Calculate impulse response when output contributes in input
It should be pretty clear that just computing the root of the characteristic equation can't be enough to find the impulse response. Because in that case the impulse responses of all systems described by
$$y[n]=y[n-1]+\sum_{k=0}^Nb_kx[n-k]\tag{1}$$
would be the same, irrespective of the values of the coefficients $b_k$.
For the given example, the most straightforward (and most instructive) way to determine the impulse response is simply to go through the given iteration until a pattern emerges:
$$\begin{align}h[0]&amp;=1\\h[1]&amp;=\frac32\\h[2]&amp;=\frac32\cdot\frac12\\h[3]&amp;=\frac32\left(\frac12\right)^2\\\vdots\end{align}$$
Can you see any pattern?