Equation of common tangent of y=x^2, y=-x^2+4 x-4 is

The equation of tangent to $y=x^2$, be $y=m x-\frac{m^2}{4}$, putting in $y=-x^2+4 x-4$, we should only get one value of $x$ i.e. Discriminate must be zero. $\begin{array}{rrr} \therefore & m x-\frac{m^2}{4} & =-x^2+4 x-4 \\ \Rightarrow & x^2+x(m-4)+4-\frac{m^2}{4} & =0 \\ \Rightarrow & m(m-4) & =0 \\ & \therefore y=0 & \text{ or } m=0,4 \end{array}$ \therefore y=0 \text{ and } y=4(x-1) \text{ are tangents. }