 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 a quadratic equation 2x square minus 3x minus 5 is equal to 0 for a quadratic equation whose roots are alpha cubed and beta cubed. 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 a, b, c are the constants. Now let p and q are the roots of this equation. Then some of the roots that is p plus q is equal to minus b over a which means p plus q will be equal to minus coefficient of x over coefficient of x square in the given equation and product of the roots that is p into q or pq is equal to c over a which means product of the roots is equal to the absolute term over the coefficient of x square in the given equation. Also if roots of the equation are given then the 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. Now these results will work out as a key idea for solving out this question. And now we will start with the solution. Here the equation is given as 2x square minus 3x minus 5 is equal to 0. Now comparing this with the standard form of quadratic equation here a is equal to 2, b is equal to minus 3 and c is equal to minus 5. Now let alpha and beta are the roots of this equation. 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. Now putting the values of b and a here it will be minus of minus 3 by 2 which is further equal to 3 by 2. Now we know that the product of the roots is equal to c over a. Now the 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 minus 5 over 2. Now in the question it has asked that we have to form a quadratic equation whose roots are alpha cube and beta cube. Now if alpha cube and beta cube are the roots of an equation then sum of the roots that is alpha cube plus beta cube will be equal to according to the formula alpha plus beta whole cube minus 3 alpha beta into alpha plus beta. Now putting the values of alpha plus beta alpha beta and alpha plus beta this will be equal to 3 by 2 whole cube minus 3 into minus 5 by 2 into 3 by 2. Which will be equal to 3 by 2 whole cube will be 27 by 8. Here minus into minus will be plus and on calculating this it will be 45 by 4. Which is further equal to on taking the LCM which is 8 this will be 27 plus 90 which is equal to 27 plus 90 is 117 by 8. Now product of the roots that is alpha cube into beta cube or alpha cube beta cube can be written as alpha beta whole cube. Now putting the value of alpha beta here this will be equal to minus 5 by 2 whole cube which is further equal to minus 5 by 2 whole cube is minus 125 by 8. Now if the roots of the equation are given then we can form the equation by using this formula. So the required equation whose roots are alpha cube and beta cube is x square minus sum of the roots into x plus product of the roots is equal to 0. Now sum of the roots is 117 by 8 and product of the roots is minus 125 by 8. So putting these values here this implies x square minus 117 by 8x plus minus 125 by 8 is equal to 0. This implies x square minus 117 by 8x minus 125 by 8 is equal to 0. Now multiplying by 8 on both sides this implies 8x square minus 117x minus 125 is equal to 0. So this is the solution of the given question and that's all for this session. Hope you all have enjoyed the session.