Writing solutions to a quadratic equation using set notation $x = \{a,b\}$

Question

If I have a quadratic equation

$$(x-1)(x-2)=0$$

Is it okay to write $x = \{1,2\}$?

What about writing $x \in \{1,2\}$? Would that be better?

And if none of the above are okay, is it okay to write "the solutions of this equation are $\{1,2\}$?"

Answer 1

$x = \{1,2\}$ is not acceptable, as $(\{1,2\} - 1)(\{1, 2\} - 2)$ doesn't make sense. $x \in \{1, 2\}$ is acceptable, since the two statements $(x-1)(x-2) = 0$ and $x\in\{1, 2\}$ are equivalent. As for your last sentence, it would be slightly more correct to say either "The solution set is $\{1, 2\}$" or "The solutions are $x = 1$ and $x = 2$."

Answer 2

All three suggestions convey the desired meaning, but all three have technical deficiencies. $x$ is not a set. The issue with the second one is that you are only saying that the solutions are among the set $\{1,2\}$, not that both of those elements are solutions.

You can say the solution set of this equation is $\{1,2\}$. The set $\{1,2\}$ is singular, so it's ungrammatical to say the solutions are this set.

Answer 3

The notation $x \in \{1,2\}$ is totally correct and the notation "the solutions of this equation are $\{1,2\}$", to a certain extent is correct. However, I would modify the second notation to "the solution set is $\{1,2\}$" or "the solutions of this equation are $x=1$ and $x=2$". But the notation $x = \{1,2\}$ is wrong since it creates the illusion that $x$ is a set whereas it is not so, it is just a variable and can assume real numbers.

Answer 4

It is wrong to write $x = \{1,2\}$. The notation $x \in \{1,2\}$ is fine, but you have to be careful with the case the roots are the same, as the set $\{ 3,3 \}$ actually means $\{ 3\}$. While this is typically easy to interpret for quadratics, it leads to issues for higher degree equations; sets cannot differentiate between $x=1, x=2, x=2$ and $x=1, x=1, x=2$. This is the reason why people typically don't use sets for equations in general.