Question 1067683
 the length of a painting is 5 centimeters more than its width.
let L = the length of the painting
let w = the width
then
L = (w+5)
:
 it is matted in a frame whose length is 8 more than twice the width of the painting and width is five less than twice the width of the painting.
2w+8 = length of frame
and
2w-5 = the width of the frame
:
 the frame alone had an area of 650 square cm
 find the dimensions of the painting and the frame
:
Overall area - painting area = frame area
(2w+8)*(2w-5) - w(w+5) = 650
FOIL
4w^2 - 10w + 16w - 40 - w^2 - 5w = 650
Combine like terms to form a quadratic equation
4w^2 - w^2 + 6w - 5w - 40 - 650 = 0
3w^2 + w - 690 = 0
Use the quadratic formula to find w, but this will factor to:
(3s+46)(w-15) = 0
the positive solution is all we want here
w = 15 cm is the width of the painting
then
L = 15 + 5 = 20 cm is the length of the painting
:
Find the dimensions of the frame
2(15) + 8 = 38 cm is the length of the frame
and
2(15) - 5 = 25 cm is the width
:
;
Confirm our solutions find the overall area and painting area
38*25 = 950
20*15 = 300
-------------subtract
frame A:650 sq/cm