For the following two (02) items: Let and be the roots of the quadratic equation x^2 + ( _ 0.5 (a^2))x + ( _ 0.5 (a^2))^4 = 0 , where a^2 1 and _ 0.5 (a^2) > 0 . Further, ^2 = ( _ a^2 (0.5)) . What is the relation between and ?

Step-by-step solution

Let p = _ 0.5 (a^2) . We are given p > 0 . The given quadratic equation is: x^2 + px + p^4 = 0 Let the roots be and . Then: + = -p = p^4 We are given the relation: ^2 = _ a^2 (0.5) Using the change of base formula, _ a^2 (0.5) = 1 _ 0.5 (a^2) = 1 p . Thus, the relation becomes: ^2 = p = p ^2 Substitute = p ^2 into the product of roots: (p ^2)( ) = p^4 p ^3 = p^4 Since p > 0 , we can divide by p to get: ^3 = p^3 Assuming is a real root, we have = p . Substitute = p into the sum of roots: + p = -p = -2p Now we can fi