Question

The total number of pairs of integers (x,y) which satisfy the equation x²-4xy+5y²+2y-4 =0 is [?]

Answer 1

if we consider it a quadratic in x ,then
\[x=\frac{ -4y \pm \sqrt{16y^2-20 y^2-8y+16} }{ 2 }=\frac{ -4y \pm \sqrt{-4 y^2-8y+16} }{ 2 }\]
\[=-2y \pm \sqrt{-y^2-2y+4}\]
to get the value of x and y as integers
\[-y^2-2y+4~ should~ be ~a~ perfect ~\square.\]
which is impossible as its discriminant (-2)^2-4*-1*4=20 is not a perfect square or equal to 0.
but if y=0,then \[x=0 \pm \sqrt{4}=\pm2\]
so only integral solutions are (2,0) and (-2,0)

Answer 2

But my book says there are more than two pairs... "/

Answer 3

i say infinitely many

Answer 4

if you find a point in which it is valid, say (0,0)
and say a point which is valid at (x,y)

then we assume that a line exists between those 2 points

then there must be a related point on this line when x/2, x/4, x/8, x/16, ... etc

and since you can infinitely divide numbers into a infinitely small value, then there must be an infinitely amount of points between 0 and x with a infinitely respective y value