Question

if x=root 5-root 2 by root 5+ root 2 and y=root 5 + root 2 by root 5- root 2, find the value of x square+xy+y square

Answer 1

You can solve it by rationalising both the terms

Answer 2

Given: x=√5-√2 /√5+√2
y =√5+√2 /√5-√2

Let's first rationalise x
x =√5-√2 /√5+√2 *√5-√2 /√5-√2 

= (√5-√2)²/(√5)²-(√2 )²

=(√5)²+(√2)²-2(√5)(√2)/5-2

= 5+2-2√10 / 3 
=7-2√10 / 3 

Now we rationalise y
y =√5+√2 /√5-√2 *√5+√2 /√5+√2

= 5+2+2√10 / 3 = 7+2√10 / 3

Answer: now to find x² +y² +xy 
=(7-2√10 / 3)² +(7+2√10 / 3)² +(7-2√10 / 3)(7+2√10 / 3) 

=(49+40-28√10)/ 9 +(49+40+28√10) / 9 + [(7)²-(2√10)²]/(3)²

=(49+40-28√10)/ 9 +(49+40+28√10) / 9 + [49-40]/9

=(49+40-28√10)/ 9 +(49+40+28√10) / 9 + 9/9

= (89-28√10)/9 + (89+28√10)/ 9 + 1

= (178 -28√10+28√10) / 9 +1

=178/9 + 1

=19.777+1

=20.77