Solve the simultaneous equations x^2+y^2=10 , x^4+y^4=82

x^2+y^2=10     
(1)

x^4+y^4=82     
(2)

(x^2+y^2)^2=
x^4+2x^2y^2+y^4

Since  x^2+y^2=10  and x^4+y^4=82 we
have:

100=82+2xy^2    

then:

2x^2y^2=18  

(xy)^2=9

xy=3   
xy=-3

from (1)

x^2+y^2+2xy=
10+6=16

(x+y)^2=16     x+y=4   
xy=3

being symmetric:   x=1 y=3   x=3
y=1

 x+y=-4      xy=3       x=-1 y=-3    x=-3 
y=-1

or

x^2+2xy+y^2=
10-6=4

(x+y)^2=4        x+y=2    xy=-3     x=3 y=-1  x=-1
y=3

or

(x+y)^2=-4    
x+y=-2    xy=-3     x=-3 y=1  x=1 y=-3 

So the couples we
are searching for are:

(1,3),
(3,1),(-3,-1),(-1,-3),(3,-1),(-1,3),(-3,1),(1,-3)