If x = 3 + 2√2, find √x.

Concept / Approach

• Try to express x as a perfect square of the form (√2 ± 1)^2.

Step-by-step calculation
(√2 + 1)^2 = (√2)^2 + 2·√2·1 + 1^2 = 2 + 2√2 + 1 = 3 + 2√2Therefore, x = (√2 + 1)^2 ⇒ √x = √2 + 1

Verification / Alternative
(√2 − 1)^2 = 2 − 2√2 + 1 = 3 − 2√2 ≠ x, so the positive root is √2 + 1.

Final Answer
√2 + 1