Let , be the roots of the quadratic equation x^2 - 2x + m = 0 , where m Z . It is given that the absolute difference of the cubes of the roots satisfies 8 | ^3 - ^3| 24 . The sum of all possible values of m is :

Step-by-step solution

For the equation x^2 - 2x + m = 0 , the sum and product of the roots are: + = 2 = m We need to evaluate | ^3 - ^3| . Using the algebraic identity: | ^3 - ^3| = | - | ( ^2 + + ^2) First, find | - | : | - | = ( + )^2 - 4 = 4 - 4m = 2 1 - m Next, find ^2 + + ^2 : ^2 + + ^2 = ( + )^2 - = 4 - m Substitute these back into the expression: | ^3 - ^3| = 2 1 - m (4 - m) We are given 8 | ^3 - ^3| 24 , so: 8 2 1 - m (4 - m) 24 4 1 - m (4 - m) 12 Let t = 1 - m . Since m Z and we require real roots ( 1 - m 0 ), t must be a non-n