Are there theorems about extraneous solutions? What causes them and how many are produced? If so, which branch of mathematics would I find such theorems (and proofs)? Same thing with missing solutions.


When learning to solve equations, the first fundamental rule is "Do the same thing to both sides". However, some times you need to be careful.

When the thing you do isn't always valid (for instance, you're dividing with something that could potentially be $0$), then you might end up losing solutions. Example: $$ x(x-5)=3x\\ x-5=3\\ x=8 $$ Dividing by $x$ lost us the solution $x=0$ (because for that solution we would be dividing by $0$). In general, you potentially lose solutions at points where the operation you do is invalid. Whether you actually lose solutions, and how many you lose, differs from equation to equation, and it's difficult to give a general answer.

When the thing you do could make different numbers equal, you may introduce extra solutions. Example: $$ 2\sqrt x=x-1\\ 4x=(x-1)^2\\ 4x=x^2-4x+1 $$ which has solutions $x=3-2\sqrt2\approx 0.17$ and $x=3+2\sqrt2\approx 5.83$. Note that if you insert these two solutions into the original equation, only one of them is a solution. The other one makes the left-hand side positive and the right-hand side negative, which is not equal. However, after squaring them they become equal and thus an extra solution is born. You get extra solutions at points where the two sides are different numbers, but the thing you do makes them equal. How many times this happens (and this how many extra solutions you get) differs from equation to equation, and it's difficult to give a general answer.

You also potentially get extra solutions when your manipulations give you expressions that are valid at points where your original expressions weren't valid, and similarly you potentially lose solutions when your new expressions are more restrictive.

I don't think there is a specific theorem for this. It's just something that you learn to keep in mind as you go.


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