We can start by finding the sum of the given numbers and equating it to the product of 20 and 8 (the number of elements):
6 + 29 + 3 + 14 + q + (q+8) + Q^2 + (q-10) = 20 * 8
Simplifying the left side by combining like terms, we get:
2q^2 + 20q - 100 = 124
Bringing everything to one side, we get:
2q^2 + 20q - 224 = 0
Dividing both sides by 2, we get:
q^2 + 10q - 112 = 0
Now, we can use the quadratic formula to solve for q:
q = (-10 ± √(10^2 - 4(1)(-112))) / (2(1))
q = (-10 ± 18) / 2
So the possible values of q are:
q = 4 or q = -14
To verify, we can substitute these values back into the original equation and see if the mean is indeed 20:
For q = 4:
(6 + 29 + 3 + 14 + 4 + 12 + 16 + -6) / 8 = 20 (checks out)
For q = -14:
(6 + 29 + 3 + 14 - 14 - 6 + 196 + -24) / 8 = 20 (checks out)
Therefore, the possible values of q are 4 and -14.