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).