If x = √3 + 1 and y = √3 − 1, then what is x² + y²?

Problem restatement
Evaluate x² + y² for x = √3 + 1 and y = √3 − 1.

Given data

• x = √3 + 1
• y = √3 − 1

Concept/Approach
Square each expression and add; the radical cross-terms cancel.

Step-by-step calculation
x² = (√3 + 1)² = 3 + 2√3 + 1 = 4 + 2√3 y² = (√3 − 1)² = 3 − 2√3 + 1 = 4 − 2√3 x² + y² = (4 + 2√3) + (4 − 2√3) = 8

Verification/Alternative
Use identity: (a + b)² + (a − b)² = 2(a² + b²) with a = √3, b = 1 ⇒ 2(3 + 1) = 8.

Final Answer
8