Question

If x =
$\frac{ \sqrt{5 } - 2 }{ \sqrt{5} + 2}$
And y =
$\frac{ \sqrt{5} + 2}{ \sqrt{5} - 2}$
Find
$x ^{2} + {y}^{2} + xy$
​

Answer 1

Answer:

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Solution :

Given a = ( 2-√5)/(2+√5)

b = (2+√5)/(2-√5)

i )a+b

= [(2-√5)/(2+√5)]+[(2+√5)/2-√5)]

=[(2-√5)²+(2+√5)²]/[(2+√5)(2-√5)]

={ 2[2² + (√5)²]}/[2² - (√5 )²]

= [ 2(4 + 5 ) ]/(4-5)

[ ( x - y)² + ( x + y )² = 2 ( x² + y² ) ]

= -18 ------( 1 )

ii ) a - b

=[(2-√5)/(2+√5)]-[(2+√5)/(2-√5)]

=[(2-√5)²-(2+√5)²]/[(2+√5)(2-√5)]

= [ -4×2×(√5) ]/[ 2² - ( √5 )² ]

[ (x-y)² - ( x+y )² = -4xy ]

= ( -8√5 )/(4 - 5 )

= 8√5 ---( 2 )

Therefore ,

a² - b²

= ( a + b )( a - b )

= ( -18 )( 8√5 ) [ from (1)&(2) ]

= -144√5