No, not all can be solved by completing the square because this method requires the presence of the [tex]bx[/tex] term from [tex]a x^{2} +bx+c=0[/tex].
For example, we can not use completing the square for [tex] x^{2} -9 =0[/tex] because it does not have the middle term. However, this kind of equation may be easily solved using different methods such as extracting the square roots, etc.
