO 2 + 2C = = Si + 2CO, the oxidant is the reducing agent and the oxidant is the oxidizing agent In the reaction of SiO 2 + 2C = = Si + 2CO, the oxidant is the reducing agent, and the ratio of oxidant to reducing agent is 0
- 6. What are the uses of Si and SiO2 (for example, what are the common optical fibers and cement) and what are the uses of Si in high school chemistry?
- 7. Si (s) + O2 (g) = SiO2 (s) how to calculate the enthalpy change of the reaction seems to consider Si and SiO Si (s) + O 2 (g) = SiO 2 (s) how to calculate the enthalpy change of the reaction seems to consider the spatial structure of Si and SiO Let silicon be a mol
- 8. It is known that the kinematics equation of a particle is x = 10cos (π T), y = 10sin (π T), (SI unit)
- 9. Given the kinematics equation of a particle, how to find the trajectory equation of a particle?
- 10. On the chemical equation of Na and the equation of compound containing Na Na containing compounds can appear on both the reactant side and the product side,
- 11. The quadratic equations of one variable with two sum and two product of x ^ 2 + 3x-2 = 0 as roots are?
- 12. Judge the number of roots of equation 2 ^ x = - x + 1D
- 13. Find the equation (one variable one time), and the relationship 1、 The distance between the enemy and our army is 25 km. The enemy runs away at the speed of 5 km / h. our army pursues at the speed of 8 km / h at the same time, and fights at the distance of 1 km. How many hours after the pursuit begins? 2、 A and B practice running. A is 7 meters per second, B is 6.5 kilometers per second. If B runs 8 meters first, how many seconds can a catch up with B? 3、 A pair of students had a military exercise. They were traveling at a speed of 5 km / h. after 18 minutes, the school sent a notice to the team leader. The correspondent rode a bicycle 14 km / h to catch up with the students. How many hours can he catch up with the students? 4、 It takes 25 seconds for a train to enter a 300m tunnel at a constant speed. A fixed lamp on the top of the tunnel shines vertically on the top of the train for 10 seconds. The length of the train can be calculated 5、 A circular runway is 390m long, 6m / s in a and 7m / s in B 1. How many hours have they met for the first time in running at the same time, on the same ground and in the same direction? 2. How many hours have they met for the first time? Question 2 B run 6.5km per second
- 14. Super difficult mathematical problems (solved by equation of degree 1 with one variable) 1. The train was blocked in the middle of the way and delayed for six minutes. Then it increased its speed from 40 kilometers per hour to 50 kilometers per hour. How many kilometers can we make up for the delay? 2. A and B set out from a to B at the speed of 4 kilometers per hour at the same time. After 2.5 kilometers, a had to go back to get a document. He went back at the speed of 6 kilometers per hour and chased B at the same speed. As a result, they arrived at B at the same time. It was known that a had been delayed for 15 minutes in the office to get the document, so they could find the distance between a and B 3. Party A and Party B process 284 parts. Party A processes 48 parts per hour, and Party B processes 70 parts per hour. Party a first processes one hour, and then Party B cooperates with Party A. how many hours has Party B finished the task? Please write the process! Thank you~~~~~~
- 15. A quadratic problem of one variable, just write the equation A three digit number, its one digit number, ten digit number and hundred digit number are three consecutive increasing integers, and the sum of square of one digit number and hundred digit number is five times of ten digit number. Find the three digit number, let the ten digit number of this number be x, please list the equation according to the meaning of the question
- 16. It is known that a and B are the two roots of the equation x, where the square minus 5x minus 3 is equal to 0. A new bivariate linear equation is obtained, and its two roots are a + 2B and a + B
- 17. Headache help me The parabola y = ax ^ 2 + HX + C intersects with the negative half axis of X axis. The positive half axis intersects with A.B The positive half axis of y-axis intersects with C and ob = 2oC = 2oa 1. Find the value of the algebraic expression ABC 2. If the line Y-X + B passes through point C For all real numbers, the value of the algebraic formula ax ^ 2 + BX + C is not greater than 9 / 16 HELP I have already Give it to whoever leaves a message arbitrarily..
- 18. Speed help 2-2²-2³-…… -2 ^18 -2^19+ 2^20 ^To the power of
- 19. What skills do you have in the second chapter of the first volume of mathematics? In terms of growth rate
- 20. To solve it with a quadratic equation of one variable, as follows, we need the equation and the answer 1. A company sold 200 computers in August and 242 computers in October. What is the average monthly growth rate in these two months? 2. If the square of X - 2 (M + 1) x + m square + 5 is a complete square, then M =? 3. Cut a square with 5cm side length from each corner of the rectangular iron sheet, and then fold it up to make a box without cover. The length of the iron sheet is twice the width, and the volume of the box is 1.5dm square, then the length of the iron sheet is equal to? The width is equal to? 4. If the ratio of the two right sides of a right triangle is 8:15 and the length of the hypotenuse is 6.8cm, then the area of the right triangle is equal to? 5. There is a 60 m long and 40 m wide rectangular lawn. A small rectangular flower bed should be opened in the middle of the lawn to make the width of the lawn around the lawn the same, and make the area of the flower bed account for half of the area of the lawn around the lawn?