 Hello and welcome to the session. In this session we will discuss a question which says that if alpha, beta are the roots of the equation x2 plus 3x minus 4 is equal to 0, find the values of first part alpha square plus beta square, second part alpha minus beta, third part alpha cube minus beta cube. Now before starting the solution of this question we should know some results. Now we know that the standard form of quadratic equation is ax square plus dx plus c is equal to 0. Where a is not equal to 0 and abc are the constants. Now let p and q are the roots of this equation. Then sum of the roots that is p plus q will be equal to minus v over a. That means p plus q will be equal to minus coefficient of h over coefficient of x square in the given equation. Also product of the roots that is p into q or pq will be 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. 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 x square plus 3x minus 4 is equal to 0. Now comparing this equation with the standard form of quadratic equation. Here a is equal to 1, v is equal to 3 and c is equal to minus 4. Now let alpha beta be the roots of the given equation. Now according to the formula which is given in the key idea sum of the roots is equal to minus b over a. Now here 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 3 by 1. Which is further equal to minus 3. Now here according to the formula which is given in the key idea product of the roots is equal to c over a. Therefore product of the roots that is alpha into beta or alpha beta is equal to c over a. Putting the values of c and a here it will be minus 4 by 1 which is equal to minus 4. Now in the first part we have to find the value of alpha square plus beta square. By using the formula alpha square plus beta square is equal to alpha plus beta whole square minus 2 alpha beta. Now alpha plus beta is minus 3 and alpha beta is minus 4. Putting these values here this will be equal to minus 3 whole square minus 2 into minus 4. This will be equal to minus 3 whole square is 9 minus into minus will be plus 2 into 4 is 8. Which is further equal to 9 plus 8 is 17. Now in the second part we have to find the value of alpha minus beta. Now using the formula alpha minus beta whole square will be equal to alpha plus beta whole square minus 4 alpha beta. This implies alpha minus beta is equal to square root of alpha plus beta whole square minus 4 alpha beta. Now putting the values of alpha plus beta and alpha beta here this will be equal to square root of alpha plus beta whole square. That means minus 3 whole square minus 4 into alpha beta that is minus 4. Which is further equal to square root of minus 3 whole square is 9 minus into minus is plus and 4 into 4 is 16. Which will give square root of 25 which is equal to plus minus 5. Therefore if alpha is greater than beta then alpha minus beta is equal to 5. And if alpha is less than beta then alpha minus beta is equal to minus 5. Now in the third part we have to find the value of alpha cube minus beta cube. Now by using the formula alpha cube minus beta cube is equal to alpha minus beta whole cube plus 3 alpha beta into alpha minus beta. Now let's discuss the first case when alpha is greater than beta. That means if alpha minus beta is equal to 5 then alpha cube minus beta cube according to this formula is alpha minus beta whole cube plus 3 alpha beta into alpha minus beta. Now putting the values of alpha minus beta and alpha beta here this will be equal to 5 cube plus 3 into alpha beta is minus 4 into alpha minus beta is 5. Which is equal to 5 cube is 125 plus into minus is minus 3 into 4 into 5 is 60 so this is equal to 65. So in the first case alpha cube minus beta cube is equal to 65. Now let's discuss the second case when alpha is less than beta that means if alpha minus beta is equal to minus 5 then alpha cube minus beta cube according to the formula will be alpha minus beta whole cube plus 3 alpha beta into alpha minus beta. Now putting the values of alpha minus beta alpha beta and alpha minus beta this will be equal to minus 5 whole cube plus 3 into alpha beta is minus 4 into alpha minus beta which is minus 5. This is equal to minus 5 whole cube is minus 125 plus 3 into minus 4 into minus 5 is 60 so this will be equal to minus 125 plus 60 which is minus 65. Hence the value of alpha square plus beta square is equal to 17 alpha minus beta is equal to 5 and minus 5 alpha cube minus beta cube is equal to 65 and minus 65. So this is the solution of the given question and that's all for this session hope you all have enjoyed the session.