How can we say later that x can be equal to zero when we took x as a common factor and hence divided by it?

Question

To solve an equation like this:

$x^2+x=0$

We can take $x$ as a common factor

$\frac{x^2}{x}+\frac{x}{x}=0$

$x(x+1)=0$

However to do such a step we have to say that $x$ isn't equal to zero, because after all, if $x$ is equal to zero, then $\frac{x}{x} \neq1$, like we factorized. Instead our factorization should look like this:

$x(∞)=0$

And even though we stated clearly that $x$ can't be equal to zero above, later on to get the solutions we state:

$x=-1$ $or$ $x=0$

I know this is a very silly question and that's why I didn't want to ask my teacher about it. And I totally get that it doesn't matter in this equation anyway.

But shouldn't this be considered wrong, to state that $x$ shouldn't be a certain value to reach a certain step then when we are over that step and we reach another step we say that $x$ is that certain value we based our work on the fact that it shouldn't be it.

Answer 1

There are two ways to solve your equation. First, we could deal with the cases separately. If $x \neq 0$, then we can safely divide by $x$, meaning that $$ x^2+x=0 $$ implies $$ x+1=0 $$ which in turn implies that $x=-1$. Then, we have to deal with the case that $x=0$. This is also a solution to your equation—we can verify this by plugging it in.

There is a second, more elegant way of solving the equation, that you yourself have alluded to. If $x^2+x=0$, then $$ x(x+1)=0 \, , $$ which implies $x=0$ or $x=1$. Note that there is no division by $x$ here at all. Finally, you appear to suggest at one point in your question that $0/0 = \infty$. This is not true. In fact, $0/0$ is undefined. It is as meaningless to talk about $0/0$ as it is to talk about $\text{yellow}/\text{blue}$.

Answer 2

The 4 biggest words in math are If, Then, Or, And. For practice, write whole math arguments in complete grammatically correct sentences and do not omit the big 4. And do not omit any justifications.

Illogical: If $x^2+x=0$ then $x+1=(x^2+x)/x=0/x=0.$

Logical: If $x^2+x=0$ and if $x\ne 0$ then $x+1=(x^2+x)/x=0/x=0.$

"Illogical" includes an unwarranted hidden assumption that division by $x$ is possible. But "Logical" above still doesn't answer the whole original Q, because it includes an extra "if" (i.e. if $x\ne 0$) that's not in the original Q.

Complete: $x^2+x=0\iff x(x+1)=0$ [by the Distributive Law] $\iff (x=0\lor x+1=0)$ [by a basic axiom of arithmetic] $\iff (x=0\lor x=-1)$ [Let's not get too tedious; you can skip the justification for this].

It is especially useful when you can manage to have all "$\iff$", as then the argument reads logically in reverse order.