To find the position where a third charge does not experience any force between two charges of 1.67 µC and -0.6 µC, the net forces from both charges must balance, leading to the equation F_1 + F_2 = 0. The initial setup was incorrect as it equated the forces without considering their opposing directions. The correct approach involves setting F_1 equal to -F_2, resulting in a solvable quadratic equation. The discussion emphasizes the importance of clear communication in problem-solving, particularly when sharing calculations. The corrected equation should yield a valid solution for the position of the third charge.