Stating that either root is zero in solving a quadratic equation

Question

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.]

Answer 1

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.

Answer 2

The Two Kinds of “OR” It turns out, in mathematical logic, there are two completely different kinds of OR: the inclusive OR and the exclusive OR. The inclusive OR always includes the “both” case. For the inclusive OR, the phrase “A or B” includes the possibilities:

(a) A by itself

(b) B by itself

(c) A and B together

By contrast, the exclusive OR always excludes the “both” case. For the exclusive OR, the phrase “A or B” includes (a) & (b) above, but definitely NOT (c). In colloquial language, often context makes it clear which OR is intended. For example, if you imagine when you were, say, eight years old, and it was around dessert time, and your mother said emphatically, “You can have cake or ice cream!!!” At that moment, your mother most certainly was not expatiating on the merits of the inclusive OR. In fact, in many scenarios of ordinary life, the word OR implicitly means the exclusive OR—especially when any emotional emphasis lands on the word itself!

The Word “OR” in Mathematics Things change when we look at the use within mathematics. First of all, the word OR is very important: it is part of the logical apparatus in a problem. For example, the solution of quadratic usually yields two solutions, and these solutions are related by the word OR. In mathematics, wherever the word OR appears, it is always the inclusive OR, 100% of the time. If a mathematician says “A or B,” she always implicitly includes the case of A and B together. If the mathematician wants to specific an exclusive OR, she would have to go out of her way to say something wordier: “A alone or B alone but not both.” (In Symbolic Logic and in the Computer Science world, folks sometimes use the abbreviation XOR for the exclusive OR.)

Answer 3

If you write a polynomial $p(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0$, then by a theorem (the Fundamental Theorem of Algebra), you can completely factor the polynomial as $p(x) = a_n(x - b_1)(x-b_2)\cdots(x - b_n)$ where the $b_j$'s are the (potentially complex) roots.

Then $p(x) = 0$ precisely when $x = b_j$ for at least one $j, 1 \leq j \leq n$. With this phrasing, the "at least one" covers the issue you're talking about. Note, however, if $x = b_j$ and $x = b_k$ that it follows that $b_j = b_k$, so two or more of the linear facts $(x - b_j), (x - b_k)$ will be zero simultaneously if and only if $b_j = b_k$.

So in your example, it can never be the case that for some $x$, both $x - 0 = 0$ and $x - 3 = 0$. But if you take the polynomial $x^2 - 2x + 1 = (x-1)(x-1)$, then for $x = 1$ you have that both are simultaneously $0$.