Students you can watch my all the videos in the playlist Students you can watch my all the videos in the playlist All the playlists are arranged topic-wise so you will not get any problem. Because many students complain of not getting this topic or that topic. So, all the topic are arranged playlist wise so you can watch any topic you want Hello students, today I am here with a new topic. Here we will discuss and learn about the Reciprocal Equation under the topic Theory Of Equation the Reciprocal Equation under the topic Theory Of Equation in the previous video. In which we had discussed how to solve the reciprocal roots of an equation. Now we will learn what is a reciprocal equation. And how to solve the roots of a reciprocal equation. And how to solve the roots of a reciprocal equation. So students, I am Dr. Gajendra Purohit and here you are watching my YouTube Channel where content related to engineering, mathematics and B.Sc and competitive exams like CSR, NET, GATE, IIT JAM are available. From last 1.5 Year I am providing a successful content for you. So students, if you find my content useful then please subscribe my channel and press the bell icon to get notified. And don’t forget to comment and like so that I can share such valuable content with you. So students, now we will get to know what is a reciprocal equation and how we solve the roots of a reciprocal equation. And students you can also watch the other videos by clicking on the “i” icon And students you can also watch the other videos by clicking on the “i” icon like engineering, mathematics, B.Sc and all the related topics. And there are many other videos on the topic “Theory Of Equation” so you can get a thorough learning of this topic as well. So students, what is a reciprocal equation? An equation which remains unaltered by changing X into 1/X is called a reciprocal equation. This means in any equation if we keep 1/X instead of X there should be no change. That is a reciprocal equation. Let’s take an example. X2 + 2x + 1=0 X2 + 2x + 1=0 This is an equation. If I keep 1/X in place of X, be attentive… 1/X in place of X, so you can see here If we find our LCM and simplify X that becomes 1/X2. Then again on simplifying this equation, it becomes the original equation without any changes. The equation we have got now is as same as the previous one. So such type of equation is called a reciprocal equation. Clear students. So on keeping 1/X in place of X, there will be no change such type of equation is called reciprocal equation. But the question is how we will get to know that a particular equation is a reciprocal equation or not. So students, there is a unique way to know this. If the co-efficient of X power of 5 is equal to the co-efficient of 1. Also, the co-efficient of 5x to the power of 4 is equal to the co-efficient of 5x. And the chain goes on. I hope it is clear. The + or – difference will be discussed further. First we should understand this. Like the second one in which the co-efficient of the first value and the last value is same. And the co-efficient of second and last second values are same. Also the co-efficient of third and last third values are same. So, this is also a reciprocal equation. See the third example. The co-efficient of first and last values are same. The co-efficient of second and last second values are same. Then the co-efficient of third and last third values are same. So such type of equation is called reciprocal equation. You don’t need to put these signs and check. You can check the question by this type also and find out whether the equation is reciprocal or not. Now comes the kinds of reciprocal equations. I will explain you the 1st and 2nd kinds of reciprocal equations. For example, in the first equation the co-efficient of the first value and the last value is same but the signs are different, one side plus (+) and the other side minus (-) This is the type of 2nd kind of reciprocal equation. This is the type of 2nd kind of reciprocal equation. Because the coefficient are same but the signs are opposite. That’s why it is 2nd kind of reciprocal equation. Now see the second example. The co-efficient and signs of first and last value are same. If this is + 6 that is -6. Also if this is -25 that is +25. If this is +31 then that is -31 So, this is again second kind of reciprocal equation. So, this is again second kind of reciprocal equation. Now see the third example. Here this is 6 so that is also +6. This is +1 so that is also +1. Here it is -43 so there also -43. So the co-efficient of 6x to the power of 5 is same as constant co-efficient. So the co-efficient of 6x to the power of 5 is same as constant co-efficient. Then the just next degree and this degree are same. So, here the signs are same. Such type of reciprocal equation is of the 1st kind. So, here the signs are same. Such type of reciprocal equation is of the 1st kind. So I think now the kinds of reciprocal equation are clear to you. First we had discussed what is a reciprocal equation. An equation which remains unaltered by changing X into 1/X is called a reciprocal equation. An equation which remains unaltered by changing X into 1/X is called a reciprocal equation. Then we had discussed the kinds of reciprocal equations. If we have to check the reciprocal equation, then I had told you the co-efficient of first and last values will be same. That means alternate cycle. If the co-efficient are of same sign, the equation is of 1st kind. And in case of opposite signs, the equation is of 