Finding value of constants of quadric equation by experiment

1. Jun 3, 2015

arpon

1. The problem statement, all variables and given/known data
Suppose, the following equation describes the relation between an independent and a dependent variable physical quantities(that will be measured by experiments; for example, temperature, current, voltage etc) x & y :
$y = ax^2 + bx + c$
We have to find the values of the constants (a, b, c).

2. Relevant equations

3. The attempt at a solution
I have done this experiments for linear relations ,i.e. y = mx + c, where the y-intercept is c and the slope is m. I have also performed experiments for equations like $y = ax^2$ . In this case, I plotted a graph $y$ vs $x^2$, and the slope of the straight line is $a$.
In this case, I can easily find the values of a, b, c by taking three pairs of (x, y) value and then solving the three equations, but it gives me less accurate answer.
What is the best way to find the values of the constants in this experiment?

2. Jun 3, 2015

haruspex

Since you are given x and y values, the fact that it is a quadratic is unimportant. Rather, you have a system of linear equations in three unknowns.
Assuming you have more than three pairs of x,y values, it is overspecified. There is a standard equation for getting the least squares fit.
If you write the system in matrix form AX=Y, (X being the vector a, b, c, and A being the matrix of 1, x, x2 values) ATA ought to be invertible. The least squares solution is X=(ATA)-1ATY.