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.
Best Answer
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.