The governing body, or such committee as may be appointed by it, shall investigate and consider the matter set forth in the petition specified in section 40-39-05 and, at the time and place specified in the notice, shall hear the testimony and evidence of persons interested. After hearing the testimony and evidence or upon the report of the committee favoring the granting of the petition, the governing body, by a resolution passed by a two-thirds vote of all its members, may declare the public grounds, streets, alleys, or highways described in the petition vacated upon such terms and conditions as it shall deem just and reasonable.