consider the equation x=tanx and its 10th possible solution and use a quadratic formula to find an estimate?

Question

find an a priori estimate first then use this and the quadratic equation to find an improved estimate.

Answer 1

tan x is a function that goes from -inf to +inf periodically, with each period covering a range of pi radians.

the solutions to x = tan x are the points where the curves y = tan x and y = x intersect. The first is at x = 0; the second between x = (1/2)pi and (3/2)pi, the third is between x = (3/2)pi and (5/2)pi; etc. (in fact, given the fact that x is positive, you can halve the intervals: between (2/2)pi and (3/2)pi, between (4/2)pi and (5/2)pi, etc.

This gives you a (very) rough estimate (i.e. use the midpoint of the appropriate interval)

As for improving the estimate, there are a number of techniques, but I'm not sure which you've learned.

My first choice is Newton's method, which will converge nicely here:

http://en.wikipedia.org/wiki/Newton's_method

If you need to use a quadratic, then recall that:

tan x = x + x^3/(3!) + ...

For small values of x (less than pi/2), but you can offset the value you are looking for by multiples of (pi/2) to get in in the right range. That is, you can determine an "n" such that the problem becomes:

x = u + n(pi/2) so
u + n(pi/2) = tan(u + n(pi/2))

and knowing that tan(u + (2)pi) = tan u, etc., you can convert the right hand side into tan(u) and then use the first two terms of the series.

HTH.