Math mode and beamer

You need some math environment with multi-line support. There is align*, gather*, split, multline, aligned, alignat, and the list continues..

\documentclass[aspectratio=1610]{beamer}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{amsmath,amssymb,lmodern}

\begin{document}
\begin{frame}{Example Quadratic Formula}
    \begin{columns}
        \column{0.5\textwidth}
        Dicriminant
        \begin{align*}
            \text{function } \left[d\right]&=\text{dicriminant}(a, b, c)\\
            d &= b\wedge 2 - 4*a*c
        \end{align*}
        \column{0.5\textwidth}
        some text
    \end{columns}
\end{frame}

\end{document}

The math-environment is designed to show "inline" math formulae. The word "inline" implies, that the math is presented in the normal running text. Therefore it can't break lines by using Ret or \\. You have to encounter each line in a \begin{math} ... \end{math}environment of its own, like so:

\documentclass[aspectratio=1610]{beamer}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{amssymb,amsmath}

\begin{document}
\begin{frame}{Example Quadratic Formula}
    \begin{columns}
        \column{0.5\textwidth}
        Dicriminant
        \begin{math}
            \text{function } \left[d\right]=\text{dicriminant}(a, b, c)
        \end{math} 
        \begin{math}
            d = b\wedge 2 - 4*a*c
        \end{math}
        \column{0.5\textwidth}
        some text
    \end{columns}
\end{frame}
\end{document}

This works without compilation error.

Despite that, in your case, I would ask you, to use an environment, which was designed for displayed formulae, as you display the formulae as a paragraph of is own right.

A displayed formulae is typeset different, compared to an inline formulae. The font is bigger, the position of the sub- and superscripts is different, ... Do yourself the favour and enjoy the typographic enhancement.

Therefore, I suggest you use something like an eqnarray environment or something similar from the AMS packages, e. g. align:

\documentclass[aspectratio=1610]{beamer}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{amssymb,amsmath}

\begin{document}
\begin{frame}{Example Quadratic Formula}
    \begin{columns}
        \column{0.5\textwidth}
        Discriminant
        \begin{align}
          \text{function} \left[d\right] &= \text{discriminant}(a, b, c)\\
          \Delta &= b^2 - 4\cdot a\cdot c
        \end{align}
        \column{0.5\textwidth}
        some text
    \end{columns}
\end{frame}
\end{document}

Which compiles without error and results in