When the quadratic formula has square root of zero, how to proceed?

The square root function, in the reals, $\sqrt a$ is defined for all $a\geq 0$: that means the square root of a real number $a$ is defined strictly for all $a$ greater than or equal to $0$.

So your root becomes $\dfrac {2\pm 0}{2} = 1$. This root has multiplicity of two; indeed, $$x^2 - 2x + 1 = (x-1)^2 = (x-1)(x-1)$$