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Let's say in order to solve an equation [TEX]f(x)=0[/TEX], we multiply the equation by [TEX]x[/TEX]. Therefore we conclude that x can never be =0. But what if at the end step we conclude that [TEX]x=0[/TEX] (maybe along with other solutions)? Do we reject the answer and accept the others? Or is our method of solving the equation incorrect?

I do know that it is quite impossible for you to arrive at [TEX]x=0[/TEX] after you multiply [TEX]x[/TEX] throughout in order to solve the equation, as that would mean you had introduced an unnecessary common multiple into the equation..