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=$\frac{- \sqrt{71}- \sqrt{47} }{2}$ and $\frac{- \sqrt{71}+ \sqrt{47} }{2}$

sub those for x

x=y+√71
note: √71=(2√71)/2

x=$\frac{- \sqrt{71}- \sqrt{47} +2\sqrt{71}}{2}$ ,$\frac{\sqrt{71}- \sqrt{47}}{2}$
or
x=$\frac{- \sqrt{71}+ \sqrt{47} +2\sqrt{71}}{2}$ ,$\frac{\sqrt{71}+ \sqrt{47}}{2}$

xy=$\frac{\sqrt{71}- \sqrt{47}}{2}$  times $\frac{-\sqrt{71}- \sqrt{47}}{2}$ or $\frac{\sqrt{71}+ \sqrt{47}}{2}$  times $\frac{-\sqrt{71}+ \sqrt{47}}{2}$

the result is -6 both times

xy=-6