no real $x$ is exised for the given equation.
If $x$ is a positive non-zero real number, Now in equation $(\sqrt{243}+3)^x+(\sqrt{243}-3)^x=15^x $, left part is $(15.58...+3)^x +$ some +ve value $= 15^x$ ..here, left hand side is greater than right hand side. So, no +ve real 'x' exist here. when $x=0$, then also equation is not satisfied. when $x$ is negative then value of $15^x$ will always be less than $(\sqrt{243}+3)^x+(\sqrt{243}-3)^x$ and values of both will be tending to zero but value can't be same for any negative $x$.