A cubic is more general than a quadratic, so clearly a quadratic is going to be a special case of a cubic.

We can demonstrate this by looking at the equation for a quadratic Bézier:

     (1 - t)^2 P_0 + 2(1 - t)t P_1 + t^2 P_2

Now, let's increase the degree.

     = [ (1 - t)^2 P_0 + 2(1 - t)t P_1 + t^2 P_2 ] * [ 1 - t + t ]

clearly this equation is equal to the above one because 1-t+t == 1 and multiplying by one does nothing.

     = [ (1 - t)^2 P_0 + 2(1 - t)t P_1 + t^2 P_2 ] * [ (1 - t) + t ]
     = (1 - t)^3 P_0 + t(1 - t)^2 P_0 + 2(1 - t)^2t P_1 + 2(1 - t)t^2 P_1 + (1 - t)t^2 P_2 + t^3 P_2

Group by powers of t:

     = (1 - t)^3 P_0 + t(1 - t)^2 [P_0 + 2*P_1] + (1 - t)t^2 [2P_1 + P_2] + t^3 P_2

Which is exactly a cubic Bezier curve.

Thus, all quadratic Beziers can represented cubic ones, but the reverse does not hold, thus quadratics are specializations of cubics.

QED.

This is known as degree elevation.