Given x = (√2 + 1)/(√2 − 1) and x − y = 4√2, determine the exact value of x^2 + y^2.

Introduction / Context:This problem combines rationalizing a radical fraction and manipulating sums/differences of two variables. The key is to simplify x first, then exploit the given difference x − y to identify y and compute x^2 + y^2 efficiently.

Given Data / Assumptions:

• x = (√2 + 1)/(√2 − 1).
• x − y = 4√2.
• All square roots are principal (non-negative) values.

Concept / Approach:Rationalize x by multiplying numerator and denominator by the conjugate (√2 + 1). Then, given the numerical value of x, use x − y = 4√2 to obtain y. Finally, compute x^2 + y^2 by squaring and adding; conjugate-like structures will cancel mixed terms.

Step-by-Step Solution:

Verification / Alternative check:Observe that y = 3 − 2√2 satisfies x − y = 4√2 directly; the conjugate structure guarantees mixed terms cancel when squaring and adding.

Why Other Options Are Wrong:38, 32, 30, and 28 are typical miscomputations from squaring or handling radicals improperly; only 34 is consistent.

Common Pitfalls:Not rationalizing correctly, or attempting to compute y via a guess without ensuring the exact difference equals 4√2.

Final Answer:34