Mở đầu về phương trình - Bài 1 - Toán học 8 - Cô Phạm Thị Huệ Chi (HAY NHẤT)

Hello everyone! On the board, I have an old calculation that we are familiar with since 6th grade. It is both chicken and dog. 36 chickens, 100 dogs. How many chickens and dogs are there? In 6th grade, we made this exercise quite difficult. However, with the 3rd grade of 8th grade, we will learn a knowledge to bring this difficult calculation back to the basic calculation. That is the content of the 3rd grade. In the 3rd grade, we will learn about the most complex hidden equation. In this class, we will learn about the following contents. The first is the most complex hidden equation and how to solve it. The second content is to learn about the equation given in the form ax plus b equals 0. With our condition that a is different from 0. The third content is the TIC equation. The fourth content is the hidden equation in the model. And the last content is to solve the equation by setting the equation. At that time, you will be able to solve the previous old equation in a very simple way. Now, let's go to today's lesson. The first lesson in this class is to start with the equation. Of course, we have to understand what the equation is. Let's go to the small part, which is the hidden equation. I give the following equation. Find x. Know 2x plus 5 equals 3 times x minus 1 plus 2. To solve this equation, how do we do it? Can we find it easily? Obviously, these are very basic finding equations that we have learned before, right? We will find it by multiplying it, right? Multiply with the right side. We will get 2x plus 5. Multiply with the left side. 3x minus 3. 3 times 1. Take 3 times x minus 3 times 1 plus 2. After that, we will convert all x to 1 and all numbers to 1, right? Convert all x to 1, we will get 2x. Convert 3x from the right to the left, we will get minus 3x. Minus 3 plus 2 is minus 1. Convert 5 from the left to the right, we will get minus 5. Then we get 2x minus 3x is 2. 2 minus 3 is minus x. Minus 1 is minus x. Minus 1 minus 5 is minus 6. So x is equal to 6. And then we will call the equation 2x plus 5 equals 3 times x minus 1 plus 2. It is an equation with hidden x. We call it hidden number x. And we summarize. The equation with hidden x will be ax equals bx. In it, the left side ax and the right side bx are two expressions of the same variable. Remember, the same variable. In addition, we can handle equations with hidden y, z, t, etc. Then we will replace it. This will be at, bt. If we set the equation with hidden t, it will be ai, bi. If we set with hidden y, that is, where there is x. For example, if I replace x with y, I will say that this is the hidden y equation. Do you understand? Let me remind you again. With the hidden x equation, we remember that there will be an equal sign in the middle. The two sides will be the left and right sides. The left and right sides are all expressions with hidden x. Then we call the equation with hidden x. Can you give me some examples of other hidden x equations? For example, the equation 2x plus 1 equals x. This is called the hidden x equation. Because we only have x, we need to find it. The second equation is 2t times 5 equals 3 times 4 times 7. Here we see that there is a hidden t, right? I will call this hidden t equation. Or now I don't like hidden t anymore. I will give an equation with hidden y. For example, 3y plus 3 equals 2 times y minus 1 plus 4. This is the equation with hidden x, hidden t, hidden y in this order. Let's go to the second example. I have 3 equations here. Let's find out which of these 4 equations is the hidden equation. First, 2x plus 3 equals 4x minus 2. Is this the hidden equation? Yes, it is. Because there is only one hidden t, hidden x. Let's go to the second equation, y plus 3 equals 2y. This is also the hidden equation. And this is the hidden equation y. Let's go to the third equation, 3u minus 1 equals u plus 5. Is this the hidden equation? Yes, it is. It is still the hidden equation. But now we set with hidden y. Let's go to the third equation, 3x plus y equals 5x minus 3. This is not the hidden equation. Why? Because we see that there are 2 hidden t, hidden x and hidden y. So this is no longer the hidden equation. So in summary, we have 3 first equations, which are the hidden equations. And the last equation is not the hidden equation. The small b part is the derivative of the equation. Let's look at the following example. For example, 3, the derivative of the equation is 2x plus 5 equals 3 times x minus 1 plus 2. It is still the hidden equation, right? We will set with x equals 6. Please calculate the value of each side of the equation. The left side is our left hand side. It is called the left side. And the right side is our right hand side. Now when x equals 6, can you calculate the value of each side? Can you? It's simple. We just need to replace x with 6, right? Now with x equals 6, replace x with 6. On the left side, we will replace each side. On the left side and on the right side, we can write brackets and write flowers. On the left side. 2x plus 5 equals 2. 