What would a solution to the equation 100 = (8 + 2x) (10 + 2x) mean in this situation?

Question

A framed picture has a total area y, in square inches. The thickness of the frame is represented by x, in inches. The equation y = (8 + 2x) (10 + 2x) relates these two variables.

What would a solution to the equation 100 = (8 + 2x) (10 + 2x) mean in this situation?

Answer 1

It is the frame thickness in inches that would result in a picture with 100 square inches of surface area and one side length of 2x + 8 and the other side length of 2x +10. So using the quadratic formula the two possible answers are (-9+√101)/2 or (-9-√101/2). The second answer when plugged into the formula results in negative side lengths for the frame which is impossible. So the answer is (-9+√101)/2. This is approx. 1.05/2. So a frame thickness of very close to 1/2 inch will frame a picture that has one side length of 2(~1/2) + 8 or ~9 and the other side length of 2(~1/2) + 10 or ~11. The surface area of the picture is ~9*~11 = 100.

Answer 2

the picture is 100 square inches in total area

the width is 8+2x

the length is 10+2x

the solution to the equation will give the value of x=the thickness of the frame

100= (8+2x)(10+2x) = 80 +36x + 4x^2

4x^2 + 36x -20=0

x^2 + 9x -5 = 0

x^2 + 9x + 81/4 - 5 - 81/4 = 0

(x+9/2)^2 = 101/4

x+9/2 = sqr(101/4) = about sqr25 = 5

x = about -4 1/2 + 5 = 1/2 inch frame or slightly bigger than 1/2

8 inch width + 2(1/2) inch frame = 9 inches

10 inch length + 2(1/2) inch frame = 11 inches

9x11 = 99 or almost 100 square inch area

x = slightly more than 1/2 inches

since square root of 101/4 is slightly more than 5

sqr(101/4) = 5.025 making x = 5.025-4.5 = 0.475 inches, slightly less than 1/2 inch