 Hello and welcome to the session. In this session we will discuss a question which says that if x is real prove that x-1 the whole x-2 the whole plus 1 is always positive. Now we will start with the solution. Here the expression is given as x-1 the whole into x-2 the whole plus 1. Which is equal to on multiplying these two we will get x square minus x minus 2x plus 2 plus 1 which is further equal to x square minus 3x plus 3. Now by the method of completing the squares we will complete the square of this expression. For this we will add and subtract the square of half the coefficient of x. So that will be equal to x square minus 3x plus 3 and here half the coefficient of x is 3 by 2 and its square is 9 by 4 so it will be plus 9 by 4 minus 9 by 4. Further this is equal to x square minus 3x can be written as 2 into 3 by 2 into x plus 9 by 4 plus 3 minus 9 by 4. So this is equal to x square minus 2 into 3 by 2 into x plus 9 by 4 can be written as 3 by 2 whole square plus 3 minus 9 by 4 will give plus 3 by 4. Now here the square is completed so this will be equal to x minus 3 by 2 whole square plus 3 by 4. Since x is real then x minus 3 by 2 whole square is greater than equal to 0 being the square for all real values of x. Now here adding 3 by 4 on both sides this implies x minus 3 by 2 whole square plus 3 by 4 is greater than equal to 3 by 4. As the given expression is greater than equal to 3 by 4 which is all positive therefore we can say the given expression that is x minus 1 the whole into x minus 2 the whole plus 1 is always positive. So this is the solution of the given question and that's all for this session. Hope you all have enjoyed the session.