Solving Diophantine equations involving $x, y, x^2, y^2$ My father-in-law, who is 90 years old and emigrated from Russia, likes to
challenge me with logic and math puzzles. He gave me this one:
Find integers $x$ and $y$ that satisfy both $(1)$ and $(2)$
$$x + y^2 = 8 \tag{1} $$
 $$x^2 + y = 18 \tag{2}$$
I found one solution by deriving this equation:
$$ x(x+1) + y(y+1) = 26$$
which means, I need to find two numbers that add up to $26$, where each number is the product of two consecutive numbers. Through trial and error I found a solution. From college days I seem to remember that there is a more elegant way using the theory of Diophantine equations.
Can someone remind me or point to me to an easy to follow explanation of the method for solving problems like these?

You can do without the theory of Diophantine equations.
$$x^2+y=18\implies y=18-x^2.$$
Plugged in the other equation,
$$x+(18-x^2)^2=8$$or
$$x^4-36x^2+x-316=0.$$
With a polynomial solver, you get two positive roots $x=4$ and $x\approx4.458899113761$.
