Solution to $\sqrt{x^2-2x+1} + 5 = 0$ I asked this question before, I wrote it wrong!
$$\sqrt{x^2-2x+1} + 5 = 0$$
My friends said the the solution could be :
$$|x-1| = -5$$
 So the solution is nothing!
But I say the solution is:
$$x^2-2x+1 = 25  $$
so $$x = 6\ |\ x = -4$$  
My Question here is which solution is right, and why.
Thanks in advance.
 A: By definition $\sqrt{x^2-2x+1}$ is always positive, i.e.,
$$\sqrt{x^2-2x+1} = \vert x-1 \vert$$
Hence, no solution exists.
A: Your friends are correct.
\begin{align*}
\sqrt{x^2 - 2x + 1} + 5 & = 0\\
\sqrt{x^2 - 2x + 1} & = -5\\
\sqrt{(x - 1)^2} & = -5\\
|x - 1| & = -5
\end{align*}
which has no solutions.  
By convention, the expression $\sqrt{u}$ means the non-negative square root of $u$.  Therefore, the equation 
$$\sqrt{x^2 - 2x + 1} = -5$$
has no solution.  When you squared both sides of the equation
$$\sqrt{x^2 - 2x + 1} = -5$$
to obtain 
$$x^2 - 2x + 1 = 25$$
you introduced extraneous solutions since the equation $x^2 - 2x + 1 = 25$ has solutions that the equation $\sqrt{x^2 - 2x + 1} + 5 = 0$ does not.  
A: You can simply check the solutions by inserting them.
However, in general by adding positive terms you will never reach the value zero. The same goes by adding a non-negative term to a positive term.
A: Addition of $2$ positive real numbers gives something positive. Now $5$ is positive and so is the square root of $x^2-2x+1$ i.e $|x-1|$ for all real $x$. Hence their addition must be greater than $5$ and obviously not $0$.
So your friend is right when he says that there is no solution.
