In your equations, y is the length of the string, not the extension (otherwise you should not subtract 2m).

Using your ansatz: 5N/m * (y-2m)^2 = 1/2kg*9.81m/s^2 * (y-1.5m)

(y/m-2)^2 = 0.981*(y/m-1.5)

This has the solutions y=1.64 (unphysical) and y=3.35, which is the right answer.