I would like to know how a second-degree diophantine equation with two unknows like this, ax^2 + bxy + cy^2 + dx + ey + f = 0, where a,b,c,d,e and f are integers
can be transformed using linear transformation into this, mu^2 + nv^2 + g = 0, where m,n and g are integers
For example, 9x^2 + 6xy - 13y^2 - 6x - 16y + 20 = 0
There is : 2u^2 - 7v^2 + 45 = 0, where u = 3x + y - 1 and v = 2y + 1
Someone told me that mu^2 + nv^2 + g = 0 could be found using the linear transformation method. How do we get to mu^2 + nv^2 + g = 0 using this ?
Or is there another simpler method?