2 times 6 plus 5. 2 times 6 plus 5 equals? 2 times 6 is 12 plus 5, right? 17. Continue. Replace it with the right side. Replace x with 6. On the right side. OK. On the right side. Now we will get 3 times 6 minus 1 plus 2. Wherever there is x, we replace it with 6. 6 minus 1 equals? 5. 3 times 5 equals? 15. 15 plus 2 equals? 17. So we see that when x equals 6, the left side is equal to the right side, right? When the left side is equal to the right side like this, we will say that x equals 6 satisfies this equation. And we will say that 6 is a function of that equation. Or x equals 6 is a function of that equation. Do you understand the concept of function? When that number is replaced by this equation, I will call it the function of the equation. Today we will get used to this new concept. And from now on, we will meet a very familiar equation in this lesson. That is the solution of the equation. Before that, we have to decide whether the solution of the equation is or the solution of the equation is. Example 4. Give the equation 2 times x plus 2 minus 7 equals 3x. Part A. People ask us if x equals 2 satisfies the equation or not. To check if x equals 2 satisfies the equation or not, how do we do it? We will replace it with the left side and the right side, right? To check whether the left side is equal to the right side or not. If it is equal, we will conclude that x equals 2 satisfies the equation. And if it is not equal, we will conclude that x equals 2 does not satisfy the equation, right? So we will replace part A. Replace x equals 2 with the left side. We get the left side. This is the left side, right? We will replace 2 with 2 plus 2 minus 7. 2 plus 2 equals 0. 2 minus 0 equals 0. 0 minus 7 equals 7. Then we replace x equals 2 with the right side. We get the right side. x equals 3 minus 2. Please note that I put a square here because it is a minus sign. 3 minus 2 equals 3 plus 2, right? 3 plus 2 equals 5. So we see that the left side is different from the right side. So the left side is different from the right side. And we conclude that x equals 2 does not satisfy the equation. x equals 2 does not satisfy the equation. Because we use the equation a lot, so we can satisfy it at home. But in the test, we cannot satisfy it. Part B. Part B, they asked us if x equals 2 is a function of the equation or not. Do you remember the concept of function that we just learned? To check if x equals 2 is a function of the equation or not, we still do the same as part A. If it satisfies, we call it a function. If it does not satisfy, we do not call it a function. Now we can also display it in the same way as part A. And we only change the conclusion that x equals 2 is a function or not a function of the equation. The second way we can display it like this. We replace x with 2 in the equation. This is our second way to display it. We will see where x is replaced by 2. So 2 times 2 plus 2 minus 7 is equal to 3 minus 2. And here we calculate with left and right separately and write below. 2 times 2 plus 2 is equal to 4. 2 times 4 is equal to 8. 8 minus 7 is equal to 1. Equivalent to 1 equals 3. 3 minus 2 is equal to 1. 1 equals 1 is always correct. Let's open it up for her. Always correct. Then we will conclude that x equals 2 is a function of the equation. This is our second way to display it. The first way she mentioned, we will still write that replacing x with 2 on the left side is ok. We replace it. Replacing x with 2 on the right side is ok. We replace it. After that, we calculate that the left side is equal to the right side. And we will conclude for her. So x equals 2 is a function of the equation. And the second way is that we will display it as she displays this part. We will write it down for her to pay attention later. To pay attention to some necessary things. The first is the formula x equals m. With m is some number that people can give us. It is also a formula with hidden x. Remember, hidden x, not hidden m. m is some number. At that time, people can call m with another name. That is the number. And our hidden is only hidden x. Just pay attention to her. This formula shows that m is the only function of the equation. With the formula x equals m, m is a function. For example, we have x equals 2. 