q1 = {tk - T/2, w};
q2 = {tk, ak};
q3 = {tk + T/2, w};
Solve[p == 0, t]; Solve[a1*q1[[1]]^2 + a2*q1[[1]] + a3 == q1[[2]] &&
a1*q2[[1]]^2 + a2*q2[[1]] + a3 == q2[[2]] &&
a1*q3[[1]]^2 + a2*q3[[1]] + a3 == q3[[2]], {a1, a2, a3}];
% /. Rule -> Set;