If x = 2 + √2, then the value of x² + 4/x² is

Squaring a binomial surd expression uses (a + b)2 = a2 + 2ab + b2, and simplifying the reciprocal of a surd uses rationalization — multiplying by the conjugate so the denominator becomes a whole number. Together these let x2 and 4/x2 be evaluated separately before they are added.

1. Let x = 2 + √2. Using (a + b)2 = a2 + 2ab + b2 with a = 2, b = √2: x2 = 22 + 2·2·√2 + (√2)2 = 4 + 4√2 + 2 = 6 + 4√2.

2. Rationalize 1/x by multiplying numerator and denominator by the conjugate (2 − √2): 1/x = (2 − √2) / ((2 + √2)(2 − √2)) = (2 − √2) / (4 − 2) = (2 − √2)/2.

3. Then 4/x2 = 4 × (1/x)2 = 4 × (2 − √2)2/4 = (2 − √2)2 = 4 − 4√2 + 2 = 6 − 4√2.

4. Add the two results: x2 + 4/x2 = (6 + 4√2) + (6 − 4√2) = 12; the √2 terms cancel exactly.

As a numeric check, x = 2 + √2 ≈ 3.4142, so x2 ≈ 11.657 and 4/x2 ≈ 0.343; their sum is 12.000, confirming the algebraic result.

Hence, x2 + 4/x2 = 12.