Finding roots of polynomial $X^2 - X + 19$ in $\mathbb Z/61 \mathbb Z$

Question

For $p = 61$.

I was given the roots of $X^2 + 3$ in $\mathbb Z/p \mathbb Z$, which are $\pm 27 + p\mathbb Z$.

I then must find the roots of $X^2 - X + 19$ in $\mathbb Z/p\mathbb Z$, which I have found without using the previous information.

However, I am supposed to use the given information to find the roots of $X^2 - X + 19$ in $\mathbb Z/p\mathbb Z$, to which, I have no clue how to move forward.

Answer 1

Complete the square.

$X^2-X+19\equiv0\bmod61\iff 4X^2-4X+76\equiv0\bmod61$

$\iff (2X-1)^2\equiv-75=-3\times 5^2 \bmod61$

$\iff 2X-1\equiv\pm27\times5\bmod61$.

Can you take it from here?