To find the value of c, we need the line y = 2x + c to be tangent to the curve y² = 4x. This means they intersect at exactly one point.

Substitute y = 2x + c into y² = 4x:

\((2x + c)^2 = 4x\)

Expand and simplify:

\(4x^2 + 4xc + c^2 = 4x\)

Rearrange to form a quadratic equation:

\(4x^2 + 4xc + c^2 - 4x = 0\)

For the line to be tangent, the discriminant of this quadratic must be zero:

\(b^2 - 4ac = 0\)

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

Calculate the discriminant:

\((4c - 4)^2 - 4 \times 4 \times c^2 = 0\)

\(16c^2 - 32c + 16 - 16c^2 = 0\)

\(-32c + 16 = 0\)

Solve for c:

\(-32c = -16\)

\(c = \frac{1}{2}\)