I've just been going through an equation, which is as follows:
$(x+4)^2 = 16$
Lets work through it:
$$x^2 + 8x +16 = 16$$ $$x^2 +8x = 0$$ $$x^2 = -8x$$ $$x = -8$$
However, as i've just found out, $0$ is also a solution, as $0 = -8$. Oh wait, no it doesn't.. And yet, $0^2 + 8 \times 0 + 16 = 16$.
What's going on here though? Does $0$ even matter? It's true that $16 = 16$, but it's not true that $0 =-8$. But maybe $16$ equals $16$, not because $-8^2 - 8 \times 8 = 0$, but just because $16$ equals to itself, independent of $0$.
I think it's interesting to draw a parallel between the falsity of the final solution (when $x = 0$), and the fact that nothing, represented by $0$, literally doesn't seem to exist in the real world.
Is $0$ ever used however? In either pure mathematics for a particular proof, or perhaps within the domain of science? I suppose one use of $0$ would be to do a 'hard reset', for whatever reason, but then again, you could just do that without $0$, I suppose.
Furthermore, maths in supposed to be consistent, and since the consistency of mathematics seems to be disrupted by the usage of $0$, does that mean $0$ is somehow 'dangerous'. My example here was simple, but what if you were say, launching a rocket to space, where lives are on the line. Can you trust $0$?
There is no inconsistency - you just made a mistake. You cannot conclude "$x=-8$" from "$x^2=-8x$". When you try to divide both sides by $x$, you need the hypothesis that $x\not=0$!
How your argument should go is, " . . .so $x^2=-8x$. Now, if $x\not=0$, then we can divide both sides by $x$ to get $x=-8$. So $x$ must be either $0$ or $-8$. Checking the original equation shows that both $0$ and $-8$ work, so those are our solutions."
Don't worry, math is safe. :D There's nothing weird about zero, other than that you can't divide by it. The moral is that, when solving a problem, you need to think about what you're doing - in particular, at each step, you need to understand why what you are saying is true. Otherwise, it's easy to convince yourself that an argument makes sense, or is complete, when in fact it does/is not.