If a = 1 + √2 and b = 1 - √2, find the value of a² + b².

Question: If a = 1 + √2 and b = 1 - √2, find the value of a² + b².

Solution:
Given that,
a = 1 + √2,
b = 1 - √2

∴ a + b = 1 + √2 + 1 - √2
= 2

And,
ab = (1 + √2)(1 - √2)
= 1² - (√2)²
= 1 - 2
= - 1

Now,
a² + b² = (a + b)² - 2ab
= (2)² - 2(- 1)
= 4 + 2
= 6