If x = ( √2 + 1 ) / ( √2 - 1 ) and x - y = 4√2, then the value of (x2 + y2) is?

Given that,
x = ( √2 + 1 ) / ( √2 - 1 )
Expanding given equation by multiplying and dividing with √2 + 1
we will get,
⇒ x = ( ( √2 + 1 ) / ( √2 - 1 ) ) x ( ( √2 + 1 ) / ( √2 + 1 ) )
⇒ x = ( √2 + 1 ) x ( √2 + 1 ) / ( √2 - 1 ) x ( √2 + 1 )
⇒ x = ( √2 + 1 ) ² / ( √2 - 1 ) x ( √2 + 1 ) [ Use the formula (A + B)(A - B) = A² - B² ]
⇒ x = ( √2 + 1 ) ² / ( √2² - 1² ) [ Use the formula (A + B)² = A² + 2AB + B² ]
⇒ x = ( 2 + 1 + 2√2 ) / (2 - 1)
⇒ x = 3 + 2√2 .....................(i)
and given in question x - y = 4 √2
⇒ y = x - 4√2
Put the value of x [ from Equation (i) ]
⇒ y = 3 + 2√2 - 4√2
⇒ y = 3 - 2√2
Now, x² + y² = (3 + 2√2)² + (3 - 2√2)
Use the Algebra formula and solve the equation.
[ Use the formula (A + B)² = A² + 2AB + B² ]
[ Use the formula (A - B)² = A² - 2AB + B² ]
⇒ x² + y² = 9 + 8 + 12√2 + 9 + 8 - 12√2
⇒ x² + y² = 34