Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2017 / S18 / 2 / b / Solution

Continuity has already been built into these forms. At $b$, derivative continuity requires $-q\sin (qb)=-\kappa \cos (qb)$ for even parity and $q\cos (qb)=-\kappa \sin (qb)$ for odd parity. ThusUse $q/\kappa =\sqrt{U_0/|E|-1}$ to obtain the two printed equations. The right side is positive, so only the corresponding allowed tangent/cotangent branches can describe a bound state; squaring without retaining this sign restriction adds spurious roots.