how to write a formula that branches into two?

Here is an adapted version from Claudio Fiandrino's answer to Joining parts of equations with lines or arrows using tikz.

\documentclass{article}
\usepackage{tikz}
\usepackage{amsmath}

\begin{document}

\begin{tikzpicture}[grow=right, 
                    sibling distance=50pt,
                    level distance=2cm,
                    edge from parent path={(\tikzparentnode.east) -- (\tikzchildnode.west)},
                    edge from parent/.style={draw,-latex}]
    \node {$ \dfrac{a \pm b}{c} =$}
    child {node {$ \dfrac{a-b}{c} $}}
    child {node {$ \dfrac{a+b}{c} $}}
    ;
\end{tikzpicture}
\end{document}

Without using a hammersledge to crack a nut, you already have the necessary packages for the below code.

Unrelated: epsfigure and anysize are obsolete. The functionalities of the former are achieved with graphicx and the latter can be replaced with geometry (which you use, anyway).

\documentclass[a4paper, 12pt]{article}
\usepackage[italian]{babel}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage[a4paper, margin=1.0cm]{geometry}
\usepackage{graphicx}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{multicol}
\linespread{1.5}
\frenchspacing

\usepackage{old-arrows}

\begin{document}


\[ \dfrac{a \pm b}{c} =
\begin{matrix}
\nearrow \\[-1.5ex] \searrow
\end{matrix}
\:
\begin{matrix}
 \dfrac{a+b}{c} \\[3ex]
 \dfrac{a-b}{c}
\end{matrix} \]

\end{document} 

If you want to have the arrows figure centred on the math axis, and control the gap between the arrows, you can replace the matrix environment for the arrows with \Vectorstack from the stackengine package (load it with option [usestackEOL]). Here are two examples, with a different gap:

\[\Lstackgap = 10pt\dfrac{a ± b}{c} =
\Vectorstack{%
\nearrow \\%[-1.5ex]
 \searrow }
\:
\begin{matrix}
 \dfrac{a+b}{c} \\[3ex]
 \dfrac{a-b}{c}
\end{matrix} \]

\[\Lstackgap = 18pt\dfrac{a ± b}{c} =
\Vectorstack{%
\nearrow \\%[-1.5ex]
 \searrow }
\:
\begin{matrix}
 \dfrac{a+b}{c} \\[3ex]
 \dfrac{a-b}{c}
\end{matrix} \]

Another simple possibility using tikz-cd.

\documentclass[a4paper,12pt]{article}
\usepackage{tikz-cd}
\usepackage{amsmath}

\begin{document}
\begin{tikzcd}[cells={nodes={minimum height=-2cm,minimum width=1cm}},column sep=3em,row sep=-7pt]
 & \dfrac{a+b}{c}  \\
\dfrac{a \pm b}{c} = \arrow[ru] \arrow[rd] &  \\
 & \dfrac{a-b}{c} 
\end{tikzcd}
\end{document}