Square roots modulo $p$

Squares mod 17.
- $1^2=1$, $2^2=4$, $3^2=9$, $4^2=16\equiv-1$, $5^2=25\equiv8$
- $6^2=36\equiv2$ ✓, $7^2=49\equiv15$, $8^2=64\equiv13$
- $9^2\equiv13$, $10^2\equiv15$, $11^2\equiv2$ ✓, $12^2\equiv8$
- Symmetry from $13^2$ onwards.

Solutions. $6$ and $11$ (since $6+11=17$, symmetric roots).

Check. $6^2=36\equiv2$, $11^2=121\equiv2\pmod{17}$. Correct.

Conclusion. $x^2\equiv2\pmod{17}$ has solutions $x\equiv6$ or $11\pmod{17}$.