We study the solutions $f:G \to H$ of the quadratic functional equation on $G$, where $G$ and $H$ are groups, $H$ abelian. We show that any solution $f$ is a function on the quotient group $[G,[G,G]]$. By help of this we find sufficient conditions on $G$ for all solutions to satisfy Kannappan's condition. We use this to derive explicit formulas for the solutions on various groups like, e.g.,the $(ax+b)$-group and $GL(n,\mathbb{R})$.

Keywords: Quadratic functional equation, Cauchy difference, Kannappan condition, semidirect product. 2000 AMS Subject Classification: 39B52.