One positive number is 28 less than another number. if the reciprocal of the smaller number is added to four times the reciprocal of the larger number, the sum is 1/6. find the two numbers

Let x represent the larger number.
  1/(x -28) +4/x = 1/6
Multiplying by the product of denominators, we have
  6x +24(x -28) = x(x -28)
In standard form, this is
  x^2 -58x +672 = 0
  (x -42)(x -16) = 0

The two numbers are (-12 and 16) or (14 and 42).