Question

For what integral values of k, the roots of the equation $k{x}^2+(2k-1)x+(k-2)=0$ are irrational?

Answer 1

$k{x}^2+(2k-1)x+(k-2)=0$
Consider the discriminant of the above equation
$D=B^2-4AC$
$=(2k-1{)}^2-4(k-2\left)\right(k)$
$=4k^2-4k+1-4(k^2-2k)$
$=4k^2-4k+1-4k^2+8k$
$=4k+1$
For irrational roots $4k+1$ cannot be a perfect square.
Hence $k\ne 2,6,12,\mathrm{20...}$
Any real values of 'k' not falling in the above series gives irrational roots.