 Hello and welcome to the session. In this session we will discuss a question which says that if x is real, prove that 5x square minus 8x plus 6 is always positive and find its minimum value. Now, we will start with the solution. Here the expression is given as 5x square minus 8x plus 6 which is equal to by taking 5 common within brackets x square minus 8 by 5x plus 6 by 5. Now, by the method of completing the squares we will complete the square of this. So this will be equal to 5 within brackets x square minus 8 by 5x plus 6 by 5 and adding and subtracting the square of half the coefficient of x it will be plus 16 by 25 minus 16 by 25. Further this is equal to 5 within brackets x square minus 8 by 5x plus 16 by 25 minus 6 by 5 minus 16 by 25. Now this is equal to 5 within brackets x square minus 8 by 5x can be written as 2 into 4 by 5 into x plus 16 by 25 can be written as 4 by 5 whole square. Here 6 by 5 minus 16 by 25 will give plus 14 by 25. Further this is equal to 5 within brackets here the square is completed and it is x minus 4 by 5 whole square plus 14 by 25. Now multiplying 5 inside we get 5 into x minus 4 by 5 whole square plus 14 by 5. Since x is real therefore x minus 4 by 5 whole square is always positive being the square for all real values of x thus 5 into x minus 4 by 5 whole square plus 14 by 5 is always positive. This expression is positive so multiplying with 5 which is positive and adding 14 by 5 which is also positive this beat expression will be always positive and we have done earlier that 5 into x minus 4 by 5 whole square plus 14 by 5 is equal to 5x square minus 8x plus 6. Now as this expression is always positive therefore 5x square minus 8x plus 6 is always positive. Now we have to find the minimum value of 5x square minus 8x plus 6. Now let 5x square minus 8x plus 6 is equal to y this implies 5x square minus 8x plus 6 minus 5 within brackets is equal to 0. Now if x be real then b square minus 4ac is greater than equal to 0. From here putting the values of a, b, c here this implies b here is minus 8 so it will be minus 8 whole square minus 4 into a here is 5 and c is 6 minus 5 is greater than equal to 0. Further this implies 64 minus 20 into 6 minus y the whole is greater than equal to 0. This implies taking 4 common within brackets 16 minus 5 into 6 minus y the whole is greater than equal to 0 which implies 4 within brackets 16 minus 30 plus 5y is greater than equal to 0. This implies 4 within brackets 5y minus 14 is greater than equal to 0. Now this condition will be satisfied if 4 and 5y minus 14 are of same sign. That means 4 is positive then y minus 14 should be greater than equal to 0 which implies 5y should be greater than equal to 14. This implies y is greater than equal to 14 by 5. This shows that y cannot be less than 14 by 5 as otherwise 4 into 5y minus 14 the whole will become negative. This means when y will be less than 14 by 5 then this expression will become negative. Hence minimum value of y that is the expression is 14 by 5. So this is the solution of the given question and that's all for this session. Hope you all have enjoyed the session.