# Determine $a, b$ & $c$ in a (possible) Quadratic equation [closed]

Given $$f(x)=x$$:

1 - $$f(5)=2$$

2 - $$f(3)=3$$


How to determine the coefficients $$a, b ,c$$ in the polynomial equation like: $$f(x) = ax^2 + bx + c$$

$2=a5^2+5b+c=25a+5b+c$

$3=a3^2+3b+c=9a+3b+c$

Subtracting the two equations, we have

$-1=16a+2b$

Without another point, there are infinitely many solutions. Quadratic equations are defined by $3$ points, as lines are defined by $2$.

Any two points define a straight line so allow $a=0$ the $y=x-8$ is a solution. This also implies an infinite solutions due to the free variable a. For example

$-0.153661 x^2+0.229292 x+5.69508=y$ and $y = 2.84754 + 0.229292 x - 0.307323 x^2$.

Both of the above are generated by adding a third point then calculating. Its a good exercise to see what third points can be added to the parabola with those constraints and why.