<s>[INST] 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.
 [/INST] It is $$\sqrt{x^2-2x+1} - 5 = 0 \\ \Rightarrow \sqrt{(x-1)^2} - 5 = 0 \\ \Rightarrow |x-1|-5=0 \\ \Rightarrow |x-1|=5 \\ \Rightarrow x-1=5 \text{ or } x-1=-5 \\ \Rightarrow x=6 \text{ or } x=-4$$ So, your solution is correct!! 
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If the equation is $\sqrt{x^2-2x+1} + 5 =0$ then we have the following: 
$$\sqrt{x^2-2x+1} + 5 = 0 \\ \Rightarrow \sqrt{(x-1)^2} + 5 = 0 \\ \Rightarrow |x-1|+5=0 \\ \Rightarrow |x-1|=-5$$ 
So, there is no solution. 
In this case the solution of your friend is correct. 
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