I want to fit a set of measured data $x_i$ and $y_i$ to the expression: $$ \beta^2 + \beta x_i = y_i $$ $\beta$ is my only free parameter. Although this is a really simple expression, the standard approaches of linear regression do not hold because of the $\beta^2$ term. But I can calculate a $\hat\beta$ that minimizes the squared error: $$ \varepsilon_i = \beta^2 + \beta x_i - y_i \\ \frac{1}{2}\frac{\partial}{\partial x_i}\sum \varepsilon_i^2 = \sum \varepsilon_i \beta = 0 \\ -\frac{1}{2}\frac{\partial}{\partial y_i}\sum \varepsilon_i^2 = \sum \varepsilon_i = 0 \\ \Rightarrow \qquad \hat\beta_{1/2}=-\frac{\langle x\rangle}{2}\pm \sqrt{\frac{\langle x \rangle}{4}+\langle y \rangle} $$ (Obviously the derivative with respect to $x_i$ does not give any useful information.)
My actual question is, how I can estimate the standard deviation of my least squares estimate $\hat\beta$ according to the input data $x_i$ and $y_i$. I tried to find some analogy to the linear regression case, but I am completely lost as the expression for $\hat\beta$ is so nonlinear...
Also, it seems that this is quite a trivial standard case of a nonlinear regression. However, I was not able to find any recipe on how to solve this while searching for terms like "second order regression one parameter" and the like on the net. Am I missing something? Is there a simple way to reformulate this as a first order fit problem?