Find the value of the constant c for which the line y=2x+c is a tangent to the curve y^2 =4x.

When a line is tangent to a curve, they have one point in common at which they intersect. Thus, we can equate their equations in terms of x or y. In this exampe, let's equate their y's.

y = 2x + c (equation 1)
y² = 4x
y = 2√x (equation 2)

2x + c =2√x
2x + c = 2(x)^1/2
c = 2x^1/2 - 2x
c = 2x^1/2 (1 - x^1/2)
c = 2√x (1 - √x)