Question 1002046
Vertices must be equal and this means that {{{f(x)-g(x)=0}}} JUST FOR THE difference at this common vertex.


{{{x^2+kx+3-(-x^2+4x+k)=0}}}
{{{x^2+kx+3+x^2-4x-k=0}}}
{{{2x^2+kx-4x+3-k=0}}}
{{{highlight_green(2x^2+(k-4)x+(3-k)=0)}}}


Still a quadratic equation with variables x and k.  This equation will have ONE solution or zero, IF the discriminant is zero.  The discriminant expression will involve k but NOT x.


Discriminant:  {{{(k-4)^2-4*2*(3-k)=0}}}------Solve for k.