 Hello friends, let's work out the following problem. It says what should be added to 1 upon x square minus 6x plus 8 to get 2x minus 7 upon x minus 2 into x minus 3 into x minus 4. So let's now move on to the solution. Now we have to find a rational expression So that if we add that rational expression to this rational expression we get this. So let the required rational expression be so if we add a to this we'll get this So we have a plus 1 upon x square minus 6x plus 8 is equal to 2x minus 7 upon x minus 2 into x minus 3 into x minus 4 Now a plus 1 upon x square minus 6x plus 8 is equal to this. So this implies a is equal to 2x minus 7 upon x minus 2 into x minus 3 into x minus 4 minus 1 upon x square minus 6x plus 8 Now again We'll simplify this expression We have 2x minus 7 upon x minus 2 into x minus 3 into x minus 4 minus 1 upon. Now we'll factorize this quadratic equation So we have x square minus 4x minus 2x plus 8 So this implies a is equal to 2x minus 7 upon x minus 2 into x minus 3 into x minus 4 minus 1 upon now taking x common from the first two terms we have x into x minus 4 and Taking minus 2 common from the last two terms here We have minus 2 into x minus 4. So this implies a is equal to 2x minus 7 upon x minus 2 into x minus 3 into x minus 4 minus 1 upon Taking x minus 4 common We have x minus 4 into x minus 2 Now to find the rational expression a we'll take the LCM Now LCM would be x minus 2 into x minus 3 into x minus 4 right Now since LCM is x minus 2 into x minus 3 into x minus 4 Dividing this by the denominator of this expression will get 1. So we need to multiply 2x minus 7 by 1 minus 1 into now we need to multiply minus 1 with x minus 3 Because if we divide this LCM by x minus 4 into x minus 2 we'll get x minus 3 So this implies a is equal to 2x minus 7 Minus now since there is a negative sign outside the bracket when we'll open the bracket sign will change So plus x becomes minus x and minus 3 becomes plus 3 upon x minus 2 into x minus 3 into x minus 4 So this implies a is equal to 2x minus x is x minus 7 plus 3 is minus 4 upon x minus 2 into x minus 3 into x minus 4 Now x minus 4 gets cancelled with x minus 4 and we have a is equal to 1 upon x minus 2 into x minus 3 Now we'll multiply the two brackets here. So we have a is equal to 1 upon x into x is x square x into minus 3 is minus 3x Now minus 2 into x is minus 2x minus 2 into minus 3 is plus 6 So this implies a is equal to 1 upon x square minus 3x minus 2x is minus 5x plus 6 So the required rational expression is this it is 1 upon x square minus 5x plus 6 This completes the question and the session do take care of calculations Why for now take care have a good day