How to Convert a Nonlinear Function into Linear for Quadratic to Linear Fit?

[Igor_S]

I have a function $$y = \sqrt{ax^2 + bx +c}$$, and 2 sets of points $${x_i},{y_i}$$ that need to be fit on this curve. First in this problem, I need to somehow convert this nonlinear function into linear and then apply least square methods to determine a,b,c.
What I came up is ofcouse squaring both sides, removing root. Now I have: $$y^2 = ax^2 + bx + c$$. I tried factoring this into $$a(x-x_1)(x-x_2)$$ but don't think this is better form. I'm not sure how is this even possible, there are 3 numbers to be determined as I can find only 2 equations from least square method (slope & intercept). How to determine a,b,c ? I know how to do it in eg. Mathematica, but I'm writing this as a FORTRAN program, so I need to write exact procedure. I don't know how to do nonlinear fits "by hand".

Thanks.P.S. Simpler example of what is supposed to be done (at least I think) is function y = ax^n. Here you just take log of both sides to get: log(y) = nlog(x) + log(a), then calculate log(y_i) and log(x_i) and slope of the line with log(x) and log(y) as variables is n, with intercept log(a).

 

[EnumaElish]

Somewhat similar to the Log example, you have 3 var's in the quadratic problem: y^2, x^2 and x. You can calculate a, b and c by inputting these 3 var's into a standard regression software (E.g. SAS); or you can write your own OLS routine to calculate them by using the formula $$\hat\beta = (Z'Z)^{-1}Z'(y^2)$$ where $\beta = (a,b,c)\text{ and }Z=(x^2,x,1)$.