How many real solutions of $$6x^2 -77[x] +147=0$$ are there, where $[x]$ is the integral part of $x$?
The answer says 4 solutions but I got none.
$6x^2 + 147 = 77[x]$ LHS= integer Therefore, RHS = integer
Then $6x^2$ just be integer
Then CASE 1: if $x$ is integer then $[x] = x$ By solving we get $x = 21/3$ and $7/2$ But x is integer , therefore no solution
CASE 2: if $x$ is not integer but $6x^2$ is integer I.e. $x^2= 1/6$. so $x = \pm 1/√6$ But non e satisfies the equation so no solution
Hence overall no solution.
I don't understand how there are 4 real solutions.