I'm relearning calculus but I often find myself applying algebraic operations to equations mechanically without having a solid understanding of the side-effects of those operations. Such as extra or reduced number of roots when squaring both sides or taking the square root of both sides. What book do you recommend for properly learning the theory of solving equations in depth?
A simple example is solving for y in $y^2=x^2$. I'd like a book that discusses solving such an equation and other forms starting from first principles and stating definitions and proving theorems on the way.