 Hello and welcome to the session. In this session we discussed the following question which says show that the quadratic equation a square b square x square plus b square x minus a square x minus 1 equal to 0 has real roots and find the roots. So we need to show that for this given quadratic equation the roots are real and we also have to find the roots. For this we must know the condition as to when the roots of a quadratic equation are not real. So consider a quadratic equation a x square plus b x plus c equal to 0 its discriminant d is given as b square minus 4 a c. So when the discriminant d is greater than equal to 0 then the roots of the given quadratic equation are real and the roots are given by say alpha equal to minus b plus square root of d that is discriminant which is b square minus 4 a c upon 2 a and the other root say beta is given by minus b minus square root d by 2 a. This is the key idea that we use in this question. Now we shall proceed with the solution. The quadratic equation given to us is a square b square x square plus b square x minus a square x minus 1 equal to 0 or we can rewrite this as a square b square x square plus b square minus a square this whole into x minus 1 is equal to 0. To check whether the given quadratic equation has real roots or not we need to find the discriminant that is d and it is given by b square minus 4 a c and here for this quadratic equation a is the coefficient of x square which is a square b square b is the coefficient of x that is b square minus a square and c is the constant which is minus 1. So now d is equal to b square that is b square minus a square d whole square minus 4 into a which is a square b square into c which is minus 1. So d is equal to b square minus a square d whole square plus 4 a square b square which means d is equal to b to the power 4 minus 2 a square b square plus a to the power 4 plus 4 a square b square further we get d is equal to b to the power 4 plus 2 a square b square plus a to the power 4 or we can say d is equal to a square plus b square d whole square. So this is obviously greater than 0 that is we get d is greater than 0 therefore the roots of the given quadratic equation real and its roots are given by alpha equal to minus b plus square root d by 2 a and beta is equal to minus b minus square root d by 2 a and we know that a is equal to a square b square b is equal to b square minus a square and the discriminant d is given by a square plus b square whole square. Thus using these values in alpha and beta we get alpha is equal to minus b square minus a square plus square root a square plus b square whole square upon 2 a square b square which means alpha is equal to a square minus b square plus a square plus b square upon 2 a square b square or you can say alpha is equal to a square minus b square plus a square plus b square by 2 a square b square. Now b square minus b square cancels and this is equal to 2 a square upon 2 a square b square 2 2 cancels a square a square cancels so we get alpha is equal to 1 upon b square. Now let's find out beta this is equal to as you know minus b minus square root d by 2 a so we put the values of a b and d so this gives us beta is equal to minus b square minus a square minus square root a square plus b square whole square by 2 into a square b square that is we have beta is equal to a square minus b square minus a square plus b square and this whole upon 2 a square b square further we have beta is equal to a square minus b square minus a square minus b square upon 2 a square b square a square minus a square cancels so this is equal to minus 2 b square upon 2 a square b square this 2 and 2 cancels b square b square cancels therefore we get beta is equal to minus 1 upon a square so the roots of the given quadratic equation are alpha equal to 1 upon b square and beta equal to minus 1 upon a square so this is our final answer with this we complete the session hope you have understood the solution for this question