 Hello and welcome to the session. In this session we discuss the following question which says finally roots of the following equation 1 upon x plus 4 minus 1 upon x minus 7 equal to 11 upon 30 where we have x not equal to minus 4 and 7. Now let's move on to the solution. The equation given to us is 1 upon x plus 4 minus 1 upon x minus 7 equal to 11 upon 30. Now as we have that x is not equal to minus 4 and 7 so multiplying the equation by x plus 4 into x minus 7 we get x plus 4 into x minus 7 whole into 1 upon x plus 4 minus 1 upon x minus 7 equal to x plus 4 into x minus 7 into 11 upon 30. So this gives us x minus 7 minus x plus 4 equal to x plus 4 into x minus 7 into 11 upon 30 that is we get x minus 7 minus x minus 4 equal to x plus 4 into x minus 7 into 11 upon 30 this further implies minus 11 equal to x plus 4 into x minus 7 into 11 upon 30. Now this 11 and this 11 gets cancelled so this gives us minus 30 equal to x square minus 7 x plus 4 x minus 28 that is we get x square minus 3 x plus 2 equal to 0. So the given equation reduces to this which is a quadratic equation. So now from this equation we have a equal to 1 b equal to minus 3 and c equal to 2. So let's find out b square minus 4 ac this would be equal to minus 3 the whole square minus 4 into 1 into 2 this is equal to 9 minus 8 and this is equal to 1. Now you get that b square minus 4 ac is greater than 0 since we have this one is greater than 0. Now we know that when b square minus 4 ac is greater than 0 then the roots of the equation given by x would be equal to minus b plus minus square root b square minus 4 ac upon 2 a. So using this we get x equal to minus b that is minus of minus 3 which is 3 plus minus square root b square minus 4 ac that is 1 that is square root 1 upon 2 a that is 2 into 1 and so this further implies that x is equal to 3 plus minus 1 since square root 1 is 1 upon 2 that is we have x equal to 3 plus 1 upon 2 and 3 minus 1 upon 2. So we get 2 values for x that is x equal to 2 and x equal to 1. Hence the final answer is that the roots of the given equation are 2 and 1. So now this complete session hope you have understood the solutions for this question.