Question

Given x^2 + y^2 = 36 and xy = -10 find x + y.

Answer 1

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=
`(-b+/- √(b^2-4ac) )/(2a)`
x=
`(-(-36)+/- √((-36)^2-4(1)(100)) )/(2(1))`
x=
`18+/- 4√(14)`

sub
y=-10/x
y=
`(-10)/(18+/- 4√(14))`

so x+y=
`18+/- 4√(14)`+
`(-10)/(18+/- 4√(14))`
multiply first number by
`(18+/- 4√(14))/(18+/- 4√(14))` and add them

x+y=
`((18+/- 4√(14))(18+/- 4√(14))-10)/(18+/- 4√(14))`
or
x+y=
`(81+22 √(14) )/(5)` or
`(81-22 √(14) )/(5)`