Aroom has an area of 60 square meters and a perimeter of 32 meters. what are the length and width of the room?

Area is L*W = 60 m^2, so that L=(60 m^2)/W.

Perimeter is P = 2W + 2L, or   P = 2W + 2[60 m^2]/W

Since we know that the perimeter is 32 meters,  2W + 120/W = 32   (meters)

Multiplying all terms by W:    2W^2 + 120 = 32W,
or

w^2 + 60 = 16w, or w^2 - 16w + 60 = 0.

Using the Quadratic Formula to find w:

         -(-16) plus or minus sqrt(20)
w =
                            2
         16 plus or minus 2sqrt(5)
    =
                            2
             
     = 8 plus or minus sqrt(5).  This produces 2 positive solutions.  Since w represents width, discard any negative result.  You may have to check twice, since you will have 2 different widths.