Rmo 1993 ^hot^ -

Second radical: ( x+8-6\sqrtx-1 = (t^2+1)+8 - 6t = t^2 - 6t + 9 = (t-3)^2 ).

[ \sqrt(t-2)^2 + \sqrt(t-3)^2 = 1. ] That is ( |t-2| + |t-3| = 1 ), with ( t \ge 0 ).

[ \sqrtx+3-4\sqrtx-1 ;+; \sqrtx+8-6\sqrtx-1 ;=; 1. ] 1. Domain: ( x-1 \ge 0 \Rightarrow x \ge 1 ).

Let ( t = \sqrtx-1 \ge 0 ). Then ( x = t^2 + 1 ).

( x = t^2 + 1 ), with ( t \in [2,3] ). So ( x \in [5, 10] ).

First radical: ( x+3-4\sqrtx-1 = (t^2+1)+3 - 4t = t^2 - 4t + 4 = (t-2)^2 ).