In your last statement, you didn't show why x ≠ 0 . The principal square root operation allows the square root of 0 , so there must be a reason for the exclusion.
The 2nd line is the square of the 1st line. If we had shown the expansion of the square before simplification, we would have: (2*√x)² + 1² - 2*(2*√x) = (√(2*x + 1))² With x = 0 , we have: (2*√0)² + 1² - 2*(2*√0) = (√(2*0 + 1))² 1² = (√1)² The inverse of the square isn't a 1-to-1 function, as there are the + and - branches, when describing the roots using the principal square root function: 1 = ±√1 1 = ±1 Thus, squaring the equation allowed the negative branch as a solution. That's why we must check our initial results against the original equation to eliminate spurious solutions introduced in the process of solving the problem.
He lept to the conclusion that x ≠ 0 without a valid reason. But, if you use x = 0 in the original equation, then you'll see that it fails: 0 - 1 - 1 = 0 is false.
