Question

If x3 - y3 = 513 and xy =54 then find x-y

Answer 1

x3+y3=72
x3+y3=72
xy=8
xy=8
Notice,

(x+y)3=x3+y3+3xy(x+y)
(x+y)3=x3+y3+3xy(x+y)
(x+y)3=72+3(8)(x+y)
(x+y)3=72+3(8)(x+y)
(x+y)3−24(x+y)−72=0
(x+y)3−24(x+y)−72=0
Above is the cubic equation in terms of (x+y)(x+y) which has one real root 66 Hence, we get
x+y=6
x+y=6
Now,
(x−y)2=(x+y)2−4xy
(x−y)2=(x+y)2−4xy
(x−y)=±(x+y)2−4xy−−−−−−−−−−−−√
(x−y)=±(x+y)2−4xy
x−y=±(6)2−4×8−−−−−−−−−−√
x−y=±(6)2−4×8
=±4–√=±2
=±4=±2
Hence, we have
x−y=±2