The equations of the common tangents to the parabola $\mathrm{y}={\mathrm{x}}^2$ and $\mathrm{y}=-(\mathrm{x}-2{)}^2$ is/are

If $\mathrm{y}=\mathrm{mx}+\mathrm{c}$  is tangent to $\mathrm{y}={\mathrm{x}}^2,$ then ${\mathrm{x}}^2-\mathrm{mx}-\mathrm{c}=0$ has equal roots. So
${\mathrm{m}}^2+4\mathrm{c}=0$
or $\mathrm{c}=-\frac{{\mathrm{m}}^2}{4}$
So, the  tangent ,to $\mathrm{y}={\mathrm{x}}^2$ is
$\mathrm{y}=\mathrm{mx}-\frac{{\mathrm{m}}^2}{4}$
Since  this is also  tangent to $\mathrm{y}=-(\mathrm{x}-2{)}^2$,
$\mathrm{mx}-\frac{{\mathrm{m}}^2}{4}=-{\mathrm{x}}^2+4\mathrm{x}-4$
has equal  roots.  So,
${\mathrm{x}}^2+(\mathrm{m}-4)\mathrm{x}+\left(4-\frac{{\mathrm{m}}^2}{4}\right)=0$
has equal  roots.  So,
$(\mathrm{m}-4{)}^2-4\left(4-\frac{{\mathrm{m}}^2}{4}\right)=0$
or ${\mathrm{m}}^2-8\mathrm{m}+16+{\mathrm{m}}^2-16=0$
or $\mathrm{m}=0,4$
So, $\mathrm{y}=0$ and $\mathrm{y}=4\mathrm{x}-4$ is the tangent.