 Hello friends welcome again to yet another session on quadratic equation and continuing with our concepts on the same topic we are now taking up some and products of roots of a quadratic equation. So far we have discussed quadratic equation how to solve quadratic equations and then we also commented upon what are the nature what type of roots are there of quadratic equation whether real unreal yep so you know real or complex roots are there whether they are equal they are not equal all of them we have seen. Now we are going to take up this exercise we are going to discuss some and product of roots and before that we will just do a recap of what was Sridharacharya's rule or what was quadratic formula. So if you remember if I have a quadratic equation ax square plus bx plus c equals 0 let us say I have this you know equation where a is not equal to 0 and a b and c are real numbers right this is what we have been you know discussing about quadratic equations. So a is not equal to 0 a b and c are real numbers and by our Sridharacharya's rule Sridharacharya's rule either you say Sridharacharya's rule or you say let's say quadratic formula quadratic quadratic formula if you know this what is it we know there are two roots to a quadratic equation one is alpha let us say alpha which is equal to minus b plus under root d which is b square minus 4 ac upon twice a or beta which is nothing but minus b minus under root b square minus 4 ac by 2 a. These are the two roots by quadratic formula now in this session we are going to take up what sum of the roots so if you see sum of sum of roots sum of roots of a quadratic a quadratic equation guys let me do this you know so when I'm doing this I'm writing alpha plus beta which is equal to c minus b plus under root b square minus 4 ac upon twice a and plus minus b minus under root b square minus 4 ac upon 2 a is it it now in the right hand side if you see what is it the common denominator is 2 a so let us take that as common yeah and here it is minus b plus under root b square minus 4 ac and then minus b minus under root b square minus 4 ac is it so you can keep this in brackets like that okay so hence you can see clearly that this term and this term is getting cancelled so hence I get minus 2 b by 2 a which is equal to minus b upon a wow this is beautiful why because if I know a b and c then sum of roots also I know so sum of roots is nothing but minus I can write coefficient coefficient of x upon coefficient of x square right that is sum of roots so hence I again so alpha plus beta is equal to minus b upon a right now let us try to find out product of root guys so product of roots will be product product of roots product of roots of a quadratic quadratic equation so we are now going to discuss product of roots now alpha again is nothing but minus b minus under root sorry plus b square minus 4 ac upon twice a and beta is equal to minus the minus under root b square minus 4 ac by twice a so guys alpha beta will be equal to minus b plus under root b square minus 4 ac by 2 a this is factor number 1 alpha and beta is minus b minus under root b square minus 4 ac by 2 a this is factor number 2 okay so if you see one you can you know 2 a and 2 a multiplied will will give you 1 upon 4 a square so you take that as separate okay and then in the in the remaining terms will be minus b plus root b square minus 4 ac and minus b minus root b square minus 4 ac is it this is what is left now if you see you know carefully this term looks like capital A plus capital B then this term again a minus capital B right so there are 2 a plus b times a minus b and you know it very you know easily that a plus b a minus b is a square minus b square so hence it will be 1 upon 4 a square which is taken out already and then this will become minus b square minus under root b square minus 4 ac which happens to be my capital B isn't it whole square so a square minus b square form so let's simplify it further it is 1 upon 4 a square within brackets minus b whole square is simply b square and then when you square root or square a square root the square root just disappears so hence becomes minus b square minus 4 ac isn't it so hence further simplifying you will get 1 upon 4 a squared b square minus b square plus 4 ac right and you see b square b square will go so it is nothing but 4 ac upon 4 a squared which is equal to c upon a again a very beautiful result because this is nothing but constant term constant term divided by coefficient of x square right so hence alpha beta is equal to c by a to summarize this discussion what do we get if alpha beta r roots of equation ax square plus bx plus c equals 0 then then alpha plus beta is equal to minus b upon a and alpha beta is equal to c upon a okay this is a good result to be remembered all the time okay let's check it whether it is actually whether it actually works so let us take an equation let's say x square minus 5x plus 4 is equal to 0 so we are now dealing with an example okay so clearly if you see the this particular equation can be reduced as minus x minus 4 equals 0 so this will be x times x minus 4 and then minus 1 times x minus 4 this is plus here this is equal to 0 so hence what will you get you will get x minus 4 times x minus 1 is equal to 0 this implies x is either 4 or x is 1 right so hence you can say alpha is equal to 4 and beta is equal to 1 2 roots so what is alpha beta guys check alpha plus beta is equal to 5 right and from the result we have alpha plus beta must be equal to minus b by a so minus b by a if you see is nothing but minus of minus 5 because b is minus 5 here divide by a which is 1 so hence it is 5 so it works right so alpha plus beta is equal to minus b by a now what is alpha beta alpha beta is 4 into 1 which is 4 and our result says alpha beta will be equal to c by a so let's evaluate c by a c is 4 divided by 1 is equal to 4 so this also works correct so alpha plus beta is minus b by a and alpha beta is c by a please remember this result right okay now in in if you see this equation let's take this equation once again ax square plus bx plus c equals 0 if you see this can I divide the entire equation by a we have done this before so this will be x square plus b by a plus c by a equals 0 is it now can the same thing can be written as x square minus minus b by ax plus c by a equals 0 why did I do this because if you notice we know this quantity and this quantity as well and what is that simply this will be x square minus alpha plus beta x plus alpha beta equals to 0 now this is again a very important result which is equal to this can be alternately written as x square minus sum of roots sum of roots you write and then multiply this with x plus product of product of roots is equal to 0 so this is a quadratic equation so hence again very vital result you must note this x square is sum of x square minus sum of roots times x plus product of roots equals to 0 what does it mean it means that if roots of a quadratic roots of a quadratic equation are known are known then then the equation can be obtained isn't it equation can be obtained how so let's take an example let's take an example let us say we have roots minus 3 and 2 these are the two roots of a given equation so let us say alpha is equal to minus 3 and beta is equal to 2 so the this for this set of roots what will be the equation equation will be simply x square minus sum of the roots so minus 3 plus 2 sum of the roots times x plus product of the roots so minus 3 times 2 and this equated to 0 will be the equation let's check it is nothing but x square and minus 3 plus 2 is minus 1 so x square plus x and this is minus 6 equals 0 right so if you see for this particular quadratic equation the roots are minus 3 and 2 you can check that when you solve this by all the methods which we have learned you will get the two roots as minus 3 and 2 so in this session we learned couple of vital concepts one is the product of the sum of the roots is minus b by a and product of roots is c by a and we can write an equation if the roots are known by which principle that is x square minus sum of roots times x plus product of roots is equal to 0 okay so these are the vital learnings of this session