Let S = α : log 2 9 2 α - 4 + 13 - log 2 5 2 · 3 2 α - 4 + 1 = 2 . Then the maximum value of β for which the equation x 2 - 2 ∑ α ∈ s α 2 x + ∑ a ∈ s α + 1 2 β = 0 has real roots, is _____ .

Step-by-step solution

Given, log 2 9 2 α - 4 + 13 - log 2 3 2 α - 4 · 5 2 + 1 = 2 Now let 3 2 α - 4 = t , so the equation becomes, log 2 t 2 + 13 - log 2 5 t 2 + 1 = 2 ⇒ log 2 t 2 + 13 5 t 2 + 1 = 2 ⇒ t 2 + 13 5 t 2 + 1 = 2 2 ⇒ t 2 + 13 = 10 t + 4 ⇒ t 2 - 10 t + 9 = 0 ⇒ t = 1   or   9 So, 3 2 α - 4 = 1   or   9 ⇒ 3 2 α - 4 = 3 0   or   3 2 ⇒ 2 α - 4 = 0   or   2 ⇒ α = 2 ,   3 Now, x 2 - 2 ∑ α &#