Question

If x=root 5- root 2/root 5+ root 2 and y=root 5+root 2 /root 5-root 2 find value of x square +xy +y square

Answer 1

x=√5-√2/√5+√2
  =(√5-√2)(√5-√2)/(√5+√2)(√5-√2)
  ={(√5)²-2.√5.√2+(√2)²}/{(√5)²-(√2)²}
  =(5-2√10+2)/(5-2)
  =(7-2√10)/3
y=√5+√2/√5-√2
  =(√5+√2)(√5+√2)/(√5-√2)(√5+√2)
  ={(√5)²+2.√5.√2+(√2)²}/{(√5)²-(√2)²}
  =(5+2√10+2)/(5-2)
  =(7+2√10)/3
Now, x²+xy+y²
=x²+y²+xy
={(x+y)²-2xy}+xy
=(x+y)²-xy
={(7-2√10)/3+(7+2√10)/3}²-{(7-2√10)/3×(7+2√10)/3}
={(7-2√10+7+2√10)/3}²-(7-2√10)/(7+2√10)
=(14/3)²-{(7-2√10)(7-2√10)/(7+2√10)(7-2√10)}
=196/9-(7-2√10)²/{(7)²-(2√10)²}
=196/9-(49-2.7.2√10+40)/(49-40)
=196/9-(9-28√10)/9
=(196-9+28√10)/9
=(187+28√10)/9