Explanation:

- The quadratic equation is \(2x^2 - 4x + k = 0\).

- For real roots, the discriminant (\(b^2 - 4ac\)) should be non-negative.

- Here, \(a = 2\), \(b = -4\), and \(c = k\).

- Calculating the discriminant: \((-4)^2 - 4 \times 2 \times k \geq 0\).

- Simplifies to: \(16 - 8k \geq 0\).

- Solve for \(k\): \(8k \leq 16\) leading to \(k \leq 2\).

- Hence, the greatest value of \(k\) for real roots is 2.

- Option 2: 2 is the correct answer.