 Hi friends welcome again to this session on certs and in this session we are going to discuss an important topic which is finding square root of a plus root b so you would have encountered these kind of problems where a cert is given and the compound cert is given and you have to find out the square root of that cert now where a and b are rational numbers please be careful and b is not a perfect square because if b is a perfect square then it is no more a cert now how to find out square root of a plus root b so we are saying let us say a plus root b square root that is what we have to find out is root x plus root y okay root x plus root y so it is natural to expect that square root of a cert would be another cert okay so what do we get if we square both sides square both sides now you'd have known by now that we have to find out x and y and if I find out x and y we get the square root of a plus root b so squaring both sides what do we get you get a plus root b on the left hand side and on the right hand side it would be root x plus root y whole square which is nothing but root x square plus 2 root x y plus root y square right so you'll get a plus root b is equal to x plus y plus 2 root x y isn't it so from the equality of two certs which we learned in the previous sessions what do we now infer we infer a must be equal to x plus y and b must be equal to and b must be root b sorry root b must be equal to root x y so we find a is equal to x plus y and root b is equal to 2 x y now how to find out x minus y y is it important you will see that if I know x minus y it will be very easy to find x and y now x minus y square can be written as x square plus y square minus 2 x y which can further be used as x minus y whole square is equal to x square plus y square plus 2 x y and then minus 4 x y correct so hence x minus y whole square is x plus y whole square minus 4 x y right now hence x minus y whole square is what is x plus y whole square guys so if you see x plus y is a so it is a square and what is minus 4 x y so if you see if you square these this relationship you'll get b is equal to 4 x y if you square both sides right now so 4 x y can be written as b so hence what is x minus y x minus y will be nothing but under root a square minus b correct no and what do I what did we have as x plus y above x plus y was nothing but a so if you see let us say this is equation number one and this is equation number two add one plus two one plus two what will you get you'll get 2x is equal to a plus under root a square minus b so what is x x is a plus under root a square minus b by 2 and now if you do one sorry 2 minus 2 minus 1 if you do this 2 minus 1 the equation once subtracted from 2 you will get 2y is equal to a minus under root a square minus b so hence y is equal to a minus under root a square minus b by 2 okay this is what so hence we could eventually find out x and y right so now let us take an example and see how to find out square root so let us say you have to find out and let me again uh yeah so let it be this let it be here so what I'm saying is let us now find out square root of square root of what 7 plus 2 root 10 this is why this is a question 7 plus 2 root 10 right so what am I saying what am I saying I'm saying 7 plus 2 root 10 square root is let us say root x plus y okay so when you square it you will get 7 plus 2 root 10 is equal to x plus y plus 2 root x y after squaring is it after squaring you will get that that means what x plus y must be equal to 7 and x y must be equal to 10 okay so you can you can further solve this but if you see x y if I take as 2 into 5 then x plus y is equal to 2 plus 5 and both the equations are solved right so hence if you see x is equal to 2 and y is equal to 5 so if you see you will get root 7 plus 2 root 10 is nothing but root 5 plus root 2 right this is the solution