Putting this in gathered inside equation* removes the whitespace:

\documentclass[preview,border=12pt,varwidth]{standalone}
\usepackage{amsmath}
\begin{document}
\abovedisplayskip=0pt\relax
\begin{equation*}
\begin{gathered}
6x^2-30x+36=0\\
6(x^2-5x+6)=0\\
6(x-2)(x-3)=0\\
x-2=0 \quad \text{or} \quad x-3=0\\
x=2 \quad \text{or} \quad x=3
\end{gathered}
\end{equation*}
\end{document}
The problem comes from the varwidth package, used by standalone. Its minimal "documentation" in the header of varwidth.sty says:
% AMSmath environments have not been tried, and undoubtedly fail.
A small test document demonstrates the behaviour we are seeing above:

\documentclass{article}
\usepackage{varwidth,amsmath}
\begin{document}
\fbox{\begin{varwidth}{15cm}
\begin{equation*}
\begin{gathered}
6x^2-30x+36=0\\
6(x^2-5x+6)=0
\end{gathered}
\end{equation*}
\end{varwidth}}
\fbox{\begin{varwidth}{15cm}
\begin{gather*}
6x^2-30x+36=0\\
6(x^2-5x+6)=0
\end{gather*}
\end{varwidth}}
\end{document}