I have data $\left\{ q_i, p_i \right\}_{i = 1}^{N}$ from a model $p_i = a q_i^2 + b q_i + c$:
I know data is always in the positive quadrant as $q_i \geq 0, \, p_i \geq 0$ and I also know $a > 0$ (Smiling parabola).
I am after a parameteric model for $q_i = g_{\boldsymbol{\theta}} (p_i)$:
Polynomial regression doesn't work well here as it requires high degree.
I wonder if there are known parametric models, preferably linear, for such data.
I'd guess it is something with sqrt yet I am not familiar with parametric family built on it.
Remark
I am looking for a parametric model, to be estimates using linear least squares / nonlinear least squares. I am not after solving for each point the quadratic equation.