 Hello and welcome to the session. Here we will discuss the following question which says that solve x plus 1 the whole into x plus 2 the whole into x plus 3 the whole into x plus 4 the whole is equal to 120. Now before starting the solution of this question we should know about the formula for finding out the roots of the quadratic equation. Now we know that the standard form of quadratic equation is a x square plus b x plus c is equal to 0 where a b and c are the constants and a is not equal to 0. Now this equation will have two roots and for finding out the value of the roots the formula is x is equal to minus b plus minus square root of b square minus 4ac whole upon 2a or you can say x is equal to minus b plus square root of b square minus 4ac whole upon 2a and minus b minus square root of b square minus 4ac whole upon 2a. Now this formula will work as the key idea for solving out this question. Now we will start with the solution. Here we have to solve x plus 1 the whole into x plus 2 the whole into x plus 3 the whole into x plus 4 the whole is equal to 120. Now for solving this equation we have to convert it to quadratic form. Now we can write this equation as x plus 1 the whole into x plus 4 the whole into x plus 2 the whole into x plus 3 the whole is equal to 120. Now we have written the above equation in such a way that the sum of the constants in the first two expressions is equal to sum of the constants in the last two expressions which means sum of the constant here will be 1 plus 4 is equal to sum of the constants in the last two expressions that is 2 plus 3 which means 5 is equal to 5. Now multiplying the first two expressions and last two expressions with each other this implies x square plus x plus 4x plus 4 the whole into x square plus 2x plus 3x plus 6 the whole is equal to 120. Now this implies x square x plus 4x is 5x plus 4 the whole into x square plus 2x plus 3x is equal to 5x plus 6 the whole is equal to 120. Now here putting the value of x square plus 5x as y put x square plus 5x is equal to y this implies y plus 4 into y plus 6 is equal to 120. Now this implies multiplying these two expressions with each other it will y square plus 4y plus 6y plus 24 and moving this 120 on this side it will be minus 120 is equal to 0. Now this implies y square plus 10y minus 96 is equal to 0. Now comparing this with the standard form of public equation here a is equal to 1, b is equal to 10 and c is equal to minus 96. Now applying the formula for finding out the roots y will be equal to minus b plus minus square root of b square minus 4 ac whole upon 2a. Now putting the values of a, b and c here it will be minus 10 plus minus square root of 10 square minus 4 into 1 into minus 96 whole upon 2 into 1. Now this will be equal to minus 10 plus minus square root of 10 square is 100 and minus into minus is plus and 4 into 1 into 96 is 384 whole upon. Now this will be equal to minus 10 plus minus square root of 484 whole upon 2. Now this will be equal to minus 10 plus minus square root of 484 is 22 whole upon 2. Therefore y is equal to minus 10 plus 22 by 2 and minus 10 minus 22 by 2. This will be equal to minus 10 plus 22 is 12 over 2 and minus 10 minus 22 is minus 32 upon 2. Now here 2 into 6 is equal to 12 so it will be 6 and 2 into 16 is equal to 32 so it will be minus 16. Now if y is equal to 6 then x square plus 5x is equal to 6 which implies x square plus 5x minus 6 is equal to 0. Now comparing this with the standard form of quadratic equation here a is equal to 1, v is equal to 5 and c is equal to minus 6. Now using the formula for finding out the roots x is equal to minus v plus minus square root of v square minus 4ac whole upon 2a. So this will be equal to putting the values of a, v and c here it will be minus 5 plus minus square root of 5 square minus 4 into 1 into minus 6 whole upon 2 into 1 which will be equal to minus 5 plus minus square root of 5 square is 25 here minus into minus will be plus 4 into 1 into 6 will give 24 whole upon 2 equal to minus 5 plus minus square root of 49 whole upon. Now this will be equal to minus 5 plus minus square root of 49 is 7 whole upon 2 therefore x is equal to minus 5 plus 7 by 2 and minus 5 minus 7 by 2. Now this will be equal to minus 5 plus 7 is 2 over 2 and minus 5 minus 7 is minus 12 over 2. Now 2 will be considered 2 so this will be equal to 1 and 2 into 6 is 12 so it will be equal to minus 6. Also if y is equal to minus 16 then x square plus 5x is equal to minus 16 which will be plus 5x plus 16 is equal to 0. Now here comparing this with the standard form of gravity equation here a is equal to 1, v is equal to 5 and c is equal to 16. Now putting these values in the formula x is equal to minus v plus minus square root of v square minus 4 is equal to 1 upon 2 a. This will be equal to minus 5 plus minus square root of 5 square minus 4 into 1 into 16 whole upon 2 into 1. Further this will be equal to minus 5 plus minus square root of 5 square is 25 and here minus 4 into 1 into 16 that is equal to 64 whole upon now this will be equal to minus 5 plus minus square root of minus 39 whole upon 2. Now this is equal to minus 5 plus minus square root of minus 1 into 39 whole upon 2. Now this will be equal to minus 5 plus minus square root of minus 1 is equal to iota so it will be iota square root of 39 whole upon 2. Therefore x is equal to minus 5 plus iota root 39 over 2 and minus 5 minus iota root 39 by 2. Hence the values of x are 1 minus 6 minus 5 plus iota root 39 by 2 and minus 5 minus iota root 39 by 2. So this is the solution of the given question and that's all for the session. Let you all have enjoy the session.