Confidence intervals around functions of estimated parameters Suppose I estimate the following model,
$$
 y = \hat{\beta_0} + \hat{\beta_1} X_1 + \hat{\beta_2} X_2\ ,
$$
and am additionally interested in the quantity
$$
 \gamma = \sqrt{\beta_0^2 - 4\beta_1\beta_2}\ .
$$
An estimate of the above, $\hat{\gamma}$, is clearly given by inserting the estimated $\hat\beta_i$'s in the above formula.
But how does one go about computing the confidence interval around such a "compound" quantity that is not directly estimated?
My attempt.
My "empirical" approach is as follows. I generate a multivariate normal distribution of size $n$ using the estimated means $\hat\beta_i$'s and the covariance matrix of the fit. I consequently have a distribution of size $n$ for the $\gamma$. The mean of this distribution $\bar\gamma$ is naturally then close to the $\hat\gamma$ computed above, and I now additionally have a standard deviation of this distribution $\sigma$.
Using $\sigma$ to generate a confidence interval.
If I use the standard definition $\bar\gamma \pm t(\alpha, n-1) \sigma / \sqrt{n}$, where $t$ is the PPF of the $t$-distribution, I run into the following quandary: since I pick $n$ myself, I can in fact make the standard error as small as I like. This is not what I want: I am looking for a "natural" confidence interval which depends solely on the uncertainty in my estimates of the $\beta$'s. Therefore, my idea is the following: $\sigma$ stays positive, and seems to "converge" to its true value for large $n$: hence, 95% of the values in my $\gamma$ distribution lie within the interval $\bar\gamma \pm 2\sigma$.
Is this approach valid?

It seems like you are estimating the discriminant of a quadratic function, ie. your function is
$$y = \hat{\beta_0} + \hat{\beta_1} X_1 + \hat{\beta_2} X_2 =
 \hat{\beta_0} + \hat{\beta_1} X + \hat{\beta_2} X^2$$
We can rewrite this in terms of the roots
$$y = \hat{\beta_2} (X-r_1)(X-r_2)$$
And now we have $$\gamma = \beta_2 (r_2-r_1)$$
If you estimate $\beta_2$, $r_1$, and $r_2$ with a non-linear estimation method and obtain an estimate for the approximate multivariate normal distribution of the estimates (based on an estimate of the Fisher information matrix), then you can describe an estimate distribution for $\gamma$ as a product of two correlated non central normal distributions.
