Let $a,b,c,d$ be integers such that $GCD(a,b,c,d)=1$. Assume that the diophantine equation $ax^2+bxy+cxz+dyz-x=0$ has a non-zero solution.Can we assert that it admits infinitely many solutions?


Thanks in advance



infinite solution of a diophantine quadratic equations
Let $k$ be integer and $f(x,y,z)=ax^2+bxy+cxz+dyz-x$.


Unless $b=1,d=0$ then $f(x,y,z)=0$ has infinitely many solutions via
the parametrization


$$X= -dk,Y=adk-ckb+ck+1,Z=k(b-1)$$ and $f(X,Y,Z)=0$.


If $b=1,d=0$ parametrization is $y=-ax-cz+1$ for integer $x,z$.


