Let's call the two numbers x and y. The first sentence, one is 6 less than twice another, translates to
We also know the sum of their squares:
So, we have the following system:
We can use the expression for x in terms of y from the first equation to turn the second equation into something involving y alone:
Expand the square:
Subtract 680 from both sides:
Using the quadratic equation
we find the two solutions
Since we know that both numbers are integer we can only accept the second solution. It yields the following x value: