We consider the equation:
$Ax^2+Bxy+Cy^2+Dx+Ey-F=0$ with $A,B,C,D,E,F \in \mathbb{Q}$
If one has a rational point (a point whose coordinates are both rational) on the curve described by the equation, then one can find infinitely many rational points on it. (We ignore the case where it's a degenerate equation, such as a single point, or two lines and so on.)
My question is: Is it always possible to find such a point? And if so, is there some algorithm to find or construct it?
So far, all I came up with is pure brute force, i.e. checking for all rational $x$ if the associated $y$ (such that $(x,y)$ satisfies the equation) is rational too. That would work, and it would give me a guaranteed answer in a finite time (since $\mathbb{Q}$ is countable), but only if I could say with absolute certainty, that there was at least one rational point (and hence infinitely many) on the curve.
I read something about the Legendre theorem, but it was extremely confusing for me. I would appreciate it, if someone could help me with that issue.
EDIT: With "we ignore degenerate equations", I meant actual degenerate equations and the case where it's no point at all (when it has only complex solutions). So just the regular conics: circle, ellipsis, hyperbola, parabola.
$x^2+y^2=3$ has no rational points.