 Hello and welcome to the session. In this session we will discuss a question which says that if x square plus p x plus q is equal to 0 and x square plus q x plus p is equal to 0 have a common root, root that either p is equal to q or 1 plus p plus q is equal to 0. Now we will start with the solution. Here the two equations are given as x square plus p x plus q is equal to 0 and x square plus q x plus p is equal to 0. And let us take this equation as 1 and this equation as 2. Now let alpha be the common root of the given equations. If alpha be the common root of the given equations then equation number 1 will become alpha square plus p alpha plus q is equal to 0 and equation number 2 will become alpha square plus q alpha plus p is equal to 0. Now let us take this equation as 3 and this equation as 4. Now subtracting equation number 4 from equation number 3 we get alpha square minus alpha square plus p alpha minus q alpha plus q minus p is equal to 0. Now here alpha square will be cancelled with alpha square so this implies p alpha minus q alpha plus q minus p is equal to 0. Now this implies combining these two terms and these two terms it will give p minus q the whole into alpha from here taking minus common it will be minus common bracket p minus q is equal to 0. This implies p minus q the whole into alpha minus 1 the whole is equal to 0. Now either minus q is equal to 0 or alpha minus 1 is equal to 0. Now if p minus q is equal to 0 then this implies p is equal to q if alpha minus 1 is equal to 0 then this implies alpha is equal to 1. Now putting alpha is equal to 1 in equation number 3 we get 1 square plus p into 1 plus q is equal to 0. So we have substituted the value of alpha is equal to 1 in equation number 3. This implies 1 square is 1 plus p into 1 is p plus q is equal to 0 therefore p is equal to q or 1 plus p plus q is equal to 0 hence proved. So this is the solution of the given question and that's all for this session. Hope you all have enjoyed the session.