Question

If (x-y)^2=71 and x^2+y^2=59 what is the value of xy?

Answer 1

solve the equations
first one
take the sqrt of both sides
x-y=√71
add y to both sides
x=y+√71
sub y+√71 for x in other part


(y+√71)²+y²=59
y²+2y√71+71+y²=59
2y²+2y√71+71=59
minus 59 both sides
2y²+2y√71+12=0
factor out 2
2(y²+y√71+6)=0
use quadratic equation or something
y=
`(- √(71)- √(47) )/(2)` and
`(- √(71)+ √(47) )/(2)`

sub those for x

x=y+√71
note: √71=(2√71)/2

x=
`(- √(71)- √(47) +2√(71))/(2)` ,
`(√(71)- √(47))/(2)`
or
x=
`(- √(71)+ √(47) +2√(71))/(2)` ,
`(√(71)+ √(47))/(2)`

xy=
`(√(71)- √(47))/(2)` times
`(-√(71)- √(47))/(2)` or
`(√(71)+ √(47))/(2)` times
`(-√(71)+ √(47))/(2)`

the result is -6 both times

xy=-6