Problem #PRU-100281

Problem

Take $x=-\frac{1}{2}$. It solves $$2x+1=0.$$ Add $x^2$ to both sides of the above equation $$x^2+2x+1=x^2.$$ Completing the square, we rewrite the equality as $$(x+1{)}^2=x^2.$$ Now, taking the square root from the both sides of the equality we get $$x+1=x.$$ Subtracting $x$ from both sides we get $$1=0$$ Can you explain what went wrong in our reasoning? Why?