Question

Verify that :xy[(x+y)(1/x+1/y)-4]=(x-y) 2

Answer 1

$\mathrm{xy((x+y)( \frac{1}{x} + \frac{1}{y} )-4)=(x-y)2} \\\ \mathrm{xy((1+ \frac{x}{y} + \frac{y}{x} +1)-4)=(x-y)2} \\\ \mathrm{(xy+x^2+y^2+xy-4xy)=(x-y)2} \\\ \mathrm{(x^2+y^2-2xy)=(x-y)2} \\\ \mathrm{(x-y)^2=(x-y)2} \\\ \boxed{\mathrm{x-y=2}} \Leftarrow$

Answer 2

Answer:

xy[(x+y)[1/x+1/y]-4]

= xy[(xy+x^2+y^2)/(xy)] - 4xy      (using (1/x+1/y) = (x+y)/(xy))

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

= x^2 - 2xy + y^2

= (x-y)^2