Sorry if the question is a little misleading, but I have no better way to express it. The text below should clarify.
Suppose I have the equation of a conic: $x^2+y^2+z^2-2x+3y+z+2=0$, with this I need to complete the squares and then rewrite it in an equation such as the equation of the parabola/hyperbola/etc.
If I expand the expression with the squares completed, should it necessarily be equal to $$x^2+y^2+z^2-2x+3y+z+2\stackrel{?}{=}0$$
- With equal I mean that one expression is transformable into the other via the common notions given in Euclid's book $1$.
- Things which equal the same thing also equal one another.
- If equals are added to equals, then the wholes are equal.
- If equals are subtracted from equals, then the remainders are equal.
- Things which coincide with one another equal one another.
- The whole is greater than the part.
- EDIT: It can also be the field axioms, as pointed out by Rory Daulton. I've reread the field axioms and think they are more appropriate to what I'm looking for.