 Hello friends, so let us continue with our topic on quadratic equation and in this session I have picked up a topic where usually in standard textbooks we don't deal with such things. So the question here is we have been talking so far about roots of the equation and we have time and again said that there are at max two roots to a quadratic equation. Now in this session we are going to take up how exactly do we prove that there are only two roots to a quadratic equation and not more and we will take up our conventional method of contradiction. So we will try to attempt this proof by assuming that there are more than two roots of our quadratic equation. So I am saying let us say there are three roots three roots of equation ax square plus bx plus c equals 0, where a is not equal to 0, okay, three where a is not equal to 0 and the roots are and let us say let us say that the roots are alpha beta and gamma, okay, so three roots alpha beta and gamma. So hence if alpha beta gamma are the roots of the given equation then we can say if alpha beta gamma are roots of ax square plus bx plus c equals 0 then we will have what all the three should satisfy the equation so hence we can say alpha square a alpha square plus b beta sorry b alpha b alpha plus c will be equal to 0, right and similarly let us say this equation is equation number one and then similarly a beta square plus b beta plus c equals 0 let us say equation number two and a gamma square plus b gamma plus c equals 0 let us say this is equation number three, okay, so hence what will happen now let us do this these you know calculations so what I am doing is I am doing subtracting subtracting two from one so if I subtract the equation two from equation one in the LHS what will I get I will get a common alpha square minus beta square then b common alpha minus beta and then plus c minus c equals 0 so if you see if you subtract it you will get this a alpha square minus a beta square I have written simultaneously together and b alpha minus b beta is b alpha times sorry b times alpha minus beta and c minus c equals to 0 so what will happen this will be nothing but a alpha square minus beta square plus b alpha minus beta will be 0 now we have okay so when we are considering alpha beta and gamma are three roots then automatically alpha beta and gamma we should add one more thing here let us say that the roots are alpha beta and gamma and all are unique all are unique that means or we can say we can use the word distinct distinct means what does this mean distinct means alpha is not equal to beta is not equal to gamma this is what it means okay so hence alpha clearly since alpha minus beta is not equal to 0 why since alpha is not equal to beta therefore we can cancel one alpha minus beta so hence this expression is nothing but a alpha minus beta times alpha plus beta plus beta times alpha minus beta equals to 0 now since alpha minus beta is not equal to 0 I can cancel out this factor correct so hence what will you get you will get a alpha plus beta plus b is equal to 0 okay a alpha plus beta plus b equals to 0 right now let us say this is equation number 4 now I am doing this subtracting subtracting what do I subtract I subtract 3 from 2 so if you see if you subtract 3 from 2 you will get alpha beta square minus gamma square plus b times beta minus gamma is equal to 0 see we will get cancelled right and again since you can write since beta minus gamma is not equal to 0 so what will happen I can write a beta minus gamma times beta plus gamma a square minus b square form plus b beta minus gamma is equal to 0 why because beta square minus gamma square is nothing but beta minus gamma times beta plus gamma okay now again I can strike this off why because it's common common to the left hand side and the right hand side there is 0 so I can strike that off and now we can say a beta plus gamma plus b is equal to 0 this is equation number 5 okay so let us now do some mathematics on 4 and 5 so I am saying now subtracting again 5 from 4 what will you get you will get a alpha plus a beta plus b this is the left hand side of 4 minus then a beta plus gamma plus b must be equal to 0 this implies this implies what a alpha plus a beta plus b minus a beta minus a gamma minus b equals to 0 on simplification so if you see a beta and a beta will get cancelled and this b and this b will get cancelled so hence I am getting a alpha minus gamma is equal to 0 now a is not equal to 0 guys because that is what you know by definition of a quadratic equation we don't we can't say a 0 so hence definitely alpha minus beta will be equal to sorry not alpha minus beta alpha minus gamma from here what will you see you will see that alpha minus gamma is 0 that means alpha is equal to gamma okay so it you know it comes out that out of the three two are same out of the three two are same that means now the two the three roots to the equation where alpha beta gamma but but alpha and gamma alpha and gamma are same so hence so hence hence our assumption our assumption that there are three there are three roots of a quadratic quadratic equation is wrong is wrong is wrong right hence hence a quadratic equation can have equation can have only two roots now these roots could be real non real whichever way but maximum possible roots are only okay it cannot have three four five six anything else right only two roots this is how we proved that a quadratic equation a quadratic equation cannot have more than two roots