2nd kind. Now here are some properties. These properties are very important and will help you when we will go on for practicals. A reciprocal equation of first kind that means when the signs are same, this example shows the first kind, where the co-efficient of first and last values are same. And the co-efficient of second and last second values are same. And the co-efficient of second and last second values are same. Also the co-efficient of third and last third values are same. These are of same signs. I hope this is clear to all. This is called the 1st kind of reciprocal equation. If there is any 1st kind reciprocal equation, then it’s root will always be X=-1 An odd degree that means a reciprocal equation with an odd degree equals to root as -1. This means when we place -1 in place of X, it satisfies the equation. This means when we place -1 in place of X, it satisfies the equation. I hope this is clear… Now we will talk about reciprocal equation of second kind. Second kind of reciprocal equation means the co-efficients will have opposite signs. Numeric values will be same. The signs will change. This is second kind of reciprocal equation with odd degree. Power odd… Highest degree is odd i.e 5. So here the root is always X=1 In the first kind root is always X=-1 and in the second kind the root is always X=1. and in the second kind the root is always X=1. A reciprocal equation of second kind of an even degree. See now comes the second kind with an even degree. It is X6. And this is of second kind. See, the first value is positive and the last value is negative. See, the first value is positive and the last value is negative. -25 and this is +25. This is +31 and that is -31. So, in this case, X=1 and X= -1, both are the roots. X= -1, both are the roots. So, when we solve a reciprocal equation, we will solve one root like this. We will also solve questions here. Because there will be questions to solve such questions on reciprocal equations. So we will solve few questions in which we have to answer which kind of reciprocal equation is this. So we will solve few questions in which we have to answer which kind of reciprocal equation is this. or whether the degree is odd or even. So students I think the concept is now clear to you. If the reciprocal equation of of first kind, then the root will be -1 If the reciprocal equation of of first kind, then the root will be -1 this is in odd degree. Now reciprocal equation of second kind with odd degree, then root will be X=1 And reciprocal equation with even degree of second kind then the root can be both X==-1 and X=1 So students now we will understand how to solve reciprocal equation. So students now we will understand how to solve reciprocal equation. Solve following reciprocal equation This is a reciprocal equation in even degree So you have to find out whether it is of first kind or second kind it is of first kind or second kind it is of first kind or second kind So students see in this there is constant co-efficient and we have +1 and -10 and -10 and then +26 like it is equal so it is of first kind of reciprocal equation of even degree and first kind of reciprocal equation with even degree has no root even degree has no root The properties that I had shown you before in that if even degree has the second kind then X=1 and X=-1 is the root but in this case it is not that situation, so we have to solve it. So what we will do is Divide by X2 by X2 by X2. What we will do is divide the complete equation by X2 so we will get X2 here we will get 10x. Here we will get 26 Here we will get -10/x +1/x2=0 All we have to do is divide the complete equation by X2 Every value, we have to divide by X2 Every value, we have to divide by X2 now when we divide, we will get (X2+1/X2) (X2+1/X2) now here we will get -10(X+1/x)+26=0 -10(X+ -10(X+1/x) -10(X+1/x)+26 -10(X+1/x)+26=0 (X2+1/X2)-10(X+1/x)+26=0 (X2+1/X2)-10(X+1/x)+26=0 Now, X2+1/x2 X2+1/x2 (X2+1/x2+2) (X2+1/x2+2) (X2+1/x2+2)-10(+1/x)-2 (2+1/x2+2)-10(x+1/x)-2+26 (2+1/x2+2)-10(x+1/x)-2+26=0 (2+1/x2+2)-10(x+1/x)-2+26=0 (2+1/x2+2)-10(x+1/x)-2+26=0 Now you might think that sir knows everything He knows what do do and which numerical to be added or subtracted which numerical to be added or subtracted So students, I just want to make a perfect square X+1/X2 i.e. a2+b2+2ab So students, I just want to make a perfect square X+1/X2 i.e. a2+b2+2ab so that gives x cut by 1/x So we will do +2 and -2 That gives (x+1/x) whole square (x+1/x) whole square- 10 (x+1/x) (x+1/x) whole square- 10 (x+1/x)+ 24 (x+1/x) whole square- 10 (x+1/x)+ 24=0 (x+1/x) whole square- 10 (x+1/x)+ 24=0 there is a quadratic equation in (x+1/x) so what we will do is now let (x+1/x)=y now we put the values square of Y – 10Y square of Y – 10Y + 24=0 and now when we solve it there is a quadratic equation in Y so we can find out the roots easily Now when we find out the roots it will come 6 and 4 here it will be (y-4) and there (y-6)=0 now we can do it by 2 types one is Y=4 and another is Y=6 Now see students, what we have to do is the value of Y is 4 so pick 4 and put it in place of Y so that will give x+1/x=4 x+1/x=4 and the another one will be x+1/x=6 now we will solve it easily. It gives square of x + 1=4x then comes square of x – 4x + 1=0 in the same way when we solve this, it gives square of x + 1 square of x + 1=6x which gives square of x – 6x + 1=0 So students, here is a quadratic equation and to solve this quadratic equation we will use a rule quare of ax + bx + c=0 value of x is -b +- under root