Derivation of range migration algorithm
So if you continue the substitution you get:

$R_t = \sqrt{((-R_BK_X/\sqrt{(K_R^2 - K_X^2)} + X_t - X_t)^2 + R_B^2)} $

$R_t = \sqrt{((-R_BK_X/\sqrt{(K_R^2 - K_X^2)} )^2 + R_B^2)} $

$R_t = \sqrt{(R_B^2K_X^2/(K_R^2 - K_X^2) + R_B^2)} $

$R_t = \sqrt{(R_B^2K_X^2/(K_R^2 - K_X^2) + R_B^2 (K_R^2 - K_X^2)/(K_R^2 -K_X^2))}$

$R_t = \sqrt{(R_B^2K_R^2/(K_R^2 - K_X^2))}$

$R_t =R_BK_R/ \sqrt{(K_R^2 - K_X^2)}$

Insert that into

$ -K_RR_t + K_X^2R_B/\sqrt{(K_R^2 - K_X^2)} - K_XX_t + K_RR_s $

you get:

$ -K_R^2 R_B/ \sqrt{(K_R^2 - K_X^2)} + K_X^2R_B/\sqrt{(K_R^2 - K_X^2)} - K_XX_t + K_RR_s $

$ -(K_R^2-K_X^2) R_B/ \sqrt{(K_R^2 - K_X^2)}  - K_XX_t + K_RR_s $

and there you are

$ -R_B\sqrt{(K_R^2-K_X^2)}   - K_XX_t + K_RR_s $