Greatset Integer function..

\hspace{-16}$Calculate all Real $\mathbf{x}$ in $\mathbf{[x]+[\sqrt{x}]=2x-2}$\\\\ Where $\mathbf{[x]=}$ Greatest Integer function.

1 Answers

262

Aditya Bhutra ·2012-03-11 22:41:21

x+âˆšx -2 < [x] +[âˆšx] â‰¤ x+âˆšx

x+âˆšx -2 < 2x-2 â‰¤ x+âˆšx

therefore x+âˆšx -2 < 2x-2 â†’x>1

and 2x-2 â‰¤ x+âˆšx â†’ xâ‰¤4

now lhs is integer therefore rhs should also be integer. hence x is integer.

form the above bound the only integers in range are 2,3,4

plugging them into the eqn we get x=3 and x=4 as the solutions.

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