2007 AIME I Problem 15

Let ABC be an equilateral triangle, and let D and F be points on sides BC and AB, respectively, with FA = 5 and CD = 2. Point E lies on side CA such that angle DEF = 60^{\circ}. The area of triangle DEF is 14\sqrt{3}. The two possible values of the length of side AB are p \pm q \sqrt{r}, where p and q are rational, and r is an integer not divisible by the square of a prime. Find r.

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.