A Value Outside the Domain Is Not a Solution at All

Both sides of that equation are built out of $\frac{1}{x-3}$, which means nothing at $x = 3$. So the question is about every real number except 3. That set is the domain; it arrives with the problem, and you do not add it at the end. Multiplying through by $x-3$ is allowed for every number the question is about, since none of them make $x-3$ zero.

$$\begin{aligned} x &= 3 + 2(x-3) \\ x &= 2x - 3 \\ x &= 3 \end{aligned}$$

The only candidate is the one number the question excluded, so nothing survives: the equation has no solution. You can see it without solving, too, since $\frac{x}{x-3} - \frac{3}{x-3} = 1$ for every $x \neq 3$, which leaves the equation asking for $1 = 2$ across its whole domain. The restriction is not a formality attached to a correct answer; it is the reason there is no answer.

Some restrictions are invisible in the first line. Take

$$\log x + \log(x-3) = 1.$$

Read $\log$ as base ten. Here $\log x$ needs $x > 0$ and $\log(x-3)$ needs $x > 3$, so the domain is $x > 3$. Combining the two logarithms gives

$$\log\big(x(x-3)\big) = 1, \qquad x(x-3) = 10, \qquad x^2 - 3x - 10 = 0,$$

which factors as $(x-5)(x+2) = 0$, so the candidates are $x = 5$ and $x = -2$.

Look at where $-2$ came from. The rewritten line is defined whenever $x(x-3) > 0$, so for $x > 3$ and also for $x < 0$. Nothing was lost; the second line is just defined on more numbers, so it can hand back answers the first never had. The product $(-2)(-5) = 10$ is fine, while $\log(-2)$ does not exist. That value was never eligible. The survivor is $x = 5$, and $\log 5 + \log 2 = \log 10 = 1$ confirms it.

Those two failures are the same kind: a number the question never asked about. An even root shows a second kind. Solve

$$\sqrt{x+2} = x.$$

The left side needs $x \geq -2$, so that is the domain. Squaring gives $x + 2 = x^2$, or $x^2 - x - 2 = 0$, which factors as $(x-2)(x+1) = 0$; the candidates are $x = 2$ and $x = -1$. But $-1$ sits inside the domain: $\sqrt{-1+2} = 1$, so the original equation reads $1 = -1$. Squaring let it through, because $a^2 = b^2$ does not give back $a = b$; it forgets the sign. The solution is $x = 2$, confirmed by $\sqrt{2+2} = 2$.

Textbooks call any candidate that survives your algebra but fails the original equation an extraneous solution, and two kinds are worth telling apart. Both $x = 3$ and $x = -2$ were outside the domain, ruled out before you wrote anything; $x = -1$ was inside it, put there by a one-way step of your own. That second kind, and the conditions your steps carry, are a subject of their own.

One study filmed three undergraduates doing their own textbook homework and looked closely at each step they chose^[1]. Most of those choices, and most of the ways they were carried out, happened without any thought about the mathematical properties of what was in front of them^[1]. Three people is a close look, not a measured rate^[1]. My own reading: a step chosen because the problem resembles an earlier one leaves its condition nothing to attach itself to.

The nearest published look at this slip is not about equations. Secondary-school students solving inequalities, including rational and square-root ones, most often ran into trouble by carrying a move over from equations as though it still applied; another source was keeping values that should have been excluded^[2]. The object had changed and the procedure had not: the same failure one topic across, not a study of our case^[2].

The question that catches it is not "what is the answer?" but "does this line have the same answers as the line above?" When 98 eighth- and ninth-graders judged whether pairs of equations had the same answers, some computed both while others looked at the step between the lines and asked whether it preserved the answers^[3]. The step-watchers were right more often, though that only shows the two went together in the same students, not that one caused the other^[3].

There is a deeper thread, and it is only my reasoning. When 271 college students and 36 junior-high teachers were asked to write down what a function is and then to use the idea, those who gave the formal definition frequently worked from a mental picture that did not match it^[4]. That study never touched domains or equation solving^[4]. But a definition names the set of inputs a function has, and a formula does not say where it stops.

In an experiment with 206 middle-school algebra students, the group that studied worked-out mistakes and explained what was wrong with them got better at solving equations than the other groups, and those who began knowing less about the parts of an algebraic expression gained more from it^[5]. They were working on quadratics, and nobody there was testing domains^[5].

Looking is not the active part. A review that pooled 42 studies found that studying somebody else's wrong solution helps only a little on its own, and more when the learner is prompted to explain the error or is given an explanation of it^[6]. A review pooling many mathematics studies found the same shape for explaining your own work: a modest gain on tests given straight afterwards, with much thinner evidence that it helps in a real classroom or that the gain lasts^[7]. Explaining why a common mistake is wrong is something those authors recommend, not something the review tested^[7].

So make the explaining cheap. Before solving anything with a fraction, an even root or a logarithm, ask three questions:

• Which numbers make a denominator zero? Those are out.
• Which numbers put a negative inside an even root? Those are out.
• Which numbers make the inside of a logarithm zero or negative? Those are out.

If none of the three turns anything up, as with a polynomial equation such as $2x + 1 = 5$ or $x^2 = 4$, there is no domain to write and writing one anyway is noise. Otherwise write what survives at the top of the page, solve, hold each candidate against that line, and put every survivor back into the original equation. When one fails, name the step that let it in.

Nothing behind this post studied domain restrictions in rational, radical or logarithmic equations, and nothing tested whether writing a restriction down changes a mark^[1][2]. The explanation is my argument from adjacent evidence: three undergraduates doing homework^[1], secondary-school students solving inequalities rather than equations^[2], and 271 college students and 36 junior-high teachers answering questions that never mentioned domains^[4].

Even the closest evidence is loose. The inequalities study names failing to reject excluded values as a difficulty among secondary-school students without saying how often it happened^[2], and the equivalence study shows two habits travelling together in the same eighth- and ninth-graders, not one producing the other^[3].

The practical advice is narrower than it sounds. The experiment behind it used 206 middle-school students on quadratics^[5], the review of 42 studies found only a small benefit overall from wrong examples^[6], and the self-explanation review rests mostly on tests given immediately^[7]. Treat "write the domain, then name the step that let the bad candidate in" as a reasonable bet, not a proven method.

None of this touches the mathematics. These studies are about how people work, not about what is true of an equation^[1][4]. That $x = 3$ and $x = -2$ are not solutions is settled by the two domains; that $x = -1$ is not one is settled by putting it back.