166 + 4c - 10(3/16) + c^2 = 0.6

Simplifying the equation:

166 + 4c - 30/16 + c^2 = 0.6
166 + 4c - 1.875 + c^2 = 0.6
165.125 + 4c + c^2 = 0.6

Rearranging the terms:

c^2 + 4c + 165.125 - 0.6 = 0
c^2 + 4c + 164.525 = 0

Now, we have a quadratic equation in the form of ax^2 + bx + c = 0 which can be solved using the quadratic formula:

c = (-b ± √(b^2 - 4ac)) / 2a

For c^2 + 4c + 164.525 = 0:

a = 1, b = 4, and c = 164.525

c = (-4 ± √(4^2 - 4(1)(164.525))) / 2(1)
c = (-4 ± √(16 - 658.1)) / 2
c = (-4 ± √(-642.1)) / 2

Since the square root of a negative number is not real, the equation has no real roots.