If possible, solve $7x^2 - 4x + 1 \equiv 0 $ (mod 11)

Question

If possible, solve $7x^2 - 4x + 1 \equiv 0 $ (mod 11)

Not sure if I'm approaching this problem correctly, any help is appreciated.

So far I have:

$7x^2 - 4x + 1 \equiv 0 $ (mod 11)

$21x^2 - 12x + 3 \equiv 0 $ (mod 11)

$-x^2 - x + 3 \equiv 0 $ (mod 11)

$-x^2 - x \equiv -3 $ (mod 11)

$-1(x^2 + x) \equiv -3 $ (mod 11)

$(x^2 + x) \equiv 3 $ (mod 11)

-Not sure if I am heading the right way, once again any help is appreciated.

Answer 1

Multiply by $8$ (which is the inverse of $7$ mod $11$) and get: $x^2+x+8=0$ (mod $11$).

Then complete the square. "Half of 1" is the same as "half of 12" (mod 11) so we can rewrite our equation as $(x+6)^2-36+8=0$ (mod $11$) which simplifies to $(x+6)^2=6$ (mod $11$).

So your equation has a solution if and only if 6 is a square (mod 11).

It isn't, so there is no solution.

To show that 6 is not a square (directly) consider: $0^2=0$, $(\pm 1)^2=1$, $(\pm 2)^2=4$, $(\pm 3)^2=9$, $(\pm 4)^2=16=5$, and $(\pm 5)^2 = 25=3$ (all mod $11$).

Answer 2

The situation is not really different than the one when solving a quadratic equations in the reals. You use the familiar formula and see if there exists a squareroot of the discriminant.

To that latter end you might use result on quadratic residues or check by hand.

Answer 3

The first thing I would have done is shift the $1$ to the other side, like so: $$7x^2 - 4x \equiv 10 \pmod{11}.$$ Then I would just have tried the eleven possible cases, from $x = 0$ to $x = 10$ (because for $x \geq 11$ the patterns just repeat). By this crude but effective method I would have found that there are no solutions. If the modulo was larger (say, $10000019$), I would want a more sophisticated method.

Answer 4

In $\mathbb F_{11}$, one has $$7x^2 - 4x + 1= 0\iff-4x^2-4x+1=0\iff x(x+1)=\frac {1}{2^2}=2^8=3$$

For all $x\in\mathbb F_{11}$ we have $x(x+1)\ne 3$ (successively one has $0,2,6,1,9,8,9,1,6,2,0$)

Hence there is not solution.