I attempted to look at the size of this sum, and the hint was to use Legendre symbols to split this sum into 2 sums. But from what I have seen so far, all I manage to do is simplify the sum rather than separate it into 2 new sums, and I have barely even used the notation.
Since $p$ is a prime number, we have that half of the congruences mod $p$ will be quadratic residues, and the other half will be nonresidues (with 0 being an outlier); moreover each quadratic residue will have 2 solutions. This gives us $$ \sum_{x=0}^{p-1}e^{2\pi ix^2/p} = 1 + \sum_{(n|p)=1}2e^{2\pi in/p}. $$ Now it looks like I haven't made much headway in arriving at the solution. Would anybody be able to show me what path to take to make it further? The hope is to come to the conclusion that the magnitude of this sum shouldn't exceed $\sqrt{p}$.