Quadratic Congruences

    Consider the congruence



where is an odd prime and Inspired by completing the square, we have











Let and then we have a simplified version of the original namely;



Solving this equation becomes the impetus.

Example (Quadratic Congruence) Solve the quadratic congruence
    Solution. Let and . So our congruence becomes We find that and Therefore we solve,
    
     and    

We find, and Solving the linear equations can be accomplished by Euler's Theorem, if needed. However, solving the quadratic congruence is something else.  

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