How are quadratic Bezier curves drawn by TikZ?

As Paul Gaborit said in his comments (a) .. controls (b) .. (c) is not a quadratic Bézier curve, but the cubic one (a) .. controls (b) and (b) .. (c).

If you want to draw a quadratic curve in TikZ you can define your own to path style. Here is an an example of how to do it using calc library.

\documentclass[tikz,border=7pt,convert={density=1400}]{standalone}
\usetikzlibrary{calc}
\tikzset{
  quadratic/.style={
    to path={
      (\tikztostart) .. controls
      ($#1!1/3!(\tikztostart)$) and ($#1!1/3!(\tikztotarget)$)
      .. (\tikztotarget)
    }
  }
}
\begin{document}
  \begin{tikzpicture}[nodes={scale=2,text opacity=1}]
    \draw[help lines] (0,0) grid (2,4);
    \filldraw[fill opacity=.07]
        (0,0) coordinate[label=center:.](A)
      --(1,0) coordinate[label=center:.](B)
      --(2,4) coordinate[label=center:.](C);
    \draw[blue] (A)..controls (B)..(C);
    \draw[red,very thick] (A) to[quadratic={(B)}] (C);  % <- The quadratic curve
    \draw[white,domain=0:2] plot (\x,\x^2);
  \end{tikzpicture}
\end{document}

And if you prefer a code that do not use calc and use the PGF command \pgfpathquadraticcurveto you can replace the style definition with :

\makeatletter
\def\pt@get#1{%
  \tikz@scan@one@point\pgfutil@firstofone#1\relax%
  \pgfpoint{\the\pgf@x}{\the\pgf@y}%
}
\tikzset{
  quadratic/.code={
    \def\tikz@to@path{}% <- TikZ do nothing
    \pgfpathquadraticcurveto{\pt@get{#1}}{\pt@get{(\tikztotarget)}}% <- PGF insert the quadratic curve
  }
}
\makeatother