This paper investigates the effect of gain and phase margins (GPMs) on stability delay margin of a two-area load frequency control (LFC) system with constant communication delay. A gain-phase margin tester (GPMT) is introduced to the LFC system as to take into GPMs in delay margin computation. A frequency domain exact method, Rekasius substitution, is proposed to compute the GPMs-based stability delay margins. The method aims to calculate all possible purely complex roots of the characteristic equation for a finite positive time delay. The approach first transforms the characteristic polynomial of the LFC system with transcendental terms into a regular polynomial. Routh-Hurwitz stability criterion is then implemented to compute the purely imaginary roots with the crossing frequency and stability delay margin. For a wide range of proportional-integral controller gains and GPMs, time delay values at which LFC system is both stable and has desired stability margin measured by GPMs are computed. The accuracy of complex roots and delay margins are verified by using an independent algorithm, the quasi-polynomial mapping-based root finder and time-domain simulations. Simulation studies indicate that delay margins must be determined considering GPMs to have a better dynamic performance in term of fast damping of oscillations, less overshoot and settling time.