For the following two (02) items: Let α and β be the roots of the...

Let $p = \log_{0.5}(a^2)$.

The given quadratic equation is $x^2 + px + p^4 = 0$.

The product of the roots is $\alpha \beta = p^4$.

We are given the condition $\beta^2 = \alpha (\log_{a^2}(0.5))$.

Using the properties of logarithms, $\log_{a^2}(0.5) = \dfrac{1}{\log_{0.5}(a^2)} = \dfrac{1}{p}$.

Substituting this into the given condition gives $\beta^2 = \dfrac{\alpha}{p}$, which implies $\alpha = p\beta^2$.

Substituting $\alpha = p\beta^2$ into the product of roots equation gives:

$(p\beta^2)\beta = p^4$

$p\beta^3 = p^4$

Since $p = \log_{0.5}(a^2) > 0$, we have $p \neq 0$. Dividing by $p$ yields:

$\beta^3 = p^3$

Considering real roots, we obtain $\beta = p$.

Substituting the value of $p$ back gives $\beta = \log_{0.5}(a^2)$.