Homogeneous diophantine equation

Question

Here I have a diophantine equation featuring a homogeneous polynomial:

$$x^2+5y^2+34z^2+2xy-10xz-22yz=0; x, y, z\in\mathbb{Z}$$

I have no idea how to approach this, I've tried various substitutions like $x=py, x=qz$ but then I get a non-homogeneous polynomial of 2 variables which is no better than this. Wolfram|Alpha finds a solution easily, but I still don't understand how.

For reference, W|A claims that the solution is $x=7n, y=3n, z=2n; n\in\mathbb{Z}$

Answer 1

Hint: Factor as $$(2y-3z)^2+(x+y-5z)^2 = 0$$

Answer 2

assuming

$x^2+5y^2+2xy=c$

and solving the equation,

$34z^2-(10x+22y)z+c=0$

one can find that

$z=(10x+22y∓(√(-4(3x-7y)^2 ))/(2*34)$

which mean that the discriminant D is negative and z is complex but z∈Z then $d=0$

or ,$3x=7y$

and $34z=5x+11y$

suppose $x=7n$ ∀ n ∈ Z

then $y=3n$ and $34z=35n+33n$ or $z=2n$