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We derived an algorithm to find the real roots of the homogeneous quadratic equation, Ax2+2Bxw+Cw2=0. Because the equation is homogeneous, a root consists of an [x, w] pair where any nonzero multiple represents the same root. We strove to find an algorithm that didn't blow up no matter what values of A, B, and C we were given, including various combinations of zeroes. At the end of the article the author wrote the final algorithm in tabular form.