 Hi and welcome to the session. I am Asha and I am going to help you with exercise number 4.3, question number 34 on page number 73. The question says, for each of the following determine whether the given quadratic equation has roots. If so, find the roots. The given quadratic equation is 3x square plus 2 root 5x minus 5 is equal to 0. Let us start with the solution and the given quadratic equation is 3x square plus 2 root 5x minus 5 is equal to 0. Let this be equation number 1. On comparing it with the general type of quadratic equation which is of the form ax square plus bx plus c is equal to 0. We see that a is equal to 3, b is equal to 2 root 5 and c is equal to minus 5. Let us now find the value of d which is equal to b square minus 4ac. On substituting the values of b, a and c we have 2 root over 5 whole square minus 4 into 3 into minus 5. We have 20 plus 60 which is equal to 80. Thus we have d is equal to 80 which is greater than 0 and thus we can say that 1 has real roots and let alpha and beta with the 2 roots which are given by alpha equal to minus b plus root over d upon 2a and beta is equal to minus b minus root over d upon 2a. Let us now substitute the value of b d and a in alpha and beta this gives minus 2 root 5 plus root over 80 upon 2 into a which is 3. This is further equal to minus 2 root 5 plus 4 root over 5 upon 2 into 3. This further gives 2 root 5 upon 2 into 3 and 2 cancels out with 2 and we have alpha equal to root over 5 by 3 and thus we have alpha equal to root over 5 upon 3. Let us now find the value of beta which is equal to minus 2 root 5 minus root over 80 upon 2 into 3 which is equal to minus 2 root 5 minus 4 root 5 upon 2 into 3 which is equal to minus 6 root 5 upon 6. 6 cancels out with 6 and we have minus root 5 and thus we have beta equal to minus root 5. Hence we can say that the given equation 1 has real roots which are given by root over 5 upon 3 and minus root over 5. So this completes the solution. Hope you enjoyed this session. Take care and bye for now.