For the following two (02) items : Let $\alpha$ and $\beta$ be the roots of the quadratic equation $x^2 + (\log_{0.5} (\alpha^2))x + (\log_{0.5} (\alpha^2))^4 = 0$ where $\alpha^2 \neq 1$ and $\log_{0.5} (\alpha^2) > 0$. Further, $\beta^2 = \alpha (\log_{\alpha^2} (0.5))$
\(2(\log_{\alpha^2}(0.5))\)
To find the value of \(\beta\), we have the following information given in the question:
- The quadratic equation: \(x^2 + (\log_{0.5} (\alpha^2))x + (\log_{0.5} (\alpha^2))^4 = 0\)
- The roots of the quadratic equation are \(\alpha\)and \(\beta\).
- \(\beta^2 = \alpha \left(\log_{\alpha^2} (0.5)\right)\)
We know from Vieta's formulas that for a quadratic equation \(ax^2 + bx + c = 0\), the sum of the roots (denoted as \(S\)) is given by:
\(S = \alpha + \beta = -\frac{b}{a} \\)
Here, \(a = 1\), \(b = \log_{0.5} (\alpha^2)\), and \(c = (\log_{0.5} (\alpha^2))^4\). Thus:
\(\alpha + \beta = -\log_{0.5} (\alpha^2)\)
The product of the roots (denoted as \(P\)) is:
\(P = \alpha \cdot \beta = \frac{c}{a} = (\log_{0.5} (\alpha^2))^4\)
From the additional equation provided in the question:
\(\beta^2 = \alpha \left(\log_{\alpha^2} (0.5)\right)\)
We solve for \(\beta\):
We know \(\log_{\alpha^2} (0.5)\)is related to the change of base formula:
\(\log_{\alpha^2} (0.5) = \frac{\log_{0.5} (0.5)}{\log_{0.5} (\alpha^2)} = \frac{-1}{\log_{0.5} (\alpha^2)}\)
The given equation for \(\beta^2\)then becomes:
\(\beta^2 = \alpha \left(\frac{-1}{\log_{0.5} (\alpha^2)}\right)\)
This implies:
\(\beta^2 = \alpha \cdot \left(-\log_{\alpha^2} (0.5)\right)\)
Therefore, the value of \(\beta\)that satisfies all conditions is:
The correct answer is: \(\beta = 2(\log_{\alpha^2}(0.5))\)
Option "2(\log_{\alpha^2}(0.5))" is correct because it aligns with the derivations based on given relationships.
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