2 will be the function of the equation we just said. That is the formula x equals 2. Part b. A formula can have 1 function, 2 functions, 3 functions and can also not have a function or have an infinite number of functions. Therefore, in the previous examples, she always said that x equals 2 is 1 function of the equation. She did not say that the function of the equation or only 1 function is a common function. That is, we have not found everything. In addition, it may have 1, 2, 3, 4 or so. It may also not have a function. If it does not have a function, people will call it an infinite equation. Remember, if we solve it without a function, we will conclude that it is an infinite equation. For example, with the formula x equals 1, we will solve it with the formula x equals 1. Remember, if we square it with a square, we will get 2 results, which are 2 opposites of each other. We learned that in the 7th grade. Then we will get x equals 1 and x equals negative 1. That is, our equation has 2 functions. It is not that the equation has only 1 function. The second equation is the equation x equals negative 1. Note that the square of a number is always 0. That is, it is greater than or equal to 0. Now, x equals 1 will be irrational. So this equation is an infinite equation. And we conclude that it is an infinite equation. We will not say that the equation has no function. Because we have a shorter term. And everyone knows that it is an infinite equation. Now, let's move on to the second part of solving equations. We will come up with the first knowledge, which is the knowledge of coefficients. The coefficients will be all the coefficients of that equation. What is a coefficient? We learned in part 1. And the coefficient will be marked by the letter S. Remember the letter S. For example, 5. The equation x equals 2 has a coefficient. I just finished talking. The formula x equals 2 will have a coefficient of 2, right? Therefore, the coefficient will be S times the coefficient of 2. The coefficient must have the coefficient sign in it. If there is no coefficient, and you write the coefficient, it will be wrong. So there must be a coefficient sign in it. The second example is an infinite equation. How do we write the coefficient? We will write S as a column. And we do not add the coefficient sign to this column. Because we already know that this column is a column, so we will only write the column. The coefficient S equals the column. For other numbers, we will add the coefficient sign. That's her idea. And she also explained to us a little bit why we mark the coefficient as S, but not the other letters. This comes from English. Set of solution. People will mark it as S. And then it's normal. In the book or in the reference books, even her book, she will also mark the coefficient of the equation as S, but not the other letters. And when the equation requires us to solve the equation, we will have to find all the coefficients of that equation so that we have enough points. If, for example, that equation has 3 coefficients, but we only find 1 coefficient, then we only get 1 third of the points. We have to find all the coefficients and conclude that the equation of the equation is S equals and write the combination of those coefficients. The third item is the equation of equations. Two equations of equations, when those two equations have the same equation. And we will write it with the sign like this. In the first period, when we solve the equation, we also often write the equal sign on the top. That's the reason. In this book, we will find out why I wrote the equal sign on the top like that. And we remember that two equations are only equal when they have the same experiment. So that we can go and handle the calculations. Especially for those who will take the test later, you need to pay more attention to the equation of equations. After the 9th grade, you will learn more about the equation of equations. We still understand that they have the same experiment. It will be like that. For example, the equation x equals negative 1. This equation of x equals negative 1 will have the experiment of S equals the combination of negative 1. Right? It's simple. Continue to the equation x plus 1 equals 0. We see that x plus 1 equals 0. x will be negative. x equals negative 1, right? So this equation will also have the experiment of S equals negative 1. The combination of negative 1. Then we will say that the equation x equals negative 1 is equal to the equation x plus 1 equals 0. And we will sign that x equals negative 1 is equal to x plus 1 equals 0. Let's add 0 here. I made a mistake. That's enough. So, in this lesson, we have learned the knowledge of the hidden equation. Then we learned about the concept of probability, right? The concept of probability, the error of probability and the equation of equations. Now let's go to the supplementary lesson. Apply and make the lesson 1 in Chinese Literature book 6. The question is, for each subsequent equation, let's see if x equals negative 1 is the probability. This lesson will be like the example that you have just taught us. To see if x is negative 1, we will replace x with negative 1, right? And we have two ways to prove it. First, we will replace it with the left side, then replace it with the right side and check to see if the left and right sides are the same or not. The second way is to replace it with the equation and check to see if the equation is correct or not. If it is always correct, then we conclude that it is negative. If it is irrational, then we conclude that it is not negative. And now, I will present it in two ways. And you guys on the class can depend on the teacher's request in each case. We will present it according to the teacher on the class. Part A. I will replace x with negative 1 in the equation. In the equation, it is 4x minus 1 equals 3x minus 2. Or 4 times x, which is 4 times negative 1. Then minus 1 again. Equals 3 times negative 1 minus 2. And we check to see if it is correct or not. 4 times negative 1 equals 4. 