How do I color in two different colors one square function?

In Metapost you could just wrap the colour specification in a if ... else ... fi construct. Like this:

draw ...  if odd i: dashed evenly withcolor marmelade else: withcolor sky fi;

Here's a complete version of your plot.

prologues:=3;
outputtemplate := "%j%c.eps";

beginfig(1);

vardef f(expr a,x) = a*(x*x)-3x+cosd(90x) enddef;

numeric u, v, s, a, xmin, xmax, ymin, ymax;
-xmin = xmax = 5;  u = 1cm;
-ymin = ymax = 20; v = 2mm;
s = 1/8;

color marmelade, sky;
marmelade = 0.67(red + 1/2 green);
sky = 0.54[blue,white];

for i=-8 upto 8:
  a := 0.3i;
  draw ((xmin,f(a,xmin))
     for x = xmin+s step s until xmax:
        -- (x,f(a,x))
     endfor) 
     xscaled u yscaled v
     if odd i: dashed evenly withcolor marmelade else: withcolor sky fi;
endfor
clip currentpicture to ( (xmin,ymin) -- (xmax,ymin) -- (xmax, ymax) -- (xmin,ymax) -- cycle) xscaled u yscaled v;

path xx, yy, bar;
xx = ((xmin,0)--(xmax,0)) scaled u;
yy = ((0,ymin)--(0,ymax)) scaled v;

defaultfont := "texnansi-lmr10";
string minus_sign; minus_sign = char 143;

for i=-20 step 10 until 20: 
   bar := xx shifted (0,i*v);
   draw bar if i<>0: withcolor .7 white  fi; 
   label.lft(if i<0: minus_sign & fi decimal abs(i), point 0 of bar);
endfor
for i= -4 step  2 until  4: 
   bar := yy shifted (i*u,0);
   draw bar if i<>0: withcolor .7 white fi; 
   label.bot(if i<0: minus_sign & fi decimal abs(i), point 0 of bar);
endfor     

setbounds currentpicture to bbox currentpicture scaled 1.03;
endfig;
end.

There are some ways you can do it:

1. Same as Thruston's solution to this same question, lets you use \ifodd in TikZ as well. But we add a counter using count=\xi (default starting from 1) so that it alternates between blue+solid and orange+dashed.

2. You can set a cycle list name, defined here as mycycle and there write your various style in order. Since we write two, they will alternate. Don't forget to write your command as \addplot+[...], note the + and the optional arguments [...].

Here's the output, it's the same for all the solutions provided. The codes for the various solutions are provided below.

Output

Using \ifodd

\documentclass[tikz,margin=15pt]{standalone}
\usepackage{pgfplots}

\pgfplotsset{compat=1.12}
\begin{document}
\begin{tikzpicture}
\begin{axis}[%
    grid=major,
    xmin=-5,xmax=5,
    ymin=-20,ymax=20,
    xlabel=$x$,
    ylabel=$y$
    ]
\foreach \a [count=\xi] in {-2.4,-2.1,...,2.4}
{%
    \ifodd\xi
        \addplot[blue, line width=.5pt] plot expression {\a*(\x^2)-3*\x +cos(90*\x)};
    \else
        \addplot[orange, dashed, line width=.5pt] plot expression {\a*(\x^2)-3*\x+cos(90*\x)};
    \fi
}
\end{axis}
\end{tikzpicture}
\end{document}

Using cycle list name

\documentclass[tikz,margin=15pt]{standalone}
\usepackage{pgfplots}

\pgfplotscreateplotcyclelist{mycycle}{%
    {blue}, 
    {dashed,orange}% <--- % must be there for this to work properly!
}

\begin{document}
\begin{tikzpicture}
\begin{axis}[%
    grid=major,
    xmin=-5,xmax=5,
    ymin=-20,ymax=20,
    xlabel=$x$,
    ylabel=$y$,
    cycle list name=mycycle
    ]

\foreach \a in {-2.4,-2.1,...,2.4}
{%
    \addplot+[smooth,line width=.5pt] plot expression {\a*(\x^2)-3*\x +cos(90*\x)};
}
\end{axis}
\end{document}