Let and be the roots of the quadratic equation 5x^2 - 8x + 2 = 0 . Then the quadratic equation whose roots are ^2 5 - 8 and ^2 5 - 8 is :

Step-by-step solution

Given the quadratic equation 5x^2 - 8x + 2 = 0 , we have the sum of roots + = 8 5 and the product of roots = 2 5 . Since is a root of the equation, it satisfies 5 ^2 - 8 + 2 = 0 . This can be rewritten as (5 - 8) = -2 , which gives 5 - 8 = - 2 . Substitute this into the first new root: ^2 5 - 8 = ^2 - 2 = - ^2 2 = - ( ) 2 Using = 2 5 , the root simplifies to: - ( 2 5 ) 2 = - 5 By symmetry, the second new root is: ^2 5 - 8 = - 5 Now, find the sum and product of the new roots. Sum of new roots: S = - 5 - 5 = - + 5 =