This paper proposes closed form solutions to the L₂-sensitivity minimization subject to L₂-scaling constraints for second-order state-space digital filters with real poles. We consider two cases of second-order digital filters: distinct real poles and multiple real poles. The proposed approach reduces the constrained optimization problem to an unconstrained optimization problem by appropriate variable transformation. We can express the L₂-sensitivity by a simple linear combination of exponential functions and formulate the L₂-sensitivity minimization problem by a simple polynomial equation. As a result, L₂-sensitivity is expressed in closed form, and its minimization subject to L₂-scaling constraints is achieved without iterative calculations.