this looks fun
ok so
we will use subsitution
xy=-10
divide both sides by x or y (I will choose y)
x=-10/y
sub -10/y fo x
(-10/y)^2+y^2=36
100/(y^2)+y^2=36
times both sides by y^2
100+y^4=36y^2
minus 36y^2 from both sides
y^4-36y^2+100=0
(y^2)^2-36(y^2)+100=0
quadratic formula
for
ax^2+bx+c=0
x=\(\frac{-b+/- \sqrt{b^2-4ac} }{2a}\)
x=\(\frac{-(-36)+/- \sqrt{(-36)^2-4(1)(100)} }{2(1)}\)
x=\(18+/- 4\sqrt{14}\)
sub
y=-10/x
y=\(\frac{-10}{18+/- 4\sqrt{14}}\)
so x+y=\(18+/- 4\sqrt{14}\)+\(\frac{-10}{18+/- 4\sqrt{14}}\)
multiply first number by \(\frac{18+/- 4\sqrt{14}}{18+/- 4\sqrt{14}}\) and add them
x+y=\(\frac{(18+/- 4\sqrt{14})(18+/- 4\sqrt{14})-10}{18+/- 4\sqrt{14}}\)
or
x+y=\(\frac{81+22 \sqrt{14} }{5}\) or \(\frac{81-22 \sqrt{14} }{5}\)
