 Hello and welcome to the session. In this session we will discuss a question which says that if alpha and beta are the roots of the quadratic equation x square minus px plus q is equal to 0 for the equation whose roots are alpha beta plus alpha plus beta and alpha beta minus alpha minus beta. Now before starting the solution of this question we should know some results. Now the standard form of quadratic equation is ax square plus bx plus c is equal to 0 where a is not equal to 0 and abc are the constants. Now here we take the roots of the equations as p and q then sum of the roots that is p plus q is equal to minus b over a. That is sum of the roots is equal to minus the coefficient of x over coefficient of x square and the product of the roots that is p into q or pq is equal to c over a which means pq is equal to absolute term over the coefficient of x square in the given equation. And if we have to form a quadratic equation whose roots are p and q then that equation can be formed by using the formula x square minus sum of the roots into x plus product of the roots is equal to 0 where sum of the roots is equal to p plus q and product of the roots is equal to pq. Now these results will work out as a key idea for solving out this question. Now we will start with the solution. Here it is given alpha and beta are the roots of the quadratic equation x square minus px plus q is equal to 0. So given alpha and beta are the roots of x square minus px plus q is equal to 0. Now comparing this equation with the standard form of quadratic equation here a is equal to 1, b is equal to minus p and c is equal to q. Now we know that the sum of the roots is equal to minus b over a. Therefore sum of the roots that is alpha plus beta is equal to minus b over a. Putting the values of b and a here it will be minus of minus p over 1 which is equal to p over 1 which is further equal to p. Also product of the roots is equal to c over a. Therefore product of the roots that is alpha beta is equal to c over a. Putting the values of c and a here it will be q over 1 which is equal to q. Now we have to form an equation whose roots are alpha beta plus alpha plus beta and alpha beta minus alpha minus beta. Therefore for the roots which are alpha beta plus alpha plus beta and alpha beta minus alpha minus beta the sum of the roots will be equal to alpha beta plus alpha plus beta plus alpha beta minus alpha minus beta. On solving here alpha will be cancelled with alpha, beta will be cancelled with beta so it will be 2 alpha beta. Putting the value of alpha beta here it will be equal to 2q. Now the product of the roots is equal to alpha beta plus alpha plus beta into alpha beta minus alpha minus beta. Now this can also be written as alpha beta plus alpha plus beta the whole into alpha beta combining these two terms and taking minus common it will be alpha plus beta the whole. Now here applying the formula of a plus b into a minus b which is equal to a square minus b square as you can see here a square minus b square is a plus b into a minus b. So this will be equal to here a is alpha beta and b is alpha plus beta so it will be alpha beta whole square minus alpha plus beta whole square. Now putting the value of alpha beta and alpha plus beta here this will be equal to q square minus b square. Now we know that if the roots of the equation are given then that equation can be formed by using this formula. Therefore the required equation is x square minus sum of the roots into x plus product of the roots is equal to 0. Now here sum of the roots is 2q and product of the roots is q square minus b square. So putting these values here the equation will be x square minus 2qx plus q square minus b square is equal to 0. So this is the solution of the given question and that's all for the session hope you all have enjoyed the session.