Step 1: Set up the modulus notation.
Let |z| = r with r>0. The given equation is |z + 2/z| = 2.
Step 2: Apply the reverse triangle inequality.
By the reverse triangle inequality, |z + 2/z| ≥ ||z| - |2/z|| = |r - 2/r|. Hence 2 ≥ |r - 2/r|, which is equivalent to |r - 2/r| ≤ 2.
Step 3: Convert to polar representation for an exact solution.
Write z = re^(iθ). Then 2/z = (2/r)e^(-iθ). Their sum is re^(iθ) + (2/r)e^(-iθ). The squared modulus is |z + 2/z|² = r² + 4/r² + 4cos 2θ. Setting this equal to 4 gives r² + 4/r² + 4cos 2θ = 4.
Step 4: Determine bounds on r using the range of cosine.
Since -1 ≤ cos 2θ ≤ 1, the equation requires r² + 4/r² - 4 ≤ 4, which simplifies to r² + 4/r² ≤ 8. Multiplying by r²: r⁴ - 8r² + 4 ≤ 0. Substitute u = r² to obtain u² - 8u + 4 ≤ 0. The quadratic roots are u = 4 ± 2√3. Thus r² ≤ 4 + 2√3, and taking square roots gives r ≤ √(4 + 2√3).
Step 5: Simplify the radical.
Recognize that 4 + 2√3 = (√3 + 1)², so r ≤ √3 + 1.
Step 6: Final conclusion.
The maximum possible value of |z| is √3 + 1.