1.  This is a mistake, but not because it is mathematically illegal,
for it is not mathematically illegal to take positive square roots 
of both sides.  However it gets nowhere and makes matters worse.  So 
what the student should have done instead was to have isolated the x²
term, since x is what is being solved for, and x is just the square 
root of x².

2.  However, since the student took positive square roots of both sides,
look at what illegal thing he then did: 

b + x = a 

He erroneously thought "the square root of a sum is the sum of the square
roots".  But there is no such rule.  

What happened was that he mis-remembered this correct rule:

The square root of a product is the product of the square roots.

and thought he could just replace the word "sum" for the word "product"
and he'd still have a correct rule.  But that doesn't work.  You cannot
replace "sum" for "product" in a rule and expect the result to be a 
correct rule.

Here is how he should have done it:

 

Isolate the  term:

 

Since x is squared, x could be either positive or negative,
so we must take both the positive and the negative square roots 
of the right side. That's called "the principle of square roots".
So the two solutions are:



Edwin