TITLE: Why must a radical be isolated before squaring both sides?
QUESTION [13 upvotes]: In the following equation:
$$\sqrt{2x + 1} + 1 = x$$
You are supposed to isolate the radical:
$$\sqrt{2x + 1} = x - 1$$
And then proceed by squaring both sides.
If you start by solving the equation this way, you will eventually complete the square and get an answer of: $$4$$
However, why must the radical be isolated before squaring both sides?
Why can't you do, for example...
$$(\sqrt{2x + 1} + 1)^2 = x^2$$
I know this would lead you down the wrong path, but I don't know why. It doesn't make sense to me because I can (once I isolate the radical) square both sides when one side $$x-1$$
involves addition/subtraction. Is there some special property of radicals that makes them have to be completely alone before they can be squared?
Thank you.

REPLY [3 votes]: The main problem with your approach is that squaring will still leave a radical, whereas isolating the radical won't. So it's mostly about simplicity, rather than doing some compulsory transformation.
However, squaring is not sufficient. You can only do
$$
(\sqrt{2x+1}+1)^2=x^2
$$
under the assumption that $x\ge0$, or you may add some spurious solutions.
There is a different approach, though. You can set $t=\sqrt{2x+1}$, with $t\ge0$, and so $2x+1=t^2$ and
$$
x=\frac{t^2-1}{2}
$$
so your equation becomes
$$
t+1=\frac{t^2-1}{2}
$$
that simplifies to
$$
t^2-2t-3=0
$$
The roots are $t=-1$ and $t=3$, but the negative root must be discarded. Therefore we get $\sqrt{2x+1}=3$, that easily gives $x=4$.

The alternative method of isolating the radical is, however, simpler. 
We get $\sqrt{2x+1}=x-1$, which, under the condition $x-1\ge0$, can be squared:
$$
2x+1=x^2-2x+1
$$
and so $x^2-4x=0$. The roots are $0$ and $4$, but only the latter satisfies $x\ge1$.