Answer:
2 inches
Explanation:
The area of the frame will be the difference between the area of the picture with frame, and the area of the picture without. We want that difference to be equal to the area of the picture without the frame.
Let x represent the width of the frame. The overall length and width of the picture will be increased by 2x when the frame is added. Then we have ...
a = LW = (8+2x)(12+2x) . . . . area of picture and frame
b = LW = (8)(12) . . . . . . . . . . area of picture
a -b = b . . . . . . desired relationship between areas
Substituting and expanding the expressions for 'a' and 'b', we find ...
a -2b = 0 . . . . subtract b
(8 +2x)(12 +2x) -2(8)(12) = 0
4x² +40x -96 = 0 . . . simplify
x² +10x -24 = 0 . . . . . divide by 4
(x -2)(x +12) = 0 . . . . factor
The positive value of x that makes this true is x = 2.
The width of the frame is 2 inches.