Your original speed was 45.668 mph.
To find seconds per mile, divide 3600 by speed in miles per hour. Let x represent the original speed. Then 3600/x is the original time per mile.
The new time per mile is 3600/(x +8.2), and that is 12 seconds less than the original time per mile:
... 3600/(x +8.2) = 3600/x -12
Multiplying by the product of the denominators, we have ...
... 3600(x) = 3600(x +8.2) -12(x)(x +8.2)
Dividing by 12 and putting this into standard form, we have
... x^2 +8.2x -300(8.2) = 0
... x = (-8.2 +√(8.2² -4(-300·8.2)))/2 = -4.1 +√2476.81 . . . . solved using the quadratic formula
... x ≈ 45.6676 . . . mi/h
Your original speed was about 45.67 mph.
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Check
Your original time per mile was about 3600/45.66756 ≈ 78.8306 seconds. Your shorter time per mile is about 3600/53.86756 ≈ 66.8306 seconds, which is 12 seconds less.
