A awesome mathematics problem | Olympiad Question | can you solve this problem | x=? 

Good solution, thank you. I went a different way with _a_ and _b_ as follows. _a + b = 2, ab = -1_ _a² - (a + b)a + ab = 0_ ∴ _a² - 2a - 1 = 0_ ⇒ _a = 1 + √2_ (since _a > b)_ ∴ _¹⁵√(x + √(x² + 1)) = √2 + 1_ ⇒ _x + √(x² + 1) = (√2 + 1)¹⁵_ ⇒ _x² + 1 = [ (√2 + 1)¹⁵ - x ]² = (√2 + 1)³⁰ - 2x(√2 + 1)¹⁵ + x²_ ⇒ _2x(√2 + 1)¹⁵ = (√2 + 1)³⁰ - 1_ ⇒ _x = ½[(√2 + 1)¹⁵ - 1/(√2 + 1)¹⁵]_ ⇒ _x = ½[(√2 + 1)¹⁵ - (√2 - 1)¹⁵]_ _(√2 + 1)¹⁵ - (√2 - 1)¹⁵ = 2[ ¹⁵C₁(2⁷) + ¹⁵C₃(2⁶) + ¹⁵C₅(2⁵) + ¹⁵C₇(2⁴) ¹⁵C₉(2³) + ¹⁵C₁₁(2²) + ¹⁵C₁₃(2¹) + ¹⁵C₁₅(2⁰) ]_ ∴ *_x = (15)(128) + (455)(64) + (3003)(32) + (6435)(16) + (5005)(8) + (1365)(4) + (105)(2) + (1)(1) = 275,807_*

i think 0 would be an obvious solution

x + √(x² + 1) = u¹⁵ x - √(x² + 1) = v¹⁵ u + v = 2 u¹⁵ + v¹⁵ = 2x (uv)¹⁵ = x² - (x² + 1) (uv)¹⁵ = -1 => uv = -1 u + v = 2 uv = -1 t² - 2t - 1 = 0 t = (2 ± 2√2)/2 t = 1 ± √2 u¹⁵ + v¹⁵ = 2x 2x = (1 + √2)¹⁵ + (1 - √2)¹⁵ *x = [(1 + √2)¹⁵ + (1 - √2)¹⁵]/2* = 275807 [(1 + √2)¹⁵ + (1 - √2)¹⁵]/2 = 1(2⁰) + 105(2¹) + 1365(2²) + 5005(2³) + 6435(2⁴) + 3003(2⁵) + 455(2⁶) + 15(2⁷) = 275807

that is an error : you wrote (-1)^2 is equal to 2 ! This is incorrect it should be +1 (i.e -1 x-1 =1 NOT 2)