When you square both sides, you introduce potential solutions which may not necessarily satisfy the original equation. This occurs because the square root is not unique. For example, if x=-2 then x^2 = 4 so x = -2, 2 so an extra value of x is incorrectly introduced which fails to satisfy the original statement.

Notice that your extra solutions don't work on the original equation but work on the version where you square both sides. You need to recognise this limitation in your operations and explain that the extra solutions need to be removed.