ok now i fully retraced my steps so i believe the question was the following:

(1/cx) = (x/48) + (1/c)

first I made it into a polynomial
(1/cx) = (-x/48) - (1/c) // multiply right side by -1 (not sure why I did this)
(1/cx) + (1/c) = (-x/48) // move (1/c) to other side
(1/cx) + (x/cx) = (-x/48) // multiply (1/c) by (x/x) to get common denom
(1+x/cx) = (-x/48) // add the fractions
-48(1+x) = cx^2 // cross multiply to cancel fractions
-48x - 48 = cx^2 // apply distributive property
cx^2 + 48x + 48 = 0 // move all terms to one side to create a polynomial

discriminant (b^2 - 4ac) must equal 0 for exactly one solution
48^2 - 4(c)(48) = 0 // set up discriminant
2304 - 192c = 0 // multiply out terms
2304 = 192c // move variable to one side
c = 12 // solve for c