Solve Quadratic Congruence Equation How to solve $3x^2 - 5x + 5 \equiv 0 \pmod 7$? In general, how to approach this kind of problem? Any help is appreciated.

As the characteristic isn't two you can directly use the quadratic formula:
$$\Delta=(-5)^2-4\cdot3\cdot5=25-60=-35=0\pmod 7\implies\text{ there's a single double root:}$$
$$x_{1,2}=\frac{5}{2\cdot3}=\frac{5}{-1}=-5=2\pmod 7$$
Another way: Factor out  the quadratic coefficient and try to complete the square:
$$3x^2-5x+5=3\left(x^2-4x+4\right)=3\left(x-2\right)^2$$
and we get the same as in the first part, of course.
