If the summation of roots of two integers is √(11+8√2). What is the summation of the squares of these two integers?

Question: If the summation of roots of two integers is √(11+8√2). What is the summation of the squares of these two integers?

Solution:
Let,
The two integers are x and y.
ATQ,
√x + √y = √(11 + 8√2)
⇒ (√x + √y)² = {√(11 + 8√2)}²
⇒ x + 2√x . √y + y = 11 + 8√2
⇒ x + y + 2√(xy) = 11 + 8√2

So, x + y = 11

And,
2√(xy) = 8√2
⇒ √(xy) = 4√2
⇒ xy = 32

Now,
(x + y)² = 11²
⇒ x² + y² + 2xy = 121
⇒ x² + y² = 121 - 2 . 32
⇒ x² + y² = 57