square of b – 4ac / 2a -b +- under root square of b – 4ac / 2a so therefore the value of x is 4 4 +- under root 16 – 4 4 +- under root 16 – 4 4 +- under root 16 – 4 4 +- under root 16 – 4 4 +- under root 16 – 4 4 +- under root 16 – 4 divided by 2 so we get x=4+- that gives 12 so we can write 2 root 3 divided by 2 so we will get the value x=2+- x=2+- root 3 now we will solve the other part so that gives x=6+- square of B under root 36-4. A and C are 1 and 1 divided by 2 so, x=6+- that gives 32 we can write 32 as 4 root 2 4 root 2 4 root 2 4 root 2 divided by 2 then we get x=3+- x=3+- 2 root 2 This way we can find all the 4 roots. so students here we can solve this question by a very simple method what we have to do is 2+- root 3 and 3+- 2 root 2 will be the answers. so this way we can solve the reciprocal equations. Here we have done an even degree question. Now we will understand the odd degree question. and how to solve it. So students, here is a question on odd degree how to solve a reciprocal equation with odd degree Solve the reciprocal equation x power 5 – 5x power 4 + 9x power 3 – 9x power 2 + 5x – 1=0 x power 5 – 5x power 4 + 9x power 3 – 9x power 2 + 5x – 1=0 x power 5 – 5x power 4 + 9x power 3 – 9x power 2 + 5x – 1=0 x power 5 – 5x power 4 + 9x power 3 – 9x power 2 + 5x – 1=0 this is second degree reciprocal equation. because the co-efficient are equal but have opposite signs. see every co-efficient has opposite sign. So, this is second kind of reciprocal equation with odd degree. This is the second kind reciprocal equation of odd degree one of its root will be 1 So as you remember from the properties here X=1 is the root. so students now we know that root is X=1 now what we have to do is divide the complete equation with X=1 and after solving the left over equation will be in 4 degrees. We will solve that equation as we had solved the last problem with even degree. I had told you the rule of division that is called “SYNTHETIC DIVISION” by which we can easily divide any equation. And students, if you want to learn more about Synthetic Division tap the “i” icon above and you can easily watch my previous videos on Synthetic Division under Theory Of Equation playlist. So now let’s see how we can solve this. Write this equation as 1-5 then 9-9 this is synthetic division. then 5-1 Here it’s factor is 1 we will write like this then comes a ‘dash’ Then we add 1 then multiply 1 by 1 then we add -5 and 1, we get -4 then we multiply 1 by -4, we get -4 then add 9 and -4, we get 5 then multiply and 5, we get 5 then add -9 and 5, we get -4 then multiply 1 and -4 we get -4 then add 5 and -4, we get 1 and here comes 0. So students, this is our remainder. Remainder is 0 because it is dividing it perfectly. Now we know we have A0, A1, A2, A3, and this is our A4. SO the equation which we get is, A0x power 4 + A1x cube + A2x square + A3x + A4=0 Now keep the values here. That gives x power 4 x power 4 – x power 4 – 4 x cube x power 4 – 4 x cube + 5x square x power 4 – 4 x cube + 5x square. Here come -4x 4x + 1 4x + 1=0 This is reciprocal equation of power 4 as I told you we will solve this. Now what we will do is divide by x square on both side of equation. on both side of equation. When we divide it, it gives, x square – 4x x square – 4x + 5 – 4/x x square – 4x + 5 – 4/x + 1/x square=0 Then we will write it completely. (x square + 1/x square) -4 as common then it gives (x + 1/x) + 5=0 students, here we will do +2 and -2 because we have to make a perfect square. this gives, (x square + 1/x square + 2 ) -4 (x+1/x) there it was +2 so here it will be -2 so, -2 + 5=0. Now we know this equation will be, (x+1/x) whole square – 4 (x + 1/x) (x+1/x) whole square – 4(x + 1/x) +3=0 So, let x+1/x be Y this equation will be, Y square – 4 Y Y square – 4 Y + 3=0 Now we will take out the factors. Factors will be (Y – 1) and (Y – 3) (Y – 1) and (Y – 3)=0 1 and 3 are the roots. Now we will keep Y=1 and there Y=3 Keep this Y=1 in the equation. It gives, X+1/x=1 and here it will be, x+1/x=3 now we will do cross-multiplication. This gives, x square – x + 1=0 the roots will be 1 +- under root B square, i.e. under root 1-4 divided by 2 so that gives, x=1 +- root 3 divided by 2 Now we come on this, we will multiply x which gives x square – 3x + 1=0 x square – 3x + 1=0 x square – 3x + 1=0 now we will find out the value of x the value of x will be 3 +- B square that gives under root 9 – 4 divided by 2 and we know the root will come 3 +- 3 +- root 5 divided by 2 this way we find out the roots. So students, I think you have understood the concept and this way we solve out the reciprocal equation all the questions available in your exercises can be easily done with the help of this understanding. I hope the content provided by me is helping you a lot. and I hope you are understanding it completely. So students, if the content provided by me is useful to you then do subscribe my channel and press the bell icon to get all the notifications. Do like my video and don’t forget to comment. I might have not replied to all the comments. But I have seen all your comments and this truly motivates me to make another useful video for you all. Don’t worry about the mathematics course, engineering course I will regularly update content on every topic. Engineering, B.Sc or M.Sc mathematics will be uploaded on my YouTube channel soon. I will try to be more regular in uploading the videos. I will try to be more regular in uploading the videos. 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