 Hello friends, so welcome to another session on problem solving and we are dealing with quadratic equation and it is a famous saying that to learn mathematics you have to do mathematics. So hence let's do mathematics, let's continue with our session on more and more problem solving. So here we have taken up one problem which was also asked in one of the exams in board exams rather. So what is this question? This question says 1 upon A plus B plus X equals 1 upon A plus 1 upon B plus 1 upon X and it is given and A plus B is not equal to 0. So why is this condition given? You will realize in a little while but this is the equation given and you have to solve it. Clearly this is not in this current form we can't call it as a linear or quadratic equation but it can certainly be reduced to a quadratic form and how do I know it? So whenever you see two terms or where in the variables are in the reciprocal let's say 1 upon X plus let's say another term is 1 upon 2X plus 1 and there is some constant on the other side you can fairly guess that this is going to be a quadratic equation. So that's where if you see here in the denominator there is X term and there is X here. So this will lead to or this will end up being a quadratic equation. Let us try and solve this now. So the first step is to get all the variable terms on one side of the equation. So let me take 1 upon X on the left hand side you will get 1 upon A plus B plus X minus 1 upon X. So 1 upon X on the right hand side comes to the left hand side and this is equal to 1 upon A plus 1 upon B. Next you can multiply the denominators to get the common denominator A plus B plus X in the LHS I am talking about. Then what should be written first? First it will be X how do I get that? So you multiply these two denominators and you leave the denominator whatever is the other term you have to write it first like that. Then you clearly know the other term will be A plus B plus X and this is equal to again common denominator is AB and hence it will be B plus A. So let's go further. So if you simplify this you will get X minus A minus B minus X on the top in the numerator and you open up the denominator you will get AX plus BX plus X squared and this is equal to A plus B upon AB. Now what if you see this X minus X and X goes so hence I have minus A minus B upon what will it be? It will be X squared so I am rearranging the terms in the denominator here it is AX plus BX is equal to A plus B and AB. Now here is the catch. Now if you see minus A plus I can take minus common in the numerator and I can write this as X squared plus AX plus B and this is equal to A plus B divided by AB. Now guys if you see there is A plus B on both the left hand side and right hand side so I can cancel it but the word of caution here is A plus B must not be equal to 0 because if it is 0 you cannot cancel the terms because in a way you are dividing both sides by A plus B then only you are canceling A plus B but division by 0 is not possible so A plus B must not be 0 hence the condition given here A plus B is not 0 already we know that so we can cancel it if the condition was not given then you cannot write that or you cannot cancel it. Now next step clearly will be nothing but minus of X squared plus AX plus BX sorry this was X so BX is equal to AB now I have taken the reciprocals so hence if you see the entire equation will be reduced to X squared plus AX plus BX and plus AB is equal to 0. So I have taken all the terms negative from this term left hand side on the right hand side and then I have swapped the places right hand side to left hand side now this will be nothing but now if you see there are four terms and clearly we can see some common terms so let us group them so X times X plus A plus B times X plus A is equal to 0 so hence it becomes X plus A times X plus B is equal to 0. So clearly either X plus A is equal to 0 or X plus B is equal to 0 right either of the factors will be 0 then only the product is 0 so hence X is equal to minus A or X is equal to minus B why are we saying or not and is because if any of the two equations are satisfied then this will hold true this particular equation will hold true it need not X need not be minus A and minus B simultaneously even if any of them is true let's say if this is true then also it will be 0 or this is true that that then also this particular equation is going to be 0 so hence we say or and not and so either of them will be sufficient okay so this is how we got the solution so hence what is the solution for this equation X is equal to minus A and why sorry X is equal to minus B you can check it by deploying X equals to minus A and minus B you will get LHS equal to RHS and hence you know how to solve these particular quadratic equation so hence what is the learning learning is it might not appear to be quadratic but if you simplify it you will end up getting a quadratic equation you have to just be you know sure that the cancelling factors need not be or must not be 0 if it is 0 you can't cancel right so these are the learnings from this problem