Thread: Polynomial Congruences

How do you find all the solutions of the congruence

7x^2-41x+48=0(mod333)

Multiplying the congruence by 28 gives

Note that and so the second congruence implies the first as well: i.e.

The reason for multiplying by 28 is to make the leading coefficient a perfect square and the coefficient of even so that we can complete the square. Completing the square gives Hence, letting we have to solve the congruence

Clearly is a solution, so one solution to the original congruence is found by solving for I can tell you that one possiblity is and substituting into the original congruence shows that is indeed one solution.

Another solution to is Find all solutions to and for each check and see if the corresponding solution for satisfies the original congruence. Good luck.