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

Let $p = \log_{0.5}(a^2)$. We are given $p > 0$.

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

Let the roots be $\alpha$ and $\beta$. Then:
$\alpha + \beta = -p$
$\alpha \beta = p^4$

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

Using the change of base formula, $\log_{a^2}(0.5) = \dfrac{1}{\log_{0.5}(a^2)} = \dfrac{1}{p}$.
Thus, the relation becomes:
$\beta^2 = \dfrac{\alpha}{p} \implies \alpha = p\beta^2$

Substitute $\alpha = p\beta^2$ into the product of roots:
$(p\beta^2)(\beta) = p^4 \implies p\beta^3 = p^4$

Since $p > 0$, we can divide by $p$ to get:
$\beta^3 = p^3$

Assuming $\beta$ is a real root, we have $\beta = p$.
Substitute $\beta = p$ into the sum of roots:
$\alpha + p = -p \implies \alpha = -2p$

Now we can find the relation between $\alpha$ and $\beta$:
$\alpha = -2p = -2\beta$

Thus, $\alpha = -2\beta$.

Answer: $\alpha = -2\beta$