I'm trying to wrap my head around the Muller method for approximating the roots of non-linear equations.
The method is quite clear, my main concern is the part where you need to construct the quadratic equation given three points on your $f$ function.
So let's say I have three points; $(x_0, f(x_0))$, $(x_1,f(x_1))$ and $(x_2,f(x_2))$.
I want to construct a quadratic equation $P(x)$, that passes through these three points.
My lecturer used the following steps:
The quadratic equation is:
$P(x)=A(x-x_2)^2+B(x-x_2)+C$
So we start by solving the equations:
$f(x_0)=A(x_0-x_2)^2+B(x_0-x_2)+C$
$f(x_1)=A(x_1-x_2)^2+B(x_1-x_2)+C$
$f(x_2)=C$
While it is well understood why the three above equations hold, I don't get where did $P(x)=A(x-x_2)^2+B(x-x_2)+C$ came from and why is it true?
Isn't the quadratic equation should be $P(x)=A(x)^2+B(x)+C$?
I mean, where did the $x_2$ came from all of a sudden?