Does the phrase "quadratic formula" (the formula for the solutions of ax2 + bx + c = 0 [EDIT: I mean the famous formula with the square root of b2-4ac as part of the numerator]) not really get directly translated into Russian? It seems that this is described by phrases like формула корней (или решений) квадратного уравнения or теорема Виета, but not квадратная формула
You're right. Квадратная формула is not used in Russian. You seem to have listed the available translations yourself.
On a related issue, I have encountered both квадратный and квадратичный, but the latter seems to be less often used (e.g., квадратичное расширение or квадратичный вычет).
Квадратный is usually translated as simply
square, квадратичный - as
quadratic. The former term sounds less scientific than the latter, so I suppose that it's part of the reason why in some more basic expressions we use квадратный rather than perhaps the more appropriate but not established квадратичный. An example would be квадратное уравнение (if I were to establish the term anew, I'd probably go with квадратичное уравнение, but the term is already established).
Is квадратичный allowed only in a more limited range of expressions, so that terms like квадратичный многочлен, квадратичный корень, or квадратичная матрица would all sound weird?
Квадратичная матрица sounds weird because it is a square matrix (the form), not quadratic (i.e. it's not about exponent = 2). Квадратичный многочлен doesn't sound weird at all. Квадратичный корень does sound weird, because perhaps the established term квадратный корень is very widespread