a picture frame is 18cm by 14cm. If 165 cm^2 of a picture shows, find the width of the frame.

OK...assuming the frame width is uniform..let's call this "x"

So....the total area that the frame encloses is 18 * 14.

So......the area occupied by the picture itself can be expressed as

(18-2x)(14-2x) = 165       

252 - 64x + 4x^2 = 165    subtract 165 from both sides

4x^2 - 64x + 87 = 0          factor this.....

(2x - 29) (2x -3) = 0   set each factor to 0

2x - 29 = 0      x =14.5 cm  reject this.........the frame width would cover the picture !!!

2x - 3 = 0        x = 1.5 cm......that seems reasonable

Also (18 - 2(1.5)) * ( 14 - 2(1.5))  =  (18 -3) * ( 14 -3) =  (15) * (11) = 165 cm

This seems to "sync-up" with the info given

So the frame is 1.5 cm wide.