Question

Let $S=\left\{\alpha: \log _2\left(9^{2 \alpha-4}+13\right)-\log _2\left(\frac{5}{2} \cdot 3^{2 \alpha-4}+1\right)=2\right\}$. Then the maximum value of $\beta$ for which the equation $x^2-2\left(\displaystyle\sum_{\alpha \in s} \alpha\right)^2 x+\displaystyle\sum_{\alpha \in s}(\alpha+1)^2 \beta=0$ has real roots, is _____

Answer 1

$ \log _2\left(9^{2 \alpha-4}+13\right)-\log _2\left(\frac{5}{2} \cdot 3^{2 \alpha-4}+1\right)=2 $
$\Rightarrow \frac{9^{2 \alpha-4}+13}{\frac{5}{2} 3^{2 \alpha-4}+1}=4 $
$\Rightarrow \alpha=2 \text { or } 3$
$ \displaystyle\sum_{\alpha \in S} \alpha=5 \text { and } \displaystyle\sum_{\alpha \in S}(\alpha+1)^2=25 $
$ \Rightarrow x^2-50 x+25 \beta=0 \text { has real roots } $
$ \Rightarrow \beta \leq 25 $
$ \Rightarrow \beta_{\max }=25$