To find the value of K, we need to find the point at which the line Y=2x+K is tangent to the parabola Y=x^2-4x+4. This means that the line and the parabola will have the same y-value at this point, and the line will be tangent to the parabola (meaning it will just touch the parabola at this point, without intersecting it).
To find the point at which the line and parabola are tangent, we can set Y=2x+K equal to Y=x^2-4x+4 and solve for x:
2x + K = x^2 - 4x + 4
x^2 - 4x + 4 - 2x - K = 0
(x-2)^2 = K
x = 2 +/- sqrt(K)
Since the line is tangent to the parabola, it will just touch the parabola at this point, which means that the parabola will have a minimum value at this point. This means that the value of K must be nonnegative, because the square root of a negative number is not a real number. Therefore, K must be equal to or greater than 0.
So, the value of K for which Y=2x+K is tangent to Y=x^2-4x+4 is K >= 0.