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?

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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?

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Correct answer: B. _ 0.5 (a^2)

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. Substituting the value of p back gives = _ 0.5 (a^2).

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