Posted by ayan on .
How many ordered pairs of integers (x,y) are there such that 2x2−2xy+y2=225?

math 
Reiny,
using the quadratic equation, I solved for y
where a = 1 , b = 2x and c = 2x^2225
y = (2x ± √(4x^2  4(4x^2  225) )/2
= (2x ± √(900  20x^2)/2
= x ± √(225  x^2)
now we want x and y to be integers, so
225  x^2 must be a perfect square
so possible results for 225 a perfect square giving us a perfect square are
225  0 = 225 , good one
225  1
225  4
225  9
225  16
225  25
225  36
225  49
225  64
225  81 = 144 ahhhh
225  100
225 121
225  144 = 81  another one
225  169
225  196
225 225 = 0  that one works
so we have x = 0, ±9, ±12, ±15
each one will give an integer for y
(0, ±15) , (9, 21), (9, 3), (9, 3), (9, 21) ....
so I count 14 such ordered pairs 
math 
hmmm,
brilliant says its wrong... 14 is wrong... it meant 2x^2−2xy+y^2=225