How do you solve this equation?
[1-(1/1.05)^30]0.0635 = [(1/1.05)^30] * [(1-(1/1.05)^(x-65)] * [30/x]
See how x is both a denominator, and an exponent?
An approximation method approximates x, as you cannot get the precise answer. It gets a decimal approximation down to however many decimal places of accuracy you want.
The easiest one I can think of is the Newton Raphson method, but finding the derivative of the function isn't too easy.
Well the Maxima simplifies (evaluates coefficients numerically because of decimal coeficients):
Which now gives a root at ~=-183.5, but no other (why I now have a root I don't know but appears to be something to do with using 20/21 for 1/1.05), but I think it is wrong - more checking needed me-thinks
(Switching back to 1/1.05 takes us back to the previous situation with no roots and plots now plot what the function evaluates to
I have changed the definition of KK to functional form which gives LaTeX to compare with your's:
Which looks OK to me)