If x = 1/(√2 + 1), then what is the exact value of x^2 + 2x − 1? Rationalize the denominator first so that x is written in a simpler surd form, then simplify fully.

Introduction / Context:
This problem tests surd manipulation and simplification after rationalizing a denominator. Expressions like 1/(√2 + 1) become much easier to work with once you remove the surd from the denominator by multiplying numerator and denominator by the conjugate (√2 − 1). After rationalization, x becomes a simple surd expression, and then evaluating x^2 + 2x − 1 is straightforward by expansion and careful cancellation. Students often make mistakes by using the wrong conjugate, forgetting that (a + b)(a − b) = a^2 − b^2, or expanding x^2 incorrectly when x contains √2. The key to speed here is recognizing that x becomes (√2 − 1), which leads to neat cancellations in the expression.

Given Data / Assumptions:

• x = 1/(√2 + 1) • Required: x^2 + 2x − 1 • Use conjugate rationalization

Concept / Approach:
Rationalize: x = 1/(√2 + 1) * (√2 − 1)/(√2 − 1) = (√2 − 1)/(2 − 1) = √2 − 1. Then compute x^2, compute 2x, and add them with −1. Surd terms often cancel if you do it correctly.

Step-by-Step Solution:
1) Rationalize x using the conjugate: x = 1/(√2 + 1) * (√2 − 1)/(√2 − 1) 2) Simplify the denominator: (√2 + 1)(√2 − 1) = (√2)^2 − 1^2 = 2 − 1 = 1 3) Therefore: x = √2 − 1 4) Compute x^2: x^2 = (√2 − 1)^2 = 2 − 2√2 + 1 = 3 − 2√2 5) Compute 2x: 2x = 2(√2 − 1) = 2√2 − 2 6) Combine x^2 + 2x − 1: (3 − 2√2) + (2√2 − 2) − 1 = 0

Verification / Alternative check:
Approximation check: x = 1/(√2 + 1) ≈ 1/2.414 ≈ 0.4142. Then x^2 + 2x − 1 ≈ 0.1716 + 0.8284 − 1 ≈ 0. This supports the exact result 0 and confirms the cancellation is correct, not accidental.

Why Other Options Are Wrong:
• 2 or 4: come from ignoring the −1 term or expanding (√2 − 1)^2 incorrectly. • 2√2: results from adding surd terms without cancelling the −2√2 from x^2. • 1: indicates incomplete simplification of constants (3 − 2 − 1).

Common Pitfalls:
• Using (√2 + 1) again instead of the conjugate (√2 − 1). • Forgetting that (√2)^2 = 2. • Making a sign error in (√2 − 1)^2, especially the middle term −2√2.

Final Answer:
0