What version are you using? Seems to work ok here, but w/o any references to Maple.
which -all solve
z = [-6.41 -12.4 2.143 102];
r = [13.58 15.7636 12.96 46.6];
syms A B C D;
eq1 = A*r(1)^2+B*z(1)+C*r(1)+D ==0;
eq2 = A*r(2)^2+B*z(2)+C*r(2)+D ==0;
eq3 = A*r(3)^2+B*z(3)+C*r(3)+D ==0;
eq4 = A*r(4)^2+B*z(4)+C*r(4)+D ==0;
sol = solve([eq1,eq2,eq3,eq4],[A,B,C,D])
By inspection, we see that solve returned the trivial solution. We can also verify it's the only solution
M = [r(:).^2 z(:) r(:) ones(4,1)];
null(M)