A bit of SymPy (symbolic math) awesomeness.

sympy_quadratic.py

>>> from sympy import Symbol, solve, pprint

>>> a=Symbol('a')
>>> b=Symbol('b')
>>> c=Symbol('c')
>>> expr = a*x*x + b*x + c
>>> solve(expr)
[{a: -(b*x + c)/x**2}]
>>> solve(expr,x, dict=True)
[{x: (-b + sqrt(-4*a*c + b**2))/(2*a)}, {x: -(b + sqrt(-4*a*c + b**2))/(2*a)}]
>>> pprint(solve(expr,x, dict=True))
⎡⎧           _____________⎫  ⎧    ⎛       _____________⎞⎫⎤
⎢⎪          ╱           2 ⎪  ⎪    ⎜      ╱           2 ⎟⎪⎥
⎢⎨   -b + ╲╱  -4⋅a⋅c + b  ⎬  ⎨   -⎝b + ╲╱  -4⋅a⋅c + b  ⎠⎬⎥
⎢⎪x: ─────────────────────⎪, ⎪x: ───────────────────────⎪⎥
⎣⎩            2⋅a         ⎭  ⎩             2⋅a          ⎭⎦
>>>