Why must a radical be isolated before squaring both sides?

You can of course write $$ (\sqrt{2x+1}+1)^2=x^2, $$ but when you multiply out you get $$ 2x+2\sqrt{2x+1}+2=x^2, $$ and there is still a radical in your new equation. The point in isolating the radical is that after that, as you square the equation, you get rid of it completely.

Well, it doesn't lead to the wrong path: if you square the equation right away you get that

$$(2x+1)+2\sqrt{2x+1}+1=x^2$$

And because $\sqrt{2x+1}=x-1$, you get the equation

$$(2x+1)+2(x-1)+1=x^2$$

This is $x^2-4x=0$ which has solutions $x=0,4$, but the solution $x=0$ doesn't do it because $\sqrt{1}$ is taken to be $1$ (and not $-1$).

(Sure, I'm replacing the square root by clearing $x-1$: everything you do to solve the equation will end up being equivalent. Point is that it is not wrong, just a bit more farfetched. It is interesting to note that mindlessly squaring forces one to note "clearing the square root" is necessary, and one can then realize one could have started doing this in the first place.)

You can do what ever you darned well want. But you have to do it correctly. You can do this:

$\sqrt{2x + 1} + 1 = x$

$(\sqrt{2x + 1} + 1)^2 = x^2$

$(2x + 1) + 2\sqrt{2x + 1} + 1 = x^2$

but now what? ... it's true but you've just made things more difficult.

What you can NOT do under any circumstances is this:

$(\sqrt{2x + 1} + 1)^2 = x^2$

$(2x + 1) + 1 = x^2$

The point is if you want to solve it, then you want to get rid of the radical and you can't do that is you square a sum with other terms.

$(a + \sqrt{b})^2 = a^2 + 2a\sqrt{b} + b$

So that doesn't do anything to get rid of it. But it's not wrong. It's just... not what you want.

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" Is there some special property of radicals that makes them have to be completely alone before they can be squared?"

Not really, everything has to be completely alone before you square it if you don't want the square to involve other ... things in it.

Actually, your question is a bit like asking "Why must we isolate before we divide:"

"$3x + 2 = 11$"

"Why do we isolate the $3x$

"$3x = 11 -2$

"Why don't we just divide first:

""$(3x +2)/3 = 11/3$

The answer is we can do what we darned well like:

$x + \frac 23 = \frac {11}3$

$x = \frac {11}3 - \frac 23$.

Nothing wrong with that... but nothing right with it either.

We isolate terms, for whatever operation, for the purpose of isolating them so we can work directly with them.

That's all.