Show that the equation $x^{2}ax=a^{-1}$ is solvable for $x$ in a group $G$ iff $a$ is the cube of some element in $G$. 

This is a home in the coursework of my Abstract Algebra Class. I am unable to understand how come abstract algebra could be used to solved this. Provided the solution I provided rarely used any abstract algebra concept. Kindly correct my perception regarding the use and understanding of Abstract Algebra. 

I need concepts and suggestions ( material) to improve the concept. 

N.B. Today was my third day in the Abstract Algebra Class. 

Thanks in advance!