Generating pythagorean triples in this question

Question

Find all integer solutions to $x^{2}+8xy+25y^{2}=225 \tag 1$

$(1)$ can be written as $$(x+4y)^2 + (3y)^2 = 15^2$$

This reduces the problem to a matter of generating pythagorean triples. I'm thinking that Euclid's method would be useful, but it's not immediately obvious to me how to use it.

Answer 1

Consider the quadratic equation $$x^2 + 8xy + 25y^2 - 225 = 0$$ where $x$ is the unknown. Compute $$\frac{\Delta}{4} = 16y^2 - 25y^2 + 225 = 225 - 9 y^2.$$ Then $x$ is an integer if and only if the discriminant is a perfect square.

Notice that $9 \cdot 5^2 = 225 = 0$, therefore $-5 \le y \le 5$. Now check for which $y$ the discriminant is a perfect square, and then compute the solutions for both $x$ and $y$.

Answer 2

Note that :-

$$15^2=9^2+12^2$$ is true.

So,$$(x+4y)^2+(3y)^2=9^2+12^2$$