Find condition that zeros of polynomial p(x)=ax2+bx+c are reciprocal of each other

Suppose has two roots, reciprocals of one another. Call them , with .

Let's divide through by for now. By the fundamental theorem of algebra, we can factorize as



Expand the RHS to get



so we must have




The first equation says and occur in a ratio of the negative sum of the roots of , while the third equation says that the first and last coefficients must be the same.