Q:

Stating that either root is zero in solving a quadratic equation

Let's say we have a simple quadratic equation $x^2 - 3x = 0$. To solve, we will factor $x$ out i.e. $x(x-3)=0$, after which we will state $x = 0$ or $(x-3) = 0$. My question is, why is there no third "option" where we say "or both". Isn't it possible for both "portions" (i.e. $x$ and $(x-3)$) to both be equal to zero? After all, if most quadratic equations would have two roots, then both $x = 0$ and $x = 3$ are the roots thus both $x = 0$ and $(x-3) = 0$ are true! 
[This question on the use of the word "or" applies to polynomial equations with degrees 3 and above too of course, but I'm choosing a quadratic one as it is the simplest case possible.]

A:

If $x=0$, then $x \ne 3$. If $x=3$ then $x\ne 0$. $x$ can't be two different numbers at the same time.