4 times negative 1 equals 5. So we get 5 on the left. On the right, 3 times negative 1 equals 3. 3 times negative 2 equals 5. 5 times 5 equals 5. It is always correct, right? Always correct. So we conclude that x equals negative 1 is the negative of the equation. So, x equals negative 1 is the negative of the equation. The equation has a negative sign. Part B. Similarly, can you do Part B for me? Let's go again. Ah, replace, right? I should do it myself and then compare it with your result. Replace x with negative 1 in the equation. Because there are 3 equations here, we should also write it clearly. x plus 1 equals 2 times x minus 3. OK. We replace it with minus 1 plus 1 equals 2 times minus 1 minus 3. Minus 1 plus 1 is OK. Ah, no, right? I calculate it correctly. Minus 1 minus 3 is OK. Minus 4. Minus 4 times 2 is OK. Minus 8. So, is it correct or wrong if it is not equal to 8? Ah, it's wrong. Then we will write it as irrational. And we will conclude that x equals negative 1 is not the negative of the equation. The last part, Part C. Part C, when we replace it, I will skip it. And we will display it like Part A and Part B. When we replace it, we will get 2 times minus 1 plus 1 plus 3. I see that minus 1 plus 1 is OK. No, 2 times 0 is OK. No, 0 plus 3 is OK. 3, so the left side is 3. The right side will be 2 minus x. That is, 2 minus minus 1 is also 3. So the left side is equal to the right side. So we conclude that x equals negative 1 is the negative of the equation in Part C. So, in Part A and Part C, x minus 1 is negative. In Part B, x minus 1 is not negative. Let's go to Part 2. The requirement of Part 2 is that in the values t minus 1, t minus 0, and t minus 1, which value is the negative of the equation t plus 2 equals 3t plus 4? Here we see that our equation is a hidden equation with a hidden value of t. Similarly, in Part 1, we will replace the values t minus 1 to see if the left side is equal to the right side or not. If it is equal, we will conclude that it is negative, right? And here I will do for us the part t minus 1 and t minus 0. For t minus 1, let's replace it and do it ourselves. First, we will replace t minus 1 in the equation. When replacing it, we will get minus 1 plus 2. Ah, that's right. Equal to 3 times 1 minus 1 plus 4. Minus 1 plus 2 is equal to 1. 1 square is equal to 1. Equal to 3 times minus 1 is equal to 3. Minus 3 plus 1 is equal to 1. Is that right or wrong? Ah, always right, right? And we will conclude that x is equal to t. t minus 1 is the denominator of the equation. Continue. In the case of t minus 0, we also write like that. Replace t minus 0 in our equation. I will replace it. Where is t minus 0? t minus 0 is equal to 0 plus 2. Equal to 3 times 0 plus 4. We see that 0 plus 2 is equal to 2. 2 times 2 is equal to 4. So I get 4 on the left. And on the right, 3 times 0 is 0. 0 plus 1 is 4. So I also get 4. 4 times 4 is always correct. And we also conclude. So, t minus 0 is the denominator of the equation. And in the last case, t minus 1, we will do the same. Let's do it and comment on the result below to see if t minus 1 is the denominator of this equation or not. The third lesson of our vocabulary is lesson 3 in the textbook chapter 6. The lesson says that the equation x plus 1 is equal to 1 plus x. We see that all the numbers are negative. So why are all the numbers negative in this equation? Because we see that the left and right sides are exactly the same, right? That is, we see that x plus 1 is equal to 1 plus x is always correct. Also, people will say that all the numbers are negative. Then people will say that this equation has a negative value equal to all x. And people ask us to show the negative part of the equation. When an equation is always correct like this, people say that this equation is negative equal to all x. And that is, we will understand with x how much this equation is also correct. That is, this equation is correct with all x. So what is the experiment of the equation? We have learned the combinations. The combination of natural number, original number, negative number, positive number, real number. So what do we conclude here? Ah, because it is correct with all x, so we can conclude that the experiment of the equation is S by R. R is the real number combination. And you guys remember that when we say that the equation has a negative value like this, or say that the equation is negative equal to all x, then we will conclude that the experiment of the equation is S by R. Please note that we can have a correct conclusion in other exercises. Go to