If x = 2 + √3 and y = 2 - √3, find the value of (x² + y²)²

Question: If x = 2 + √3 and y = 2 - √3, find the value of (x² + y²)²

Solution:
We are given:
x = 2 + √3, y = 2 - √3

Now,
x + y
= (2 + √3) + (2 - √3)
= 4

And,
⇒ xy
= (2 + √3)(2 - √3)
= 2² - (√3)²
= 4 - 3 = 1

We know,
x² + y² = (x + y)² - 2xy
⇒ x² + y² = (4)² - 2(1) [Substitute the values]
⇒ x² + y² = 16 - 2
⇒ x² + y² = 14

∴ (x² + y²)² = 14² = 196