The standard form of a quadratic equation ONLY defines the form of the equation,
but DOES NOT DEFINE the values of the coefficients.
The values of the coefficients can be all multiplied by any non-zero coefficient, leaving the equation EQIVALENT;
THEREFORE, the values of the coefficients in the standard form quadratic equation are defined only accurate to the common non-zero multiplier.
Therefore, all your attempts to "invent" a new class of Algebra problems are empty and make no sense.