A simplified algorithm for decoding binary quadratic residue (QR) codes is developed in this paper. The key idea is to use the efficient Euclidean algorithm to determine the greatest common divisor of two specific polynomials which can be shown to be the error-locator polynomial. This proposed technique differs from the previous schemes developed for QR codes. It is especially simple due to the well-developed Euclidean algorithm. In this paper, an example using the proposed algorithm to decode the (41, 21, 9) quadratic residue code is given and a C++ program of the proposed algorithm has been executed successfully to run all correctable error patterns. The simulations of this new algorithm compared with the Berlekamp-Massey (BM) algorithm for the (71, 36, 11) and (79, 40, 15) quadratic residue codes are shown.