Any person whose application to be registered has been denied by a part of the board of elections, may file a petition with the board, not later than the twentieth day prior to an election or primary, setting forth the ground of his complaint under oath, and praying to be registered. The board shall fix a time for a public hearing thereof at its office not later than the fifteenth day prior to the election or primary. At the time so fixed, the board shall hear and dispose of the petition, having first given at least forty-eight hours' notice of the hearing to the board members who rejected the petitioner's application for registration. The board, if satisfied that the petitioner is entitled to be registered, shall direct a board member, or clerk, to register him in the usual manner, and shall amend accordingly the records affected, but any qualified elector of the election district may appear and show cause why the petitioner should not be registered.