Consider the 1+1-dimensional quasi-linear diffusion equations with convection and source term ut=[um(ux)n]x+P(u)ux + Q(u), where P and Q are both smooth functions. We obtain conditions under which the equations admit the Lie Bǎcklund conditional symmetry with characteristic η=uxx+H(u)u2x+G(u)(ux)2-n+F(u)ux1-n and the Hamilton-Jacobi sign-invariant J=ut+A(u)uxn+1+B(u)ux+C(u) which preserves both signs, ≥0 and ≤0, on the solution manifold. As a result, the corresponding solutions associated with the symmetries are obtained explicitly, or they are reduced to solve two-dimensional dynamical systems.