  We consider the two-dimensional quasilinear wave equations with quadratic
nonlinearities. We introduce a new class of null forms and prove uniform
boundedness of the highest order norm of the solution for all time. This class
of null forms include several prototypical strong null conditions as special
cases. To handle the critical decay near the light cone we inflate the
nonlinearity through a new normal form type transformation which is based on a
deep cancelation between the tangential and normal derivatives with respect to
the light cone. Our proof does not employ the Lorentz boost and can be
generalized to systems with multiple speeds.
