Solution to the equation $\sqrt{x^2 - 2x + 1} - 5 = 0$ I had this equation on my exam :
$$\sqrt{x^2-2x+1} - 5 = 0$$
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.

Your solution is definitely correct:
\begin{align}
&\sqrt{x^2 - 2x +1} - 5 = 0    \\
\Rightarrow \; &x^2 -2x+1 = 25 \\
\Rightarrow \; &(x-6)(x+4) = 0  \\
\Rightarrow \; &x = 6 \text{  or  } x=-4
\end{align}
Your friend's approach is also correct, but he (or she) has a typo. After correcting this, you get:
\begin{align}
&\sqrt{x^2 - 2x +1} - 5 = 0    \\
\Rightarrow \; &|x-1| = 5 \; \text{  (not  }-5)\\
\Rightarrow \; &x = 6 \text{  or  } x=-4
\end{align}
