What is \beta equal to?

For the following two (02) items: Let \alpha and \beta be the roots of the quadratic equation x^2+(\log_{0.5}(a^2))x+(\log_{0.5}(a^2))^4=0 where a^2 \neq 1 and \log_{0.5}(a^2) \gt 0. Further, \beta^2=\alpha(\log_{a^2}(0.5)).

  1. A. \log_{a^2}(0.5)
  2. B. \log_{0.5}(a^2) ✓
  3. C. 2(\log_{a^2}(0.5))
  4. D. 2 \log_{0.5}(a^2)

Correct Answer: B. \log_{0.5}(a^2)

Explanation

Let B = \log_{0.5}(a^2). The product of the roots is \alpha\beta = B^4. Given \beta^2 = \alpha\log_{a^2}(0.5) = \frac{\alpha}{B}, we have \alpha = B\beta^2. Substituting this into the product equation yields (B\beta^2)\beta = B^4 \implies B\beta^3 = B^4 \implies \beta^3 = B^3. Assuming real roots, \beta = B = \log_{0.5}(a^2).

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