I have the following equation:
$$ Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \beta_3 X_3 + \beta_{11} X_{1}^2 + \beta_{22} X_{2}^2 + \beta_{33} X_{3}^2 + \beta_{12} X_{1} X_{2} + \beta_{23} X_{2} X_{3} + \beta_{31} X_{3} X_{1} $$
I know the values of all $\beta$s.
I have following constraints:
$$ X_1 = N(\mu_1, \sigma_1) \\ X_2 = N(\mu_2, \sigma_2) \\ X_3 = N(\mu_3, \sigma_3) $$
$X_1$, $X_2$ and $X_3$ are independent of each other and are Gaussian distributed.
I am not looking for optimizing the $Y$ value, but just solve the above equation for known values of $Y$ and $\beta$s to find $X_i$s.
I know about the Langrange's multipliers method but I am not using it right now because I think it is for optimization. Am I wrong?
Is there a method similar to Monte Carlo method for Ordinary Least Squares for Quadratic equations?
I am sorry if there is some confusion in the post as Maths/Stats is not my major strength. Please comment for additional information if needed.