# linear solution of curve fitting on multiple linear functions differing by a multiplier

I am facing the following problem. I know nonlinear least squares can provide a solution but I am wondering if a linear way to solve this data fitting problem may exists.

This is my input dataset: I've got three different dataset composed of scattered points

I know a linear equation in the form $$y(x) = bx^2 + cx$$

can be used to explain any of my dataset. I know how to fit this function to a given dataset (e.g the blue one) linearly, so could the above function separately for each dataset, but I am looking for something different.

In the specific case I have an additional constrain: I also know that the three functions describing the three dataset share the $b$ and $c$ parameters, while they differ by a multiplier, such that:

$$\begin{cases} y(x) = (bx^2 + cx) \; \text{explains the blue data}\\ y(x) = a'(bx^2 + cx) \; \text{for red data}\\ y(x) = a''(bx^2 + cx) \; \text{for green data} \end{cases}$$

I am looking for a (if it exists) linear way to solve for $a', a'', b$ and $c$ minimizing the overall sum of squares for the three functions. Any idea on how this problem can be approached? Also approximate solutions are welcome...