 Hi and welcome to the session. I am Deepika here. Let's discuss the question. Find the nature of the roots of the following quadratic equation. If the real roots exist, find them. Equation is 2x square minus 6x plus 3 is equal to 0. We know that the roots of the quadratic equation Ax square plus Bx plus c is equal to 0 are given by x is equal to minus B plus minus under root of B square minus 4ac upon 2a. If B square minus 4ac is greater than 0, then we have two distinct real roots. If discriminant that is B square minus 4ac is equal to 0, we get two equal real roots and row real roots exist if B square minus 4ac is less than 0. So let's start the solution. Our given equation is 2x square minus 6x plus 3 is equal to 0. Compare the above quadratic equation with Ax square plus Bx plus c is equal to 0. We get A is equal to 2, B is equal to minus 6 and c is equal to 3. Let's find out the discriminant. Now therefore B square minus 4ac is equal to minus 6 whole square minus 4 into 2 into 3. This is equal to 36 minus 24 which is equal to 12. Because B square minus 4ac which is equal to 12 is greater than 0, therefore the above quadratic equation has distinct roots real roots. At alpha and v term are the roots of the given quadratic equation. Therefore alpha is equal to minus B plus B square minus 4ac upon 2a and beta is equal to minus B minus under root of B square minus 4ac upon 2a. Therefore alpha is equal to minus of minus 6 plus under root of 12 upon 2a 2 into 2 and beta is equal to minus of minus 6 minus under root of 12 upon 2 into 2. Therefore alpha is equal to 6 plus 2 root 3 upon 4 and beta is equal to 6 minus 2 root 3 upon 4. Again alpha is equal to 2 3 plus root 3 upon 4 and beta is equal to twice 3 minus root 3 upon 4. Again we can rewrite this as that is on cancellation we get alpha is equal to 3 plus root 3 upon 2 and beta is equal to 3 minus root 3 upon 2. Hence distinct real roots exist. Roots exist for the above quadratic equation and they are 3 plus minus root 3 upon 2. This is our answer. I hope you have understood the question. Bye and take care.