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 equal to?

Step-by-step solution

Let p = _ 0.5 (a^2) . The given quadratic equation is x^2 + px + p^4 = 0 . The product of the roots is = p^4 . We are given the condition ^2 = ( _ a^2 (0.5)) . Using the properties of logarithms, _ a^2 (0.5) = 1 _ 0.5 (a^2) = 1 p . Substituting this into the given condition gives ^2 = p , which implies = p ^2 . Substituting = p ^2 into the product of roots equation gives: (p ^2) = p^4 p ^3 = p^4 Since p = _ 0.5 (a^2) > 0 , we have p 0 . Dividing by p yields: ^3 = p^3 Considering real roots, we obtain = p . Substitu