## How do you find the solution of quadratic congruence?

Thus , the congruence X2≡ ��(������ p) has exactly two solutions. The Quadratic congruence in one variable is given by ax2+bx+c ≡0(mod p) can be reduced to the form y2≡ d(mod p),where y=2ax+b & d=b2-4ac. From (1) we have a=1,b=3,c=11 , Y=2ax+b so the value of Y=2x+3, d=b2-4ac=32-4.11=-35.

### How do you solve quadratic residues?

We only need to solve, when a number (b) has a square root modulo p, to solve quadratic equations modulo p. Given a number a, s.t., gcd(a, p) = 1; a is called a quadratic residue if x2 = a mod p has a solution otherwise it is called a quadratic non-residue.

#### How do you solve quadratic variables?

1. Put all terms on one side of the equal sign, leaving zero on the other side.
2. Factor.
3. Set each factor equal to zero.
4. Solve each of these equations.

What are the solutions of quadratic equations?

The “solutions” to the Quadratic Equation are where it is equal to zero. There are usually 2 solutions (as shown in this graph). Just plug in the values of a, b and c, and do the calculations. We will look at this method in more detail now.

How many solutions does a quadratic congruence have?

two solutions
Solving the quadratic congruence x2 ≡ a (mod m) p odd: If a(p-1)/2 ≡ 1 (mod p), there are two solutions (mod pn).

## Is 0 a quadratic residue?

Modulo 2, every integer is a quadratic residue. Modulo an odd prime number p there are (p + 1)/2 residues (including 0) and (p − 1)/2 nonresidues, by Euler’s criterion. In this case, it is customary to consider 0 as a special case and work within the multiplicative group of nonzero elements of the field Z/pZ.

### For which primes p is 13 a quadratic residue?

For example when p = 13 we may take g = 2, so g2 = 4 with successive powers 1,4,3,12,9,10 (mod 13). These are the quadratic residues; to get the quadratic nonresidues multiply them by g = 2 to get the odd powers 2,8,6,11,5,7 (mod 13).