Question

x+y = 1
x^2+y^2=2
Find
x^5+y^5

EXPLAIN

Answer 1

Given :

x² + y² = 2 ..................(1)

x + y = 1

So :

( x + y )² = 1

==> x² + y² + 2 xy = 1

==> 2 + 2 xy = 1

==> 2 xy = - 1

==> xy = - 1/2 [ From (1) ]

Now :

( x² + y² )( x + y ) = 2 × 1

==> x³ + x²y + xy² + y³ = 2

==> x³ + y³ + xy ( x + y ) = 2

==> x³ + y³ + ( - 1/2 )( 1 ) = 2

==> x³ + y³ = 2 + 1/2

==> x³ + y³ = 5/2

Now :

( x³ + y³ )( x² + y² ) = 5/2 × 2

==> x⁵ + x³y² + x²y³ + y⁵ = 5

==> x⁵ + y⁵ + x²y² ( x + y ) = 5

==> x⁵ + y⁵ + ( - 1/2 )²( 1 ) = 5

==> x⁵ + y⁵ + 1/4 = 5

==> x⁵ + y⁵ = 5 - 1/4

==> x⁵ + y⁵ = ( 20 - 1 ) / 4

==> x⁵ + y⁵ = 19/4

The value of x⁵ + y⁵ = 19/4

Hope it helps !

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Answer 2

x² + y² = 2 ..................(1)

x + y = 1

So :

( x + y )² = 1

==> x² + y² + 2 xy = 1

==> 2 + 2 xy = 1

==> 2 xy = - 1

==> xy = - 1/2 [ From (1) ]

Now :

( x² + y² )( x + y ) = 2 × 1

==> x³ + x²y + xy² + y³ = 2

==> x³ + y³ + xy ( x + y ) = 2

==> x³ + y³ + ( - 1/2 )( 1 ) = 2

==> x³ + y³ = 2 + 1/2

==> x³ + y³ = 5/2

Now :

( x³ + y³ )( x² + y² ) = 5/2 × 2

==> x⁵ + x³y² + x²y³ + y⁵ = 5

==> x⁵ + y⁵ + x²y² ( x + y ) = 5

==> x⁵ + y⁵ + ( - 1/2 )²( 1 ) = 5

==> x⁵ + y⁵ + 1/4 = 5

==> x⁵ + y⁵ = 5 - 1/4

==> x⁵ + y⁵ = ( 20 - 1 ) / 4

==> x⁵ + y⁵ = 19/4

The value of x⁵ + y⁵ = 19/4

HOPE IT HELPS U ✌️✌️✌️