lesson 4. Lesson 4, we are lesson 5 in the book Giáo Khoa Trang 7. The two equations are x equals 0 and x times x minus 1 equals 0. People ask if these two equations are equivalent or not. Why? And we have to remember the concept of two equivalent equations. When are our two equations equivalent? Ah, when they have the same experiment. So for the equation x equals 0, what is the experiment of this equation? What is the experiment of the equation x equals 0? It is S by S1. S by S1 is the real number combination of 0. There is only one number 0. Next, the second equation is x times x minus 1 equals 0. We can find x equal to 0. Just like the way we find x from grade 6 to now, we can find x in this problem. We have x times x minus 1 equals 0. When is it equal to 0? When x equals 0 or x minus 1 equals 0. So we will have x equals 0 or x minus 1 equals 0. This is the way in grade 7. We often write x equals 0 or x minus 1 equals 0. In case 2, x minus 1 equals 0. This is the way to solve this equation. And in grade 8, we will learn more about this equation in the following lessons. Now, I will solve for you. x equals 0 means x equals 0. Or x minus 1 equals 0 means x equals how much? x minus 1 equals 0, then x is equal to 0. Now, I will solve for you. x minus 1 equals 1. We have found the second equation. And we can see that the second equation has the combination of 2 parts 0 and 1. Now, let's see the second equation. The second equation is S2 equals the combination of 0 and 1. And we can see that S1 and S2 are the same. They are the same when they have the same experiment. So we can see that S1 and S2 are different, right? S1 has only 1 part 0. S2 has 1 part 0 and 1 part 1. So we can see that these 2 equations are different. S1 is different from S2. Or we don't need to write this sentence. And we will solve for you. So, 2 equations are not equal. This is the way to solve a problem. Check if 2 or 3 equations are equal to each other. Then we will find the experiments of those equations. From there, we will compare whether their experiments are equal or not. If they are equal, then they are equal. If they are different, then they are not equal. The last problem is the 9th problem in the textbook. The problem for equations. We can see that there are both m and x. But people say that m is a number. So we know what is the hidden equation. Hidden x. Remember. Hidden x. It's not that there are 2 hidden equations, hidden x and hidden m. People say that m is a number. So we consider this as a hidden x equation. Then people ask us to prove that Part A is when m is equal to 4, the equation is true for all hidden values. That is, for any value of x, our equation is satisfied. That is, we will have the experiment A, F, R. Remember. Now, when people give m as 4, we will definitely have to replace it with the above equation to see if it is correct or not. This equation is quite long, so I limit the mention of the equation. I will sign this as the following equation. In Part A, I will replace m with 4 in the following equation. I will replace it. m is equal to 4, plus 5 times 4, plus 4. Close the square. Because the inside is a circle, we will replace the outside with a circle. With x, m is equal to 4. That is, minus 4 plus 4. Now let's calculate. How much is minus 4 squared? Many of you will still write the result as minus 16. But minus 4 squared, we have to take it as minus 4. Minus 4 times 4, the result is 16. Close the square. Any number that is squared, we also have to get the result as negative. Even if that number is negative, the result of the squared number is also negative. So I get 16 here. 5 times 4 is equal to minus 20, right? That is, we will add with minus 20. Or I will write minus 20 like this. Plus with open square minus is equal to minus. Plus 4. So the square becomes a circle. Minus 4 times 4 is equal to 0. Now let's calculate this side. 16 minus 20 plus 4, we also get the result as 0. That is, I will get 0 times minus 0. Is this right? Ah, always right, right? Because 0 times any number is equal to 0. So x is equal to how much? Still right. So here we will write always right. And we will conclude that. I will conclude according to the assignment requested. Not conclude that the experiment of the equation is. I will conclude that the equation is true with all values ​​of x. Because our hidden value is hidden x. The equation is true with all values ​​of x. Then we can sign like this. This is called the sign with all. Write a word like the letter V. But we pull it up. And cross it. This is called the sign with all x. Or cross it. Go to part B. Continue. When M is equal to minus 1, the equation is infinite. That is, we go to replace it, right? Part B. Replace M by minus 1. Into the following equation. We can. The rule is a little bit. M is equal to minus 1, right? Minus 1 all x. Plus 5 times minus 1. Plus 4 square. Times x square. Equal to minus 1 plus 4. Come here. You can type or use a computer. This part is the same. Minus 1 square. But I just type it. Minus 1 square is 1. 5 times minus 1 is minus 5. 1 plus minus 5. That is, 1 minus 5. Then I get minus 4. Minus 4 plus 4. No, so we get. 0 times x square. Minus 1 plus 4. Get 3, right? So is this result correct? I know that 0 times any number is equal to 0. But now people ask to find x square. To make 0 times x square equal to 3. Can I find it? Ah, no, right? So this is nonsense. And then you will conclude that So, the equation is invalid. With m equal to minus 1. This part is added with m equal to minus 4. So we have solved part A and part B. Do you understand how to do this problem? Similarly, in part C and part D, we will also replace the value of m. See what equation it is. In part C, we see that m is equal to 2 or m is equal to 3. There are two cases here. And we repeatedly replace each value. First, we replace the value with m equal to 2. Replace m with 2. In the equation, we get. When replacing, what do we get? 2. 2 square, right? And we note that we have to plug in the square. Because I have the letter in front. Plus 5 times 2 plus 4. Then the square will automatically become a square. Times x square equal to m plus 4. That is, minus 2 plus 4. Then we can calculate. Minus 2 square, how much is it? Minus 2 square, that is, minus 2 times minus 2. Then we have to get the result as a number. An unmarked square is minus 4. It must be 4, 4. 5 times minus 2 is minus 10, right? Minus 10 here plus 4 more. So the result is minus 2. Minus 2 x square. Minus 2 plus 4 is equal to 2. In addition, you can plug in this part of the computer for me. No problem at all. And now we have to remember that when a number is squared, the result is also 0. So x square is 0, right? 0 times 1 square, can it be a square? Ah, never, right? 0 times 1 square, the result must be negative, right? I have to explain why. Because x square is greater than 0, with all x, minus 2 x square, right? Ah, less than or equal to 0. But 2 is greater than 0. Remember that this is equation 1, so we get equation 1, infinite. No, it's irrational. So it's infinite. In equation 1, is our first equation infinite? Ah, yes, right? Because it's equal. So, equation ... Oh, what is the name of this equation? Or we will write it as the given equation. Given equation with m equals 2. Similarly, we will solve it like this, with m equals negative 3. Now, we substitute with m equals negative 3. Substitute m equals negative 3 into the equation, we get ... Substitute into what? Negative 3. Negative 3, all square, right? Plus 5 times m equals 5, minus 3 plus 4, then the square becomes a square, with x equals minus 3 plus 4. Now, let's calculate this. Minus 3 times square equals ... Ah, 9, right? Minus 3 times minus 3 equals 3 times 3. 9 minus 15 is ... Ah, 6. Minus 6. Minus 6 plus 14 is ... Minus 2. So, we get minus 2 x square. Equal ... Minus 3 plus 4 is ... Ah, 1, right? Is it similar to the previous part? Ah, very similar, right? It is still a negative number times a negative number, but the result is a positive number, so this is ... unreasonable. So, this equation is infinite. So, the equation is also infinite. And let's show it again. Because x square is greater than or equal to 0 with every x, the sign with every ... So, minus 2 x square will be less than or equal to 0. So, there is a sign here, equation 2. Equation 2, infinite. We can write that M is greater than or equal to 0, or we don't write it, it doesn't matter. Because it is clearly shown above. And we conclude that ... the equation has given ... infinite with M equal to minus 3. So, we have finished equation C. With equation D, we just need to replace M with 0, right? With equation D, we just need to replace M with 0. Even if you don't know how to handle the following equation, we will still show this part for you. Replacing it, we will get ... 0 square, right? Plus 5 times 0 plus 4, right? Times x square equals 0 plus 4. What will we get then? 0. Plus 5 times 0 is 0 again. So, there is only ... 4 x square equals 4. So, how do we calculate x square? We will take 4, divide it by 4, right? And the result is 1. x square equals 1. So, how much will x be equal to? Like the example that she has guided us, remember equation 7. x square equals 1 when and only when x shoots. Remember, x equals plus or minus 1. Because 1 square is equal to 1, minus 1 square is also equal to 1. But we often forget her sign. But we have to remember that when a square is multiplied by a diamond, we can find the value of the number we need to find, which is 2 values ​​equal to each other. So, we have to add a plus or minus sign. Then, we will get the equation x equals 1 and x equals minus 1, right? Here, plus or minus 1. And we will conclude like this. The equation means x equals 1 and x equals minus 1 with m equals 0. So, we have finished this lesson. Do you understand this function? Our lesson today is over. Thank you. If you have any questions, please comment below. I will answer as soon as possible. See you in the next lesson.