(a) Let p be an odd prime. Give an analogue of the quadratic formula mod p, and prove it. (b) Determine if x2 -x+5 0 (mod 137) has any solutions.
Rate of Change
The relation between two quantities which displays how much greater one quantity is than another is called ratio.
Slope
The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as:
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(a) Given that p is an odd prime.
The quadratic formula mod p is given by:
Here, p does not divide a.
The above equation can be rewritten as:
Multiplying both sides by ,
Now, adding to both sides,
Factor the L.H.S,
Thus, it has a solution if and only if is a quadratic residue or p divides .
If p divides , the R.H.S of the above equation will be 0 and it gives a single solution.
If it is a quadratic residue , it will have two solutions.
Hence proved.
(b)
Multiply both sides by 4,
Adding =1 on both sides,
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