4 ms·

And that's just a special case of Completing the Square (but has a very handy visualization). https://en.m.wikipedia.org/wiki/Completing_the_square https://en.

• romwell 8y ago
  Not quite. "Completing the square" is an algebraic step; the argument is geometric (translation of the parabola). One can learn how to complete the square (symbolic manipulation) and be completely unaware of the geometry. If you never learned the algebraic trick, would you invent it when solving this problem? It's not immediately clear from the algebra that a quadratic polynomial ax^2 + bx + c even can be rewritten in the form a(x-Q)^2 - R. Conversely, while this technique involves algebra that amounts to completing the square, it exhibits a general technique. If you want to say "..and that's just a special case of..", homotopy continuation methods[1][2] - - because the idea here is transforming the simpler polynomial x^2 - C into a more complicated one, and seeing what happens to the roots. [1]https://en.wikipedia.org/wiki/Numerical_continuation https://en.wikipedia.org/wiki/Numerical_continuation [2] https://en.wikipedia.org/wiki/Numerical_algebraic_geometry#Homotopy_continuation https://en.wikipedia.org/wiki/Numerical_algebraic_geometry#H...