Scalar LQR: solving the Riccati equation for the optimal gain

Consider the scalar LTI system $\dot{x} = ax + bu$ with $a = 1$, $b = 1$ (open-loop unstable). You design an infinite-horizon continuous-time LQR controller minimizing $J = \int_0^{\infty} \left( q\,x^2 + r\,u^2 \right) dt$ with $q = 3$ and $r = 1$. The optimal control is $u = -Kx$ where $K = \frac{b}{r}P$ and $P > 0$ solves the control algebraic Riccati equation (CARE). Compute the resulting closed-loop pole (the eigenvalue of $a - bK$).