Solution to Quadratic Congruence

Theorem

The quadratic congruence:

$a x^2 + b x + c \equiv 0 \pmod n$

can be solved by solving the congruence:

$y^2 \equiv \paren {b^2 - 4 a c} \pmod n$

and the linear congruence:

$2 a x + b \equiv y \pmod n$


Proof


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In particular: use the method of completing the square
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Sources

  • 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): quadratic congruence
  • 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): quadratic congruence
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