Let S = \ α: _ 2 ( 9 ^ 2α - 4 + 13) - _ 2 ((5)/(2) · 3 ^ 2α - 4 +...

Given,

$\log_{2}\left({9}^{2\alpha - 4} + 13\right) - \log_{2}\left({3}^{2\alpha - 4} \cdot \dfrac{5}{2} + 1\right) = 2$

Now let ${3}^{2\alpha - 4} = t$, so the equation becomes,

$\log_{2}\left({t}^{2} + 13\right) - \log_{2}\left(\dfrac{5t}{2} + 1\right) = 2$

$⇒ \log_{2}\dfrac{\left({t}^{2} + 13\right)}{\left(\dfrac{5t}{2} + 1\right)} = 2$

$⇒ \dfrac{\left({t}^{2} + 13\right)}{\left(\dfrac{5t}{2} + 1\right)} = {2}^{2}$

$⇒ {t}^{2} + 13 = 10t + 4$

$⇒ {t}^{2} - 10t + 9 = 0$

$⇒ t = 1\text{or}9$

So,

${3}^{2\alpha - 4} = 1\text{or}9$

$⇒ {3}^{2\alpha - 4} = {3}^{0}\text{or}{3}^{2}$

$⇒ 2\alpha - 4 = 0or2$

$⇒ \alpha = 2, 3$

Now,

${x}^{2} - 2{\left(\mathop{∑}_{\alpha \in s}\alpha\right)}^{2}x + \mathop{∑}_{a \in s}{\left(\alpha + 1\right)}^{2}\beta = 0$

$⇒ {x}^{2} - 2\left({\left(2 + 3\right)}^{2}x\right) + \left({3}^{2} + {4}^{2}\right)\beta = 0$

$⇒ {x}^{2} - 50x + 25\beta = 0$

Now for real roots

$D \ge 0$

$⇒ {50}^{2} - 4 \times 25\beta \ge 0$

$⇒ 50 - 2\beta \ge 0$

$⇒ \beta \le 25$

So, maximum value of $\beta$ is $25$.