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

The correct answer is 25.
log2​(92α−4+13)−log2​(25​⋅32α−4+1)=2
⇒25​32α−4+192α−4+13​=4
⇒α=2 or
α∈S∑​α=5 and α∈S∑​(α+1)2=25
⇒x2−50x+25β=0 has real roots
⇒β≤25
⇒βmax​=25