A Nice Algebra Problem | Math Olympiad | How to find all x values ?

[@فیروزاهنگری]

√x(2-x)=1 , -1+√x(2-x)=0
-1+2√x-x^x=0
-1+√x+√x-x+x-x√x=0
(√x-1)+(√x-x)+(x-x√x)=0
(√x-1)-√x(√x-1)-x(√x-1)=0
(√x-1)(1-√x-x)=0
√x-1=0 √x=1 x=1 or
1-√x-x=0 x+√x-1=0
√x=(-1+-√5)/2
√x=(-1+√5)/2 , x=[(-1+√5)/2]^2
x=(3-√5)/2
0

[@SALogics]

Very nice! ❤

[@prollysine]

let u=Vx , u^3-2u+1=0 , (u-1)(u^2+u-1)=0 , u= 1 , (-1+V5)/2 , (-1-V5)/2 , x=u^2 , x= 1 , (3-V5)/2 , (3+V5)/2 ,
test , x=1 , OK , (3-V5)/2 , OK , / (3+V5)/2 , not a solu / , solu , x= 1 , (3-V5)/2 ,

[@SALogics]

Very nice! ❤