In the traditional labelling of the quadratic, -b is representd by 55 in the sought base. Since midway between the two solutions is 6.5, that must be the value of -b/(2*5); that is, 55/5 = 11 must represent 13 in decimal so the base must be 12.

Does this check out? Assuming base 12, convert the equation to base 10:

5x^2 - 65x + 210 = 0

solutions:

65 +/- sqrt( 4225 - 4*1050)
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          10

          

= 6.5 +/- sqrt(25)/10

= 6.5 +/- 1/2

= 6 or 7

Base 12 checks out.