Having just one root requires
eq1 = -k^x + x^2 == 0;
eq2 = D[-k^x + x^2, x] == 0
(* 2*x - k^x*Log[k] *)
FindRoot
can solve for x
and k
simultaneously.
FindRoot[{eq1, eq2}, {{x, 2.5}, {k, 2}}]
(* {x -> 2.71828, k -> 2.08707} *)
Plot[(-k^x + x^2) /. %[[2]], {x, 2, 3}]