x^2 = (2x^2 + 8x + 8)/3 ---> distributing the 2 from 2/3
3x^2 = 2x^2 + 8x + 8 ---> mult. both sides by 3
x^2 -8x -8 = 0 ---> subtract 2x^2 + 8x + 8 from both sides
Then you can use the quadratic formula:
x = (8 +- sqrt(64+32))/2
You can throw out the negative option, since you're dealing with length.
x = (8 + sqrt(96))/2
x = (8 + 4sqrt(6))/2
x = 4 + 2sqrt(6)
Check it: The area of the picture, (4 + 2sqrt(6))^2, should be 2/3 of the total area, (6 + 2sqrt(6))^2.
(4 + 2sqrt(6))^2 = 40 + 8sqrt(6)
(6 + 2sqrt(6))^2 = 60 + 12sqrt(6)
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