A linear model is much more simple than a quadratic model, and often works just as well for most purposes.

A quadratic model such as, Y=B0 + B1X +B2X^2, violates one of the underlying assumptions of Regression: that independent variables should be independent of each other. You will find that X is highly correlated with X^2. This will make your parameter estimates, B1 and B2, unreliable.

What is the purpose of the regression?

Are you wanting to simply fit a line through the data so you can make a forecast? or are you wanting to find the nature of the relationship between X and Y. Simple forecasts mean you don't care much if your coefficients are unreliable. Nature of the relationship means that you do want a reliable estimate of the Betas.

Hume F. Winzar, thank you for your answer.

Actually, I developed a quantitation method and while validating the method I need to make a calibration curve for comparison. I tried linear regression with 1/x2 weighting factor, but in that time the method accuracy and precision were below the FDA guidelines. with quadrating regression with the same weighting factor provides very satisfactory accuracy and precision data. Also, most of the reviewer prefer linear regression but I could not do in my case....  

If the quadratic model fits better in your data than a linear regression, it is because your data is curving at one point. In quantification, curving means that your reached the quantification limit, you saturated your detector and the response is not anymore related to the concentration of your analyte. 

It is just impossible to quantify using a curved regression. Quantification is possible only in linear regression. So what you should do is: fit a linear calibration curve up to the point it is curving (to see the curving, just plot the concentration vs the response), and then dilute your samples up to the concentration in which they will fall in the linear part of your regression model. Then, you just need to multiply for the dilution factor you used to have a precise estimation of the initial concentration of your analytes of interest.

I hope to be clear. 

Thank you Luca Narduzzi, for your great answer.

I know next to nothing of the application area but a weakness of quadratic regression is its parametric form - in that it is a very restrictive shape (symmetric around the maximum or minimum). So these days  many  use generalized additive models which fit an underlying penalized smooth; see 

https://www.researchgate.net/publication/225303313_Moving_out_of_the_Linear_Rut_The_Possibilities_of_Generalized_Additive_Models

https://www.researchgate.net/publication/225303308_Generalized_Additive_Models_Graphical_Diagnostics_and_Logistic_Regresss

see Simon Wood's website for lots of resources on GAMs 

http://www.maths.bris.ac.uk/~sw15190/

Article Generalized Additive Models, Graphical Diagnostics, and Logi...

Article “Moving Out of the Linear Rut: The Possibilities of Generali...

As a former chemist, I never used an instrument that had a "linear" calibration curve. They were "approximately linear" over a small range. I always found a quadratic cal curve worked better every time. My software only gave me 2 options though, linear or quadratic. 

I always use quadratic curves and extend my cal range accordingly. I found that quadratic cal curves are more accurate and generally more precise.