For Question 3dii to get the shortest chord, x needs to be exactly half of L. So L=2x. You just sub this into the equation:
x^2 - Lx + 12 = 0
Becomes
x^2 - 2x^2 +12 = 0
-x^2 +12 = 0
x^2 = 12
Thus x is the positive square root of 12, since a length can